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Hypothesis Testing | A Step-by-Step Guide with Easy Examples

Published on November 8, 2019 by Rebecca Bevans . Revised on June 22, 2023.

Hypothesis testing is a formal procedure for investigating our ideas about the world using statistics . It is most often used by scientists to test specific predictions, called hypotheses, that arise from theories.

There are 5 main steps in hypothesis testing:

• State your research hypothesis as a null hypothesis and alternate hypothesis (H o ) and (H a  or H 1 ).
• Collect data in a way designed to test the hypothesis.
• Perform an appropriate statistical test .
• Decide whether to reject or fail to reject your null hypothesis.
• Present the findings in your results and discussion section.

Though the specific details might vary, the procedure you will use when testing a hypothesis will always follow some version of these steps.

Table of contents

Step 1: state your null and alternate hypothesis, step 2: collect data, step 3: perform a statistical test, step 4: decide whether to reject or fail to reject your null hypothesis, step 5: present your findings, other interesting articles, frequently asked questions about hypothesis testing.

After developing your initial research hypothesis (the prediction that you want to investigate), it is important to restate it as a null (H o ) and alternate (H a ) hypothesis so that you can test it mathematically.

The alternate hypothesis is usually your initial hypothesis that predicts a relationship between variables. The null hypothesis is a prediction of no relationship between the variables you are interested in.

• H 0 : Men are, on average, not taller than women. H a : Men are, on average, taller than women.

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For a statistical test to be valid , it is important to perform sampling and collect data in a way that is designed to test your hypothesis. If your data are not representative, then you cannot make statistical inferences about the population you are interested in.

There are a variety of statistical tests available, but they are all based on the comparison of within-group variance (how spread out the data is within a category) versus between-group variance (how different the categories are from one another).

If the between-group variance is large enough that there is little or no overlap between groups, then your statistical test will reflect that by showing a low p -value . This means it is unlikely that the differences between these groups came about by chance.

Alternatively, if there is high within-group variance and low between-group variance, then your statistical test will reflect that with a high p -value. This means it is likely that any difference you measure between groups is due to chance.

Your choice of statistical test will be based on the type of variables and the level of measurement of your collected data .

• an estimate of the difference in average height between the two groups.
• a p -value showing how likely you are to see this difference if the null hypothesis of no difference is true.

Based on the outcome of your statistical test, you will have to decide whether to reject or fail to reject your null hypothesis.

In most cases you will use the p -value generated by your statistical test to guide your decision. And in most cases, your predetermined level of significance for rejecting the null hypothesis will be 0.05 – that is, when there is a less than 5% chance that you would see these results if the null hypothesis were true.

In some cases, researchers choose a more conservative level of significance, such as 0.01 (1%). This minimizes the risk of incorrectly rejecting the null hypothesis ( Type I error ).

The results of hypothesis testing will be presented in the results and discussion sections of your research paper , dissertation or thesis .

In the results section you should give a brief summary of the data and a summary of the results of your statistical test (for example, the estimated difference between group means and associated p -value). In the discussion , you can discuss whether your initial hypothesis was supported by your results or not.

In the formal language of hypothesis testing, we talk about rejecting or failing to reject the null hypothesis. You will probably be asked to do this in your statistics assignments.

However, when presenting research results in academic papers we rarely talk this way. Instead, we go back to our alternate hypothesis (in this case, the hypothesis that men are on average taller than women) and state whether the result of our test did or did not support the alternate hypothesis.

If your null hypothesis was rejected, this result is interpreted as “supported the alternate hypothesis.”

These are superficial differences; you can see that they mean the same thing.

You might notice that we don’t say that we reject or fail to reject the alternate hypothesis . This is because hypothesis testing is not designed to prove or disprove anything. It is only designed to test whether a pattern we measure could have arisen spuriously, or by chance.

If we reject the null hypothesis based on our research (i.e., we find that it is unlikely that the pattern arose by chance), then we can say our test lends support to our hypothesis . But if the pattern does not pass our decision rule, meaning that it could have arisen by chance, then we say the test is inconsistent with our hypothesis .

If you want to know more about statistics , methodology , or research bias , make sure to check out some of our other articles with explanations and examples.

• Normal distribution
• Descriptive statistics
• Measures of central tendency
• Correlation coefficient

Methodology

• Cluster sampling
• Stratified sampling
• Types of interviews
• Cohort study
• Thematic analysis

Research bias

• Implicit bias
• Cognitive bias
• Survivorship bias
• Availability heuristic
• Nonresponse bias
• Regression to the mean

Hypothesis testing is a formal procedure for investigating our ideas about the world using statistics. It is used by scientists to test specific predictions, called hypotheses , by calculating how likely it is that a pattern or relationship between variables could have arisen by chance.

A hypothesis states your predictions about what your research will find. It is a tentative answer to your research question that has not yet been tested. For some research projects, you might have to write several hypotheses that address different aspects of your research question.

A hypothesis is not just a guess — it should be based on existing theories and knowledge. It also has to be testable, which means you can support or refute it through scientific research methods (such as experiments, observations and statistical analysis of data).

Null and alternative hypotheses are used in statistical hypothesis testing . The null hypothesis of a test always predicts no effect or no relationship between variables, while the alternative hypothesis states your research prediction of an effect or relationship.

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How to State the Conclusion about a Hypothesis Test

After you have completed the statistical analysis and decided to reject or fail to reject the Null hypothesis, you need to state your conclusion about the claim. To get the correct wording, you need to recall which hypothesis was the claim.

If the claim was the null, then your conclusion is about whether there was sufficient evidence to reject the claim. Remember, we can never prove the null to be true, but failing to reject it is the next best thing. So, it is not correct to say, “Accept the Null.”

If the claim is the alternative hypothesis, your conclusion can be whether there was sufficient evidence to support (prove) the alternative is true.

Use the following table to help you make a good conclusion.

The best way to state the conclusion is to include the significance level of the test and a bit about the claim itself.

For example, if the claim was the alternative that the mean score on a test was greater than 85, and your decision was to  Reject then Null , then you could conclude: “ At the 5% significance level, there is sufficient evidence to support the claim that the mean score on the test was greater than 85. ”

The reason you should include the significance level is that the decision, and thus the conclusion, could be different if the significance level was not 5%.

If you are curious why we say “Fail to Reject the Null” instead of “Accept the Null,” this short video might be of interest:  Here

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It is concluded that the null hypothesis Ho is not rejected proportion p is greater than 0.5, at the 0.05 significance

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How to Write Hypothesis Test Conclusions (With Examples)

A   hypothesis test is used to test whether or not some hypothesis about a population parameter is true.

To perform a hypothesis test in the real world, researchers obtain a random sample from the population and perform a hypothesis test on the sample data, using a null and alternative hypothesis:

• Null Hypothesis (H 0 ): The sample data occurs purely from chance.
• Alternative Hypothesis (H A ): The sample data is influenced by some non-random cause.

If the p-value of the hypothesis test is less than some significance level (e.g. α = .05), then we reject the null hypothesis .

Otherwise, if the p-value is not less than some significance level then we fail to reject the null hypothesis .

When writing the conclusion of a hypothesis test, we typically include:

• Whether we reject or fail to reject the null hypothesis.
• The significance level.
• A short explanation in the context of the hypothesis test.

For example, we would write:

We reject the null hypothesis at the 5% significance level.   There is sufficient evidence to support the claim that…

Or, we would write:

We fail to reject the null hypothesis at the 5% significance level.   There is not sufficient evidence to support the claim that…

The following examples show how to write a hypothesis test conclusion in both scenarios.

Example 1: Reject the Null Hypothesis Conclusion

Suppose a biologist believes that a certain fertilizer will cause plants to grow more during a one-month period than they normally do, which is currently 20 inches. To test this, she applies the fertilizer to each of the plants in her laboratory for one month.

She then performs a hypothesis test at a 5% significance level using the following hypotheses:

• H 0 : μ = 20 inches (the fertilizer will have no effect on the mean plant growth)
• H A : μ > 20 inches (the fertilizer will cause mean plant growth to increase)

Suppose the p-value of the test turns out to be 0.002.

Here is how she would report the results of the hypothesis test:

We reject the null hypothesis at the 5% significance level.   There is sufficient evidence to support the claim that this particular fertilizer causes plants to grow more during a one-month period than they normally do.

Example 2: Fail to Reject the Null Hypothesis Conclusion

Suppose the manager of a manufacturing plant wants to test whether or not some new method changes the number of defective widgets produced per month, which is currently 250. To test this, he measures the mean number of defective widgets produced before and after using the new method for one month.

He performs a hypothesis test at a 10% significance level using the following hypotheses:

• H 0 : μ after = μ before (the mean number of defective widgets is the same before and after using the new method)
• H A : μ after ≠ μ before (the mean number of defective widgets produced is different before and after using the new method)

Suppose the p-value of the test turns out to be 0.27.

Here is how he would report the results of the hypothesis test:

We fail to reject the null hypothesis at the 10% significance level.   There is not sufficient evidence to support the claim that the new method leads to a change in the number of defective widgets produced per month.

Additional Resources

The following tutorials provide additional information about hypothesis testing:

Introduction to Hypothesis Testing 4 Examples of Hypothesis Testing in Real Life How to Write a Null Hypothesis

10 Examples of Using Probability in Real Life

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Chapter 3: Hypothesis Testing

The previous two chapters introduced methods for organizing and summarizing sample data, and using sample statistics to estimate population parameters. This chapter introduces the next major topic of inferential statistics: hypothesis testing.

A hypothesis is a statement or claim about a property of a population.

The Fundamentals of Hypothesis Testing

When conducting scientific research, typically there is some known information, perhaps from some past work or from a long accepted idea. We want to test whether this claim is believable. This is the basic idea behind a hypothesis test:

• State what we think is true.
• Quantify how confident we are about our claim.
• Use sample statistics to make inferences about population parameters.

For example, past research tells us that the average life span for a hummingbird is about four years. You have been studying the hummingbirds in the southeastern United States and find a sample mean lifespan of 4.8 years. Should you reject the known or accepted information in favor of your results? How confident are you in your estimate? At what point would you say that there is enough evidence to reject the known information and support your alternative claim? How far from the known mean of four years can the sample mean be before we reject the idea that the average lifespan of a hummingbird is four years?

Hypothesis testing is a procedure, based on sample evidence and probability, used to test claims regarding a characteristic of a population.

A hypothesis is a claim or statement about a characteristic of a population of interest to us. A hypothesis test is a way for us to use our sample statistics to test a specific claim.

The population mean weight is known to be 157 lb. We want to test the claim that the mean weight has increased.

Two years ago, the proportion of infected plants was 37%. We believe that a treatment has helped, and we want to test the claim that there has been a reduction in the proportion of infected plants.

Components of a Formal Hypothesis Test

The null hypothesis is a statement about the value of a population parameter, such as the population mean (µ) or the population proportion ( p ). It contains the condition of equality and is denoted as H 0 (H-naught).

H 0 : µ = 157 or H 0 : p = 0.37

The alternative hypothesis is the claim to be tested, the opposite of the null hypothesis. It contains the value of the parameter that we consider plausible and is denoted as H 1 .

H 1 : µ > 157 or H 1 : p ≠ 0.37

The test statistic is a value computed from the sample data that is used in making a decision about the rejection of the null hypothesis. The test statistic converts the sample mean ( x̄ ) or sample proportion ( p̂ ) to a Z- or t-score under the assumption that the null hypothesis is true . It is used to decide whether the difference between the sample statistic and the hypothesized claim is significant.

The p-value is the area under the curve to the left or right of the test statistic. It is compared to the level of significance ( α ).

The critical value is the value that defines the rejection zone (the test statistic values that would lead to rejection of the null hypothesis). It is defined by the level of significance.

The level of significance ( α ) is the probability that the test statistic will fall into the critical region when the null hypothesis is true. This level is set by the researcher.

The conclusion is the final decision of the hypothesis test. The conclusion must always be clearly stated, communicating the decision based on the components of the test. It is important to realize that we never prove or accept the null hypothesis. We are merely saying that the sample evidence is not strong enough to warrant the rejection of the null hypothesis. The conclusion is made up of two parts:

1) Reject or fail to reject the null hypothesis, and 2) there is or is not enough evidence to support the alternative claim.

Option 1) Reject the null hypothesis (H 0 ). This means that you have enough statistical evidence to support the alternative claim (H 1 ).

Option 2) Fail to reject the null hypothesis (H 0 ). This means that you do NOT have enough evidence to support the alternative claim (H 1 ).

Another way to think about hypothesis testing is to compare it to the US justice system. A defendant is innocent until proven guilty (Null hypothesis—innocent). The prosecuting attorney tries to prove that the defendant is guilty (Alternative hypothesis—guilty). There are two possible conclusions that the jury can reach. First, the defendant is guilty (Reject the null hypothesis). Second, the defendant is not guilty (Fail to reject the null hypothesis). This is NOT the same thing as saying the defendant is innocent! In the first case, the prosecutor had enough evidence to reject the null hypothesis (innocent) and support the alternative claim (guilty). In the second case, the prosecutor did NOT have enough evidence to reject the null hypothesis (innocent) and support the alternative claim of guilty.

The Null and Alternative Hypotheses

There are three different pairs of null and alternative hypotheses:

where c is some known value.

A Two-sided Test

This tests whether the population parameter is equal to, versus not equal to, some specific value.

H o : μ = 12 vs. H 1 : μ ≠ 12

The critical region is divided equally into the two tails and the critical values are ± values that define the rejection zones.

Figure 1. The rejection zone for a two-sided hypothesis test.

A forester studying diameter growth of red pine believes that the mean diameter growth will be different if a fertilization treatment is applied to the stand.

• H o : μ = 1.2 in./ year
• H 1 : μ ≠ 1.2 in./ year

This is a two-sided question, as the forester doesn’t state whether population mean diameter growth will increase or decrease.

A Right-sided Test

This tests whether the population parameter is equal to, versus greater than, some specific value.

H o : μ = 12 vs. H 1 : μ > 12

The critical region is in the right tail and the critical value is a positive value that defines the rejection zone.

Figure 2. The rejection zone for a right-sided hypothesis test.

A biologist believes that there has been an increase in the mean number of lakes infected with milfoil, an invasive species, since the last study five years ago.

• H o : μ = 15 lakes
• H 1 : μ >15 lakes

This is a right-sided question, as the biologist believes that there has been an increase in population mean number of infected lakes.

A Left-sided Test

This tests whether the population parameter is equal to, versus less than, some specific value.

H o : μ = 12 vs. H 1 : μ < 12

The critical region is in the left tail and the critical value is a negative value that defines the rejection zone.

Figure 3. The rejection zone for a left-sided hypothesis test.

A scientist’s research indicates that there has been a change in the proportion of people who support certain environmental policies. He wants to test the claim that there has been a reduction in the proportion of people who support these policies.

• H o : p = 0.57
• H 1 : p < 0.57

This is a left-sided question, as the scientist believes that there has been a reduction in the true population proportion.

Statistically Significant

When the observed results (the sample statistics) are unlikely (a low probability) under the assumption that the null hypothesis is true, we say that the result is statistically significant, and we reject the null hypothesis. This result depends on the level of significance, the sample statistic, sample size, and whether it is a one- or two-sided alternative hypothesis.

Types of Errors

When testing, we arrive at a conclusion of rejecting the null hypothesis or failing to reject the null hypothesis. Such conclusions are sometimes correct and sometimes incorrect (even when we have followed all the correct procedures). We use incomplete sample data to reach a conclusion and there is always the possibility of reaching the wrong conclusion. There are four possible conclusions to reach from hypothesis testing. Of the four possible outcomes, two are correct and two are NOT correct.

Table 1. Possible outcomes from a hypothesis test.

A Type I error is when we reject the null hypothesis when it is true. The symbol α (alpha) is used to represent Type I errors. This is the same alpha we use as the level of significance. By setting alpha as low as reasonably possible, we try to control the Type I error through the level of significance.

A Type II error is when we fail to reject the null hypothesis when it is false. The symbol β (beta) is used to represent Type II errors.

In general, Type I errors are considered more serious. One step in the hypothesis test procedure involves selecting the significance level ( α ), which is the probability of rejecting the null hypothesis when it is correct. So the researcher can select the level of significance that minimizes Type I errors. However, there is a mathematical relationship between α, β , and n (sample size).

• As α increases, β decreases
• As α decreases, β increases
• As sample size increases (n), both α and β decrease

The natural inclination is to select the smallest possible value for α, thinking to minimize the possibility of causing a Type I error. Unfortunately, this forces an increase in Type II errors. By making the rejection zone too small, you may fail to reject the null hypothesis, when, in fact, it is false. Typically, we select the best sample size and level of significance, automatically setting β .

Figure 4. Type 1 error.

Power of the Test

A Type II error ( β ) is the probability of failing to reject a false null hypothesis. It follows that 1- β is the probability of rejecting a false null hypothesis. This probability is identified as the power of the test, and is often used to gauge the test’s effectiveness in recognizing that a null hypothesis is false.

The probability that at a fixed level α significance test will reject H 0 , when a particular alternative value of the parameter is true is called the power of the test.

Power is also directly linked to sample size. For example, suppose the null hypothesis is that the mean fish weight is 8.7 lb. Given sample data, a level of significance of 5%, and an alternative weight of 9.2 lb., we can compute the power of the test to reject μ = 8.7 lb. If we have a small sample size, the power will be low. However, increasing the sample size will increase the power of the test. Increasing the level of significance will also increase power. A 5% test of significance will have a greater chance of rejecting the null hypothesis than a 1% test because the strength of evidence required for the rejection is less. Decreasing the standard deviation has the same effect as increasing the sample size: there is more information about μ .

Hypothesis Test about the Population Mean ( μ ) when the Population Standard Deviation ( σ ) is Known

We are going to examine two equivalent ways to perform a hypothesis test: the classical approach and the p-value approach. The classical approach is based on standard deviations. This method compares the test statistic (Z-score) to a critical value (Z-score) from the standard normal table. If the test statistic falls in the rejection zone, you reject the null hypothesis. The p-value approach is based on area under the normal curve. This method compares the area associated with the test statistic to alpha ( α ), the level of significance (which is also area under the normal curve). If the p-value is less than alpha, you would reject the null hypothesis.

As a past student poetically said: If the p-value is a wee value, Reject Ho

Both methods must have:

• Data from a random sample.
• Verification of the assumption of normality.
• A null and alternative hypothesis.
• A criterion that determines if we reject or fail to reject the null hypothesis.
• A conclusion that answers the question.

There are four steps required for a hypothesis test:

• State the null and alternative hypotheses.
• State the level of significance and the critical value.
• Compute the test statistic.
• State a conclusion.

The Classical Method for Testing a Claim about the Population Mean ( μ ) when the Population Standard Deviation ( σ ) is Known

A forester studying diameter growth of red pine believes that the mean diameter growth will be different from the known mean growth of 1.35 inches/year if a fertilization treatment is applied to the stand. He conducts his experiment, collects data from a sample of 32 plots, and gets a sample mean diameter growth of 1.6 in./year. The population standard deviation for this stand is known to be 0.46 in./year. Does he have enough evidence to support his claim?

Step 1) State the null and alternative hypotheses.

• H o : μ = 1.35 in./year
• H 1 : μ ≠ 1.35 in./year

Step 2) State the level of significance and the critical value.

• We will choose a level of significance of 5% ( α = 0.05).
• For a two-sided question, we need a two-sided critical value – Z α /2 and + Z α /2 .
• The level of significance is divided by 2 (since we are only testing “not equal”). We must have two rejection zones that can deal with either a greater than or less than outcome (to the right (+) or to the left (-)).
• We need to find the Z-score associated with the area of 0.025. The red areas are equal to α /2 = 0.05/2 = 0.025 or 2.5% of the area under the normal curve.
• Go into the body of values and find the negative Z-score associated with the area 0.025.

Figure 5. The rejection zone for a two-sided test.

• The negative critical value is -1.96. Since the curve is symmetric, we know that the positive critical value is 1.96.
• ±1.96 are the critical values. These values set up the rejection zone. If the test statistic falls within these red rejection zones, we reject the null hypothesis.

Step 3) Compute the test statistic.

• The test statistic is the number of standard deviations the sample mean is from the known mean. It is also a Z-score, just like the critical value.

• For this problem, the test statistic is

Step 4) State a conclusion.

• Compare the test statistic to the critical value. If the test statistic falls into the rejection zones, reject the null hypothesis. In other words, if the test statistic is greater than +1.96 or less than -1.96, reject the null hypothesis.

Figure 6. The critical values for a two-sided test when α = 0.05.

In this problem, the test statistic falls in the red rejection zone. The test statistic of 3.07 is greater than the critical value of 1.96.We will reject the null hypothesis. We have enough evidence to support the claim that the mean diameter growth is different from (not equal to) 1.35 in./year.

A researcher believes that there has been an increase in the average farm size in his state since the last study five years ago. The previous study reported a mean size of 450 acres with a population standard deviation ( σ ) of 167 acres. He samples 45 farms and gets a sample mean of 485.8 acres. Is there enough information to support his claim?

• H o : μ = 450 acres
• H 1 : μ >450 acres
• For a one-sided question, we need a one-sided positive critical value Z α .
• The level of significance is all in the right side (the rejection zone is just on the right side).
• We need to find the Z-score associated with the 5% area in the right tail.

Figure 7. Rejection zone for a right-sided hypothesis test.

• Go into the body of values in the standard normal table and find the Z-score that separates the lower 95% from the upper 5%.
• The critical value is 1.645. This value sets up the rejection zone.

• Compare the test statistic to the critical value.

Figure 8. The critical value for a right-sided test when α = 0.05.

• The test statistic does not fall in the rejection zone. It is less than the critical value.

We fail to reject the null hypothesis. We do not have enough evidence to support the claim that the mean farm size has increased from 450 acres.

A researcher believes that there has been a reduction in the mean number of hours that college students spend preparing for final exams. A national study stated that students at a 4-year college spend an average of 23 hours preparing for 5 final exams each semester with a population standard deviation of 7.3 hours. The researcher sampled 227 students and found a sample mean study time of 19.6 hours. Does this indicate that the average study time for final exams has decreased? Use a 1% level of significance to test this claim.

• H o : μ = 23 hours
• H 1 : μ < 23 hours
• This is a left-sided test so alpha (0.01) is all in the left tail.

Figure 9. The rejection zone for a left-sided hypothesis test.

• Go into the body of values in the standard normal table and find the Z-score that defines the lower 1% of the area.
• The critical value is -2.33. This value sets up the rejection zone.

Figure 10. The critical value for a left-sided test when α = 0.01.

• The test statistic falls in the rejection zone. The test statistic of -7.02 is less than the critical value of -2.33.

We reject the null hypothesis. We have sufficient evidence to support the claim that the mean final exam study time has decreased below 23 hours.

Testing a Hypothesis using P-values

The p-value is the probability of observing our sample mean given that the null hypothesis is true. It is the area under the curve to the left or right of the test statistic. If the probability of observing such a sample mean is very small (less than the level of significance), we would reject the null hypothesis. Computations for the p-value depend on whether it is a one- or two-sided test.

Steps for a hypothesis test using p-values:

• State the level of significance.
• Compute the test statistic and find the area associated with it (this is the p-value).
• Compare the p-value to alpha ( α ) and state a conclusion.

Instead of comparing Z-score test statistic to Z-score critical value, as in the classical method, we compare area of the test statistic to area of the level of significance.

The Decision Rule: If the p-value is less than alpha, we reject the null hypothesis

Computing P-values

If it is a two-sided test (the alternative claim is ≠), the p-value is equal to two times the probability of the absolute value of the test statistic. If the test is a left-sided test (the alternative claim is “<”), then the p-value is equal to the area to the left of the test statistic. If the test is a right-sided test (the alternative claim is “>”), then the p-value is equal to the area to the right of the test statistic.

Let’s look at Example 6 again.

A forester studying diameter growth of red pine believes that the mean diameter growth will be different from the known mean growth of 1.35 in./year if a fertilization treatment is applied to the stand. He conducts his experiment, collects data from a sample of 32 plots, and gets a sample mean diameter growth of 1.6 in./year. The population standard deviation for this stand is known to be 0.46 in./year. Does he have enough evidence to support his claim?

Step 2) State the level of significance.

• For this problem, the test statistic is:

The p-value is two times the area of the absolute value of the test statistic (because the alternative claim is “not equal”).

Figure 11. The p-value compared to the level of significance.

• Look up the area for the Z-score 3.07 in the standard normal table. The area (probability) is equal to 1 – 0.9989 = 0.0011.
• Multiply this by 2 to get the p-value = 2 * 0.0011 = 0.0022.

Step 4) Compare the p-value to alpha and state a conclusion.

• Use the Decision Rule (if the p-value is less than α , reject H 0 ).
• In this problem, the p-value (0.0022) is less than alpha (0.05).
• We reject the H 0 . We have enough evidence to support the claim that the mean diameter growth is different from 1.35 inches/year.

Let’s look at Example 7 again.

The p-value is the area to the right of the Z-score 1.44 (the hatched area).

• This is equal to 1 – 0.9251 = 0.0749.
• The p-value is 0.0749.

Figure 12. The p-value compared to the level of significance for a right-sided test.

• Use the Decision Rule.
• In this problem, the p-value (0.0749) is greater than alpha (0.05), so we Fail to Reject the H 0 .
• The area of the test statistic is greater than the area of alpha ( α ).

We fail to reject the null hypothesis. We do not have enough evidence to support the claim that the mean farm size has increased.

Let’s look at Example 8 again.

• H 0 : μ = 23 hours

The p-value is the area to the left of the test statistic (the little black area to the left of -7.02). The Z-score of -7.02 is not on the standard normal table. The smallest probability on the table is 0.0002. We know that the area for the Z-score -7.02 is smaller than this area (probability). Therefore, the p-value is <0.0002.

Figure 13. The p-value compared to the level of significance for a left-sided test.

• In this problem, the p-value (p<0.0002) is less than alpha (0.01), so we Reject the H 0 .
• The area of the test statistic is much less than the area of alpha ( α ).

We reject the null hypothesis. We have enough evidence to support the claim that the mean final exam study time has decreased below 23 hours.

Both the classical method and p-value method for testing a hypothesis will arrive at the same conclusion. In the classical method, the critical Z-score is the number on the z-axis that defines the level of significance ( α ). The test statistic converts the sample mean to units of standard deviation (a Z-score). If the test statistic falls in the rejection zone defined by the critical value, we will reject the null hypothesis. In this approach, two Z-scores, which are numbers on the z-axis, are compared. In the p-value approach, the p-value is the area associated with the test statistic. In this method, we compare α (which is also area under the curve) to the p-value. If the p-value is less than α , we reject the null hypothesis. The p-value is the probability of observing such a sample mean when the null hypothesis is true. If the probability is too small (less than the level of significance), then we believe we have enough statistical evidence to reject the null hypothesis and support the alternative claim.

Software Solutions

(referring to Ex. 8)

One-Sample Z

Excel does not offer 1-sample hypothesis testing.

Hypothesis Test about the Population Mean ( μ ) when the Population Standard Deviation ( σ ) is Unknown

Frequently, the population standard deviation (σ) is not known. We can estimate the population standard deviation (σ) with the sample standard deviation (s). However, the test statistic will no longer follow the standard normal distribution. We must rely on the student’s t-distribution with n-1 degrees of freedom. Because we use the sample standard deviation (s), the test statistic will change from a Z-score to a t-score.

Steps for a hypothesis test are the same that we covered in Section 2.

Just as with the hypothesis test from the previous section, the data for this test must be from a random sample and requires either that the population from which the sample was drawn be normal or that the sample size is sufficiently large (n≥30). A t-test is robust, so small departures from normality will not adversely affect the results of the test. That being said, if the sample size is smaller than 30, it is always good to verify the assumption of normality through a normal probability plot.

We will still have the same three pairs of null and alternative hypotheses and we can still use either the classical approach or the p-value approach.

Selecting the correct critical value from the student’s t-distribution table depends on three factors: the type of test (one-sided or two-sided alternative hypothesis), the sample size, and the level of significance.

For a two-sided test (“not equal” alternative hypothesis), the critical value (t α /2 ), is determined by alpha ( α ), the level of significance, divided by two, to deal with the possibility that the result could be less than OR greater than the known value.

• If your level of significance was 0.05, you would use the 0.025 column to find the correct critical value (0.05/2 = 0.025).
• If your level of significance was 0.01, you would use the 0.005 column to find the correct critical value (0.01/2 = 0.005).

For a one-sided test (“a less than” or “greater than” alternative hypothesis), the critical value (t α ) , is determined by alpha ( α ), the level of significance, being all in the one side.

• If your level of significance was 0.05, you would use the 0.05 column to find the correct critical value for either a left or right-side question. If you are asking a “less than” (left-sided question, your critical value will be negative. If you are asking a “greater than” (right-sided question), your critical value will be positive.

Find the critical value you would use to test the claim that μ ≠ 112 with a sample size of 18 and a 5% level of significance.

In this case, the critical value (t α /2 ) would be 2.110. This is a two-sided question (≠) so you would divide alpha by 2 (0.05/2 = 0.025) and go down the 0.025 column to 17 degrees of freedom.

What would the critical value be if you wanted to test that μ < 112 for the same data?

In this case, the critical value would be 1.740. This is a one-sided question (<) so alpha would be divided by 1 (0.05/1 = 0.05). You would go down the 0.05 column with 17 degrees of freedom to get the correct critical value.

In 2005, the mean pH level of rain in a county in northern New York was 5.41. A biologist believes that the rain acidity has changed. He takes a random sample of 11 rain dates in 2010 and obtains the following data. Use a 1% level of significance to test his claim.

4.70, 5.63, 5.02, 5.78, 4.99, 5.91, 5.76, 5.54, 5.25, 5.18, 5.01

The sample size is small and we don’t know anything about the distribution of the population, so we examine a normal probability plot. The distribution looks normal so we will continue with our test.

Figure 14. A normal probability plot for Example 9.

The sample mean is 5.343 with a sample standard deviation of 0.397.

• H o : μ = 5.41
• H 1 : μ ≠ 5.41
• This is a two-sided question so alpha is divided by two.

Figure 15. The rejection zones for a two-sided test.

• t α /2 is found by going down the 0.005 column with 14 degrees of freedom.
• t α /2 = ±3.169.
• The test statistic is a t-score.

Figure 16. The critical values for a two-sided test when α = 0.01.

• The test statistic does not fall in the rejection zone.

We will fail to reject the null hypothesis. We do not have enough evidence to support the claim that the mean rain pH has changed.

A One-sided Test

Cadmium, a heavy metal, is toxic to animals. Mushrooms, however, are able to absorb and accumulate cadmium at high concentrations. The government has set safety limits for cadmium in dry vegetables at 0.5 ppm. Biologists believe that the mean level of cadmium in mushrooms growing near strip mines is greater than the recommended limit of 0.5 ppm, negatively impacting the animals that live in this ecosystem. A random sample of 51 mushrooms gave a sample mean of 0.59 ppm with a sample standard deviation of 0.29 ppm. Use a 5% level of significance to test the claim that the mean cadmium level is greater than the acceptable limit of 0.5 ppm.

The sample size is greater than 30 so we are assured of a normal distribution of the means.

• H o : μ = 0.5 ppm
• H 1 : μ > 0.5 ppm
• This is a right-sided question so alpha is all in the right tail.

Figure 17. Rejection zone for a right-sided test.

• t α is found by going down the 0.05 column with 50 degrees of freedom.
• t α = 1.676

Step 4) State a Conclusion.

Figure 18. Critical value for a right-sided test when α = 0.05.

The test statistic falls in the rejection zone. We will reject the null hypothesis. We have enough evidence to support the claim that the mean cadmium level is greater than the acceptable safe limit.

BUT, what happens if the significance level changes to 1%?

The critical value is now found by going down the 0.01 column with 50 degrees of freedom. The critical value is 2.403. The test statistic is now LESS THAN the critical value. The test statistic does not fall in the rejection zone. The conclusion will change. We do NOT have enough evidence to support the claim that the mean cadmium level is greater than the acceptable safe limit of 0.5 ppm.

The level of significance is the probability that you, as the researcher, set to decide if there is enough statistical evidence to support the alternative claim. It should be set before the experiment begins.

P-value Approach

We can also use the p-value approach for a hypothesis test about the mean when the population standard deviation ( σ ) is unknown. However, when using a student’s t-table, we can only estimate the range of the p-value, not a specific value as when using the standard normal table. The student’s t-table has area (probability) across the top row in the table, with t-scores in the body of the table.

• To find the p-value (the area associated with the test statistic), you would go to the row with the number of degrees of freedom.
• Go across that row until you find the two values that your test statistic is between, then go up those columns to find the estimated range for the p-value.

Estimating P-value from a Student’s T-table

If your test statistic is 3.789 with 3 degrees of freedom, you would go across the 3 df row. The value 3.789 falls between the values 3.482 and 4.541 in that row. Therefore, the p-value is between 0.02 and 0.01. The p-value will be greater than 0.01 but less than 0.02 (0.01<p<0.02).

If your level of significance is 5%, you would reject the null hypothesis as the p-value (0.01-0.02) is less than alpha ( α ) of 0.05.

If your level of significance is 1%, you would fail to reject the null hypothesis as the p-value (0.01-0.02) is greater than alpha ( α ) of 0.01.

Software packages typically output p-values. It is easy to use the Decision Rule to answer your research question by the p-value method.

(referring to Ex. 12)

One-Sample T

Test of mu = 0.5 vs. > 0.5

Additional example: www.youtube.com/watch?v=WwdSjO4VUsg .

Hypothesis Test for a Population Proportion ( p )

Frequently, the parameter we are testing is the population proportion.

• We are studying the proportion of trees with cavities for wildlife habitat.
• We need to know if the proportion of people who support green building materials has changed.
• Has the proportion of wolves that died last year in Yellowstone increased from the year before?

Recall that the best point estimate of p , the population proportion, is given by

when np (1 – p )≥10. We can use both the classical approach and the p-value approach for testing.

The steps for a hypothesis test are the same that we covered in Section 2.

The test statistic follows the standard normal distribution. Notice that the standard error (the denominator) uses p instead of p̂ , which was used when constructing a confidence interval about the population proportion. In a hypothesis test, the null hypothesis is assumed to be true, so the known proportion is used.

• The critical value comes from the standard normal table, just as in Section 2. We will still use the same three pairs of null and alternative hypotheses as we used in the previous sections, but the parameter is now p instead of μ :

• For a two-sided test, alpha will be divided by 2 giving a ± Z α /2 critical value.
• For a left-sided test, alpha will be all in the left tail giving a – Z α critical value.
• For a right-sided test, alpha will be all in the right tail giving a Z α critical value.

A botanist has produced a new variety of hybrid soy plant that is better able to withstand drought than other varieties. The botanist knows the seed germination for the parent plants is 75%, but does not know the seed germination for the new hybrid. He tests the claim that it is different from the parent plants. To test this claim, 450 seeds from the hybrid plant are tested and 321 have germinated. Use a 5% level of significance to test this claim that the germination rate is different from 75%.

• H o : p = 0.75
• H 1 : p ≠ 0.75

This is a two-sided question so alpha is divided by 2.

• Alpha is 0.05 so the critical values are ± Z α /2 = ± Z .025 .
• Look on the negative side of the standard normal table, in the body of values for 0.025.
• The critical values are ± 1.96.

Figure 19. Critical values for a two-sided test when α = 0.05.

The test statistic does not fall in the rejection zone. We fail to reject the null hypothesis. We do not have enough evidence to support the claim that the germination rate of the hybrid plant is different from the parent plants.

Let’s answer this question using the p-value approach. Remember, for a two-sided alternative hypothesis (“not equal”), the p-value is two times the area of the test statistic. The test statistic is -1.81 and we want to find the area to the left of -1.81 from the standard normal table.

• On the negative page, find the Z-score -1.81. Find the area associated with this Z-score.
• The area = 0.0351.
• This is a two-sided test so multiply the area times 2 to get the p-value = 0.0351 x 2 = 0.0702.

Now compare the p-value to alpha. The Decision Rule states that if the p-value is less than alpha, reject the H 0 . In this case, the p-value (0.0702) is greater than alpha (0.05) so we will fail to reject H 0 . We do not have enough evidence to support the claim that the germination rate of the hybrid plant is different from the parent plants.

You are a biologist studying the wildlife habitat in the Monongahela National Forest. Cavities in older trees provide excellent habitat for a variety of birds and small mammals. A study five years ago stated that 32% of the trees in this forest had suitable cavities for this type of wildlife. You believe that the proportion of cavity trees has increased. You sample 196 trees and find that 79 trees have cavities. Does this evidence support your claim that there has been an increase in the proportion of cavity trees?

Use a 10% level of significance to test this claim.

• H o : p = 0.32
• H 1 : p > 0.32

This is a one-sided question so alpha is divided by 1.

• Alpha is 0.10 so the critical value is Z α = Z .10
• Look on the positive side of the standard normal table, in the body of values for 0.90.
• The critical value is 1.28.

Figure 20. Critical value for a right-sided test where α = 0.10.

• The test statistic is the number of standard deviations the sample proportion is from the known proportion. It is also a Z-score, just like the critical value.

Figure 21. Comparison of the test statistic and the critical value.

The test statistic is larger than the critical value (it falls in the rejection zone). We will reject the null hypothesis. We have enough evidence to support the claim that there has been an increase in the proportion of cavity trees.

Now use the p-value approach to answer the question. This is a right-sided question (“greater than”), so the p-value is equal to the area to the right of the test statistic. Go to the positive side of the standard normal table and find the area associated with the Z-score of 2.49. The area is 0.9936. Remember that this table is cumulative from the left. To find the area to the right of 2.49, we subtract from one.

p-value = (1 – 0.9936) = 0.0064

The p-value is less than the level of significance (0.10), so we reject the null hypothesis. We have enough evidence to support the claim that the proportion of cavity trees has increased.

(referring to Ex. 15)

Test and CI for One Proportion

Test of p = 0.32 vs. p > 0.32

Hypothesis Test about a Variance

When people think of statistical inference, they usually think of inferences involving population means or proportions. However, the particular population parameter needed to answer an experimenter’s practical questions varies from one situation to another, and sometimes a population’s variability is more important than its mean. Thus, product quality is often defined in terms of low variability.

Sample variance S 2 can be used for inferences concerning a population variance σ 2 . For a random sample of n measurements drawn from a normal population with mean μ and variance σ 2 , the value S 2 provides a point estimate for σ 2 . In addition, the quantity ( n – 1) S 2 / σ 2 follows a Chi-square ( χ 2 ) distribution, with df = n – 1.

The properties of Chi-square ( χ 2 ) distribution are:

• Unlike Z and t distributions, the values in a chi-square distribution are all positive.
• The chi-square distribution is asymmetric, unlike the Z and t distributions.
• There are many chi-square distributions. We obtain a particular one by specifying the degrees of freedom (df = n – 1) associated with the sample variances S 2 .

Figure 22. The chi-square distribution.

One-sample χ 2 test for testing the hypotheses:

Alternative hypothesis:

where the χ 2 critical value in the rejection region is based on degrees of freedom df = n – 1 and a specified significance level of α .

As with previous sections, if the test statistic falls in the rejection zone set by the critical value, you will reject the null hypothesis.

A forester wants to control a dense understory of striped maple that is interfering with desirable hardwood regeneration using a mist blower to apply an herbicide treatment. She wants to make sure that treatment has a consistent application rate, in other words, low variability not exceeding 0.25 gal./acre (0.06 gal. 2 ). She collects sample data (n = 11) on this type of mist blower and gets a sample variance of 0.064 gal. 2 Using a 5% level of significance, test the claim that the variance is significantly greater than 0.06 gal. 2

H 0 : σ 2 = 0.06

H 1 : σ 2 >0.06

The critical value is 18.307. Any test statistic greater than this value will cause you to reject the null hypothesis.

The test statistic is

We fail to reject the null hypothesis. The forester does NOT have enough evidence to support the claim that the variance is greater than 0.06 gal. 2 You can also estimate the p-value using the same method as for the student t-table. Go across the row for degrees of freedom until you find the two values that your test statistic falls between. In this case going across the row 10, the two table values are 4.865 and 15.987. Now go up those two columns to the top row to estimate the p-value (0.1-0.9). The p-value is greater than 0.1 and less than 0.9. Both are greater than the level of significance (0.05) causing us to fail to reject the null hypothesis.

(referring to Ex. 16)

Test and CI for One Variance

The chi-square method is only for the normal distribution.

Excel does not offer 1-sample χ 2 testing.

Putting it all Together Using the Classical Method

To test a claim about μ when σ is known.

• Write the null and alternative hypotheses.
• State the level of significance and get the critical value from the standard normal table.

• Compare the test statistic to the critical value (Z-score) and write the conclusion.

To Test a Claim about μ When σ is Unknown

• State the level of significance and get the critical value from the student’s t-table with n-1 degrees of freedom.

• Compare the test statistic to the critical value (t-score) and write the conclusion.

To Test a Claim about p

• State the level of significance and get the critical value from the standard normal distribution.

Table 4. A summary table for critical Z-scores.

To Test a Claim about Variance

• State the level of significance and get the critical value from the chi-square table using n-1 degrees of freedom.

• Compare the test statistic to the critical value and write the conclusion.
• Natural Resources Biometrics. Authored by : Diane Kiernan. Located at : https://textbooks.opensuny.org/natural-resources-biometrics/ . Project : Open SUNY Textbooks. License : CC BY-NC-SA: Attribution-NonCommercial-ShareAlike

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S.3 hypothesis testing.

In reviewing hypothesis tests, we start first with the general idea. Then, we keep returning to the basic procedures of hypothesis testing, each time adding a little more detail.

The general idea of hypothesis testing involves:

• Making an initial assumption.
• Collecting evidence (data).
• Based on the available evidence (data), deciding whether to reject or not reject the initial assumption.

Every hypothesis test — regardless of the population parameter involved — requires the above three steps.

Example S.3.1

Is normal body temperature really 98.6 degrees f section  .

Consider the population of many, many adults. A researcher hypothesized that the average adult body temperature is lower than the often-advertised 98.6 degrees F. That is, the researcher wants an answer to the question: "Is the average adult body temperature 98.6 degrees? Or is it lower?" To answer his research question, the researcher starts by assuming that the average adult body temperature was 98.6 degrees F.

Then, the researcher went out and tried to find evidence that refutes his initial assumption. In doing so, he selects a random sample of 130 adults. The average body temperature of the 130 sampled adults is 98.25 degrees.

Then, the researcher uses the data he collected to make a decision about his initial assumption. It is either likely or unlikely that the researcher would collect the evidence he did given his initial assumption that the average adult body temperature is 98.6 degrees:

• If it is likely , then the researcher does not reject his initial assumption that the average adult body temperature is 98.6 degrees. There is not enough evidence to do otherwise.
• either the researcher's initial assumption is correct and he experienced a very unusual event;
• or the researcher's initial assumption is incorrect.

In statistics, we generally don't make claims that require us to believe that a very unusual event happened. That is, in the practice of statistics, if the evidence (data) we collected is unlikely in light of the initial assumption, then we reject our initial assumption.

Example S.3.2

Criminal trial analogy section  .

One place where you can consistently see the general idea of hypothesis testing in action is in criminal trials held in the United States. Our criminal justice system assumes "the defendant is innocent until proven guilty." That is, our initial assumption is that the defendant is innocent.

In the practice of statistics, we make our initial assumption when we state our two competing hypotheses -- the null hypothesis ( H 0 ) and the alternative hypothesis ( H A ). Here, our hypotheses are:

• H 0 : Defendant is not guilty (innocent)
• H A : Defendant is guilty

In statistics, we always assume the null hypothesis is true . That is, the null hypothesis is always our initial assumption.

The prosecution team then collects evidence — such as finger prints, blood spots, hair samples, carpet fibers, shoe prints, ransom notes, and handwriting samples — with the hopes of finding "sufficient evidence" to make the assumption of innocence refutable.

In statistics, the data are the evidence.

The jury then makes a decision based on the available evidence:

• If the jury finds sufficient evidence — beyond a reasonable doubt — to make the assumption of innocence refutable, the jury rejects the null hypothesis and deems the defendant guilty. We behave as if the defendant is guilty.
• If there is insufficient evidence, then the jury does not reject the null hypothesis . We behave as if the defendant is innocent.

In statistics, we always make one of two decisions. We either "reject the null hypothesis" or we "fail to reject the null hypothesis."

Errors in Hypothesis Testing Section  

Did you notice the use of the phrase "behave as if" in the previous discussion? We "behave as if" the defendant is guilty; we do not "prove" that the defendant is guilty. And, we "behave as if" the defendant is innocent; we do not "prove" that the defendant is innocent.

This is a very important distinction! We make our decision based on evidence not on 100% guaranteed proof. Again:

• If we reject the null hypothesis, we do not prove that the alternative hypothesis is true.
• If we do not reject the null hypothesis, we do not prove that the null hypothesis is true.

We merely state that there is enough evidence to behave one way or the other. This is always true in statistics! Because of this, whatever the decision, there is always a chance that we made an error .

Let's review the two types of errors that can be made in criminal trials:

Table S.3.2 shows how this corresponds to the two types of errors in hypothesis testing.

Note that, in statistics, we call the two types of errors by two different  names -- one is called a "Type I error," and the other is called  a "Type II error." Here are the formal definitions of the two types of errors:

There is always a chance of making one of these errors. But, a good scientific study will minimize the chance of doing so!

Making the Decision Section  

Recall that it is either likely or unlikely that we would observe the evidence we did given our initial assumption. If it is likely , we do not reject the null hypothesis. If it is unlikely , then we reject the null hypothesis in favor of the alternative hypothesis. Effectively, then, making the decision reduces to determining "likely" or "unlikely."

In statistics, there are two ways to determine whether the evidence is likely or unlikely given the initial assumption:

• We could take the " critical value approach " (favored in many of the older textbooks).
• Or, we could take the " P -value approach " (what is used most often in research, journal articles, and statistical software).

In the next two sections, we review the procedures behind each of these two approaches. To make our review concrete, let's imagine that μ is the average grade point average of all American students who major in mathematics. We first review the critical value approach for conducting each of the following three hypothesis tests about the population mean $\mu$:

In Practice

• We would want to conduct the first hypothesis test if we were interested in concluding that the average grade point average of the group is more than 3.
• We would want to conduct the second hypothesis test if we were interested in concluding that the average grade point average of the group is less than 3.
• And, we would want to conduct the third hypothesis test if we were only interested in concluding that the average grade point average of the group differs from 3 (without caring whether it is more or less than 3).

Upon completing the review of the critical value approach, we review the P -value approach for conducting each of the above three hypothesis tests about the population mean \(\mu\). The procedures that we review here for both approaches easily extend to hypothesis tests about any other population parameter.

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Statistics By Jim

Making statistics intuitive

How Hypothesis Tests Work: Significance Levels (Alpha) and P values

By Jim Frost 45 Comments

Hypothesis testing is a vital process in inferential statistics where the goal is to use sample data to draw conclusions about an entire population . In the testing process, you use significance levels and p-values to determine whether the test results are statistically significant.

You hear about results being statistically significant all of the time. But, what do significance levels, P values, and statistical significance actually represent? Why do we even need to use hypothesis tests in statistics?

In this post, I answer all of these questions. I use graphs and concepts to explain how hypothesis tests function in order to provide a more intuitive explanation. This helps you move on to understanding your statistical results.

Hypothesis Test Example Scenario

To start, I’ll demonstrate why we need to use hypothesis tests using an example.

A researcher is studying fuel expenditures for families and wants to determine if the monthly cost has changed since last year when the average was $260 per month. The researcher draws a random sample of 25 families and enters their monthly costs for this year into statistical software. You can download the CSV data file: FuelsCosts . Below are the descriptive statistics for this year.

We’ll build on this example to answer the research question and show how hypothesis tests work.

Descriptive Statistics Alone Won’t Answer the Question

The researcher collected a random sample and found that this year’s sample mean (330.6) is greater than last year’s mean (260). Why perform a hypothesis test at all? We can see that this year’s mean is higher by $70! Isn’t that different?

Regrettably, the situation isn’t as clear as you might think because we’re analyzing a sample instead of the full population. There are huge benefits when working with samples because it is usually impossible to collect data from an entire population. However, the tradeoff for working with a manageable sample is that we need to account for sample error.

The sampling error is the gap between the sample statistic and the population parameter. For our example, the sample statistic is the sample mean, which is 330.6. The population parameter is μ, or mu, which is the average of the entire population. Unfortunately, the value of the population parameter is not only unknown but usually unknowable. Learn more about Sampling Error .

We obtained a sample mean of 330.6. However, it’s conceivable that, due to sampling error, the mean of the population might be only 260. If the researcher drew another random sample, the next sample mean might be closer to 260. It’s impossible to assess this possibility by looking at only the sample mean. Hypothesis testing is a form of inferential statistics that allows us to draw conclusions about an entire population based on a representative sample. We need to use a hypothesis test to determine the likelihood of obtaining our sample mean if the population mean is 260.

Background information : The Difference between Descriptive and Inferential Statistics and Populations, Parameters, and Samples in Inferential Statistics

A Sampling Distribution Determines Whether Our Sample Mean is Unlikely

It is very unlikely for any sample mean to equal the population mean because of sample error. In our case, the sample mean of 330.6 is almost definitely not equal to the population mean for fuel expenditures.

If we could obtain a substantial number of random samples and calculate the sample mean for each sample, we’d observe a broad spectrum of sample means. We’d even be able to graph the distribution of sample means from this process.

This type of distribution is called a sampling distribution. You obtain a sampling distribution by drawing many random samples of the same size from the same population. Why the heck would we do this?

Because sampling distributions allow you to determine the likelihood of obtaining your sample statistic and they’re crucial for performing hypothesis tests.

Luckily, we don’t need to go to the trouble of collecting numerous random samples! We can estimate the sampling distribution using the t-distribution, our sample size, and the variability in our sample.

We want to find out if the average fuel expenditure this year (330.6) is different from last year (260). To answer this question, we’ll graph the sampling distribution based on the assumption that the mean fuel cost for the entire population has not changed and is still 260. In statistics, we call this lack of effect, or no change, the null hypothesis . We use the null hypothesis value as the basis of comparison for our observed sample value.

Sampling distributions and t-distributions are types of probability distributions.

Related posts : Sampling Distributions and Understanding Probability Distributions

Graphing our Sample Mean in the Context of the Sampling Distribution

The graph below shows which sample means are more likely and less likely if the population mean is 260. We can place our sample mean in this distribution. This larger context helps us see how unlikely our sample mean is if the null hypothesis is true (μ = 260).

The graph displays the estimated distribution of sample means. The most likely values are near 260 because the plot assumes that this is the true population mean. However, given random sampling error, it would not be surprising to observe sample means ranging from 167 to 352. If the population mean is still 260, our observed sample mean (330.6) isn’t the most likely value, but it’s not completely implausible either.

The Role of Hypothesis Tests

The sampling distribution shows us that we are relatively unlikely to obtain a sample of 330.6 if the population mean is 260. Is our sample mean so unlikely that we can reject the notion that the population mean is 260?

In statistics, we call this rejecting the null hypothesis. If we reject the null for our example, the difference between the sample mean (330.6) and 260 is statistically significant. In other words, the sample data favor the hypothesis that the population average does not equal 260.

However, look at the sampling distribution chart again. Notice that there is no special location on the curve where you can definitively draw this conclusion. There is only a consistent decrease in the likelihood of observing sample means that are farther from the null hypothesis value. Where do we decide a sample mean is far away enough?

To answer this question, we’ll need more tools—hypothesis tests! The hypothesis testing procedure quantifies the unusualness of our sample with a probability and then compares it to an evidentiary standard. This process allows you to make an objective decision about the strength of the evidence.

We’re going to add the tools we need to make this decision to the graph—significance levels and p-values!

These tools allow us to test these two hypotheses:

• Null hypothesis: The population mean equals the null hypothesis mean (260).
• Alternative hypothesis: The population mean does not equal the null hypothesis mean (260).

Related post : Hypothesis Testing Overview

What are Significance Levels (Alpha)?

A significance level, also known as alpha or α, is an evidentiary standard that a researcher sets before the study. It defines how strongly the sample evidence must contradict the null hypothesis before you can reject the null hypothesis for the entire population. The strength of the evidence is defined by the probability of rejecting a null hypothesis that is true. In other words, it is the probability that you say there is an effect when there is no effect.

For instance, a significance level of 0.05 signifies a 5% risk of deciding that an effect exists when it does not exist.

Lower significance levels require stronger sample evidence to be able to reject the null hypothesis. For example, to be statistically significant at the 0.01 significance level requires more substantial evidence than the 0.05 significance level. However, there is a tradeoff in hypothesis tests. Lower significance levels also reduce the power of a hypothesis test to detect a difference that does exist.

The technical nature of these types of questions can make your head spin. A picture can bring these ideas to life!

To learn a more conceptual approach to significance levels, see my post about Understanding Significance Levels .

Graphing Significance Levels as Critical Regions

On the probability distribution plot, the significance level defines how far the sample value must be from the null value before we can reject the null. The percentage of the area under the curve that is shaded equals the probability that the sample value will fall in those regions if the null hypothesis is correct.

To represent a significance level of 0.05, I’ll shade 5% of the distribution furthest from the null value.

The two shaded regions in the graph are equidistant from the central value of the null hypothesis. Each region has a probability of 0.025, which sums to our desired total of 0.05. These shaded areas are called the critical region for a two-tailed hypothesis test.

The critical region defines sample values that are improbable enough to warrant rejecting the null hypothesis. If the null hypothesis is correct and the population mean is 260, random samples (n=25) from this population have means that fall in the critical region 5% of the time.

Our sample mean is statistically significant at the 0.05 level because it falls in the critical region.

Related posts : One-Tailed and Two-Tailed Tests Explained , What Are Critical Values? , and T-distribution Table of Critical Values

Comparing Significance Levels

Let’s redo this hypothesis test using the other common significance level of 0.01 to see how it compares.

This time the sum of the two shaded regions equals our new significance level of 0.01. The mean of our sample does not fall within with the critical region. Consequently, we fail to reject the null hypothesis. We have the same exact sample data, the same difference between the sample mean and the null hypothesis value, but a different test result.

What happened? By specifying a lower significance level, we set a higher bar for the sample evidence. As the graph shows, lower significance levels move the critical regions further away from the null value. Consequently, lower significance levels require more extreme sample means to be statistically significant.

You must set the significance level before conducting a study. You don’t want the temptation of choosing a level after the study that yields significant results. The only reason I compared the two significance levels was to illustrate the effects and explain the differing results.

The graphical version of the 1-sample t-test we created allows us to determine statistical significance without assessing the P value. Typically, you need to compare the P value to the significance level to make this determination.

Related post : Step-by-Step Instructions for How to Do t-Tests in Excel

What Are P values?

P values are the probability that a sample will have an effect at least as extreme as the effect observed in your sample if the null hypothesis is correct.

This tortuous, technical definition for P values can make your head spin. Let’s graph it!

First, we need to calculate the effect that is present in our sample. The effect is the distance between the sample value and null value: 330.6 – 260 = 70.6. Next, I’ll shade the regions on both sides of the distribution that are at least as far away as 70.6 from the null (260 +/- 70.6). This process graphs the probability of observing a sample mean at least as extreme as our sample mean.

The total probability of the two shaded regions is 0.03112. If the null hypothesis value (260) is true and you drew many random samples, you’d expect sample means to fall in the shaded regions about 3.1% of the time. In other words, you will observe sample effects at least as large as 70.6 about 3.1% of the time if the null is true. That’s the P value!

Learn more about How to Find the P Value .

Using P values and Significance Levels Together

If your P value is less than or equal to your alpha level, reject the null hypothesis.

The P value results are consistent with our graphical representation. The P value of 0.03112 is significant at the alpha level of 0.05 but not 0.01. Again, in practice, you pick one significance level before the experiment and stick with it!

Using the significance level of 0.05, the sample effect is statistically significant. Our data support the alternative hypothesis, which states that the population mean doesn’t equal 260. We can conclude that mean fuel expenditures have increased since last year.

P values are very frequently misinterpreted as the probability of rejecting a null hypothesis that is actually true. This interpretation is wrong! To understand why, please read my post: How to Interpret P-values Correctly .

Discussion about Statistically Significant Results

Hypothesis tests determine whether your sample data provide sufficient evidence to reject the null hypothesis for the entire population. To perform this test, the procedure compares your sample statistic to the null value and determines whether it is sufficiently rare. “Sufficiently rare” is defined in a hypothesis test by:

• Assuming that the null hypothesis is true—the graphs center on the null value.
• The significance (alpha) level—how far out from the null value is the critical region?
• The sample statistic—is it within the critical region?

There is no special significance level that correctly determines which studies have real population effects 100% of the time. The traditional significance levels of 0.05 and 0.01 are attempts to manage the tradeoff between having a low probability of rejecting a true null hypothesis and having adequate power to detect an effect if one actually exists.

The significance level is the rate at which you incorrectly reject null hypotheses that are actually true ( type I error ). For example, for all studies that use a significance level of 0.05 and the null hypothesis is correct, you can expect 5% of them to have sample statistics that fall in the critical region. When this error occurs, you aren’t aware that the null hypothesis is correct, but you’ll reject it because the p-value is less than 0.05.

This error does not indicate that the researcher made a mistake. As the graphs show, you can observe extreme sample statistics due to sample error alone. It’s the luck of the draw!

Related posts : Statistical Significance: Definition & Meaning and Types of Errors in Hypothesis Testing

Hypothesis tests are crucial when you want to use sample data to make conclusions about a population because these tests account for sample error. Using significance levels and P values to determine when to reject the null hypothesis improves the probability that you will draw the correct conclusion.

Keep in mind that statistical significance doesn’t necessarily mean that the effect is important in a practical, real-world sense. For more information, read my post about Practical vs. Statistical Significance .

If you like this post, read the companion post: How Hypothesis Tests Work: Confidence Intervals and Confidence Levels .

You can also read my other posts that describe how other tests work:

• How t-Tests Work
• How the F-test works in ANOVA
• How Chi-Squared Tests of Independence Work

To see an alternative approach to traditional hypothesis testing that does not use probability distributions and test statistics, learn about bootstrapping in statistics !

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Reader Interactions

December 11, 2022 at 10:56 am

For very easy concept about level of significance & p-value 1.Teacher has given a one assignment to student & asked how many error you have doing this assignment? Student reply, he can has error ≤ 5% (it is level of significance). After completion of assignment, teacher checked his error which is ≤ 5% (may be 4% or 3% or 2% even less, it is p-value) it means his results are significant. Otherwise he has error > 5% (may be 6% or 7% or 8% even more, it is p-value) it means his results are non-significant. 2. Teacher has given a one assignment to student & asked how many error you have doing this assignment? Student reply, he can has error ≤ 1% (it is level of significance). After completion of assignment, teacher checked his error which is ≤ 1% (may be 0.9% or 0.8% or 0.7% even less, it is p-value) it means his results are significant. Otherwise he has error > 1% (may be 1.1% or 1.5% or 2% even more, it is p-value) it means his results are non-significant. p-value is significant or not mainly dependent upon the level of significance.

December 11, 2022 at 7:50 pm

I think that approach helps explain how to determine statistical significance–is the p-value less than or equal to the significance level. However, it doesn’t really explain what statistical significance means. I find that comparing the p-value to the significance level is the easy part. Knowing what it means and how to choose your significance level is the harder part!

December 3, 2022 at 5:54 pm

What would you say to someone who believes that a p-value higher than the level of significance (alpha) means the null hypothesis has been proven? Should you support that statement or deny it?

December 3, 2022 at 10:18 pm

Hi Emmanuel,

When the p-value is greater than the significance level, you fail to reject the null hypothesis . That is different than proving it. To learn why and what it means, click the link to read a post that I’ve written that will answer your question!

April 19, 2021 at 12:27 am

Thank you so much Sir

April 18, 2021 at 2:37 pm

Hi sir, your blogs are much more helpful for clearing the concepts of statistics, as a researcher I find them much more useful. I have some quarries:

1. In many research papers I have seen authors using the statement ” means or values are statically at par at p = 0.05″ when they do some pair wise comparison between the treatments (a kind of post hoc) using some value of CD (critical difference) or we can say LSD which is calculated using alpha not using p. So with this article I think this should be alpha =0.05 or 5%, not p = 0.05 earlier I thought p and alpha are same. p it self is compared with alpha 0.05. Correct me if I am wrong.

2. When we can draw a conclusion using critical value based on critical values (CV) which is based on alpha values in different tests (e.g. in F test CV is at F (0.05, t-1, error df) when alpha is 0.05 which is table value of F and is compared with F calculated for drawing the conclusion); then why we go for p values, and draw a conclusion based on p values, even many online software do not give p value, they just mention CD (LSD)

3. can you please help me in interpreting interaction in two factor analysis (Factor A X Factor b) in Anova.

Thank You so much!

(Commenting again as I have not seen my comment in comment list; don’t know why)

April 18, 2021 at 10:57 pm

Hi Himanshu,

I manually approve comments so there will be some time lag involved before they show up.

Regarding your first question, yes, you’re correct. Test results are significant at particular significance levels or alpha. They should not use p to define the significance level. You’re also correct in that you compare p to alpha.

Critical values are a different (but related) approach for determining significance. It was more common before computer analysis took off because it reduced the calculations. Using this approach in its simplest form, you only know whether a result is significant or not at the given alpha. You just determine whether the test statistic falls within a critical region to determine statistical significance or not significant. However, it is ok to supplement this type of result with the actual p-value. Knowing the precise p-value provides additional information that significant/not significant does not provide. The critical value and p-value approaches will always agree too. For more information about why the exact p-value is useful, read my post about Five Tips for Interpreting P-values .

Finally, I’ve written about two-way ANOVA in my post, How to do Two-Way ANOVA in Excel . Additionally, I write about it in my Hypothesis Testing ebook .

January 28, 2021 at 3:12 pm

Thank you for your answer, Jim, I really appreciate it. I’m taking a Coursera stats course and online learning without being able to ask questions of a real teacher is not my forte!

You’re right, I don’t think I’m ready for that calculation! However, I think I’m struggling with something far more basic, perhaps even the interpretation of the t-table? I’m just not sure how you came up with the p-value as .03112, with the 24 degrees of freedom. When I pull up a t-table and look at the 24-degrees of freedom row, I’m not sure how any of those numbers correspond with your answer? Either the single tail of 0.01556 or the combined of 0.03112. What am I not getting? (which, frankly, could be a lot!!) Again, thank you SO much for your time.

January 28, 2021 at 11:19 pm

Ah ok, I see! First, let me point you to several posts I’ve written about t-values and the t-distribution. I don’t cover those in this post because I wanted to present a simplified version that just uses the data in its regular units. The basic idea is that the hypothesis tests actually convert all your raw data down into one value for a test statistic, such as the t-value. And then it uses that test statistic to determine whether your results are statistically significant. To be significant, the t-value must exceed a critical value, which is what you lookup in the table. Although, nowadays you’d typically let your software just tell you.

So, read the following two posts, which covers several aspects of t-values and distributions. And then if you have more questions after that, you can post them. But, you’ll have a lot more information about them and probably some of your questions will be answered! T-values T-distributions

January 27, 2021 at 3:10 pm

Jim, just found your website and really appreciate your thoughtful, thorough way of explaining things. I feel very dumb, but I’m struggling with p-values and was hoping you could help me.

Here’s the section that’s getting me confused:

“First, we need to calculate the effect that is present in our sample. The effect is the distance between the sample value and null value: 330.6 – 260 = 70.6. Next, I’ll shade the regions on both sides of the distribution that are at least as far away as 70.6 from the null (260 +/- 70.6). This process graphs the probability of observing a sample mean at least as extreme as our sample mean.

** I’m good up to this point. Draw the picture, do the subtraction, shade the regions. BUT, I’m not sure how to figure out the area of the shaded region — even with a T-table. When I look at the T-table on 24 df, I’m not sure what to do with those numbers, as none of them seem to correspond in any way to what I’m looking at in the problem. In the end, I have no idea how you calculated each shaded area being 0.01556.

I feel like there’s a (very simple) step that everyone else knows how to do, but for some reason I’m missing it.

Again, dumb question, but I’d love your help clarifying that.

thank you, Sara

January 27, 2021 at 9:51 pm

That’s not a dumb question at all. I actually don’t show or explain the calculations for figuring out the area. The reason for that is the same reason why students never calculate the critical t-values for their test, instead you look them up in tables or use statistical software. The common reason for all that is because calculating these values is extremely complicated! It’s best to let software do that for you or, when looking critical values, use the tables!

The principal though is that percentage of the area under the curve equals the probability that values will fall within that range.

And then, for this example, you’d need to figure out the area under the curve for particular ranges!

January 15, 2021 at 10:57 am

HI Jim, I have a question related to Hypothesis test.. in Medical imaging, there are different way to measure signal intensity (from a tumor lesion for example). I tested for the same 100 patients 4 different ways to measure tumor captation to a injected dose. So for the 100 patients, i got 4 linear regression (relation between injected dose and measured quantity at tumor sites) = so an output of 4 equations Condition A output = -0,034308 + 0,0006602*input Condition B output = 0,0117631 + 0,0005425*input Condition C output = 0,0087871 + 0,0005563*input Condition D output = 0,001911 + 0,0006255*input

My question : i want to compare the 4 methods to find the best one (compared to others) : do Hypothesis test good to me… and if Yes, i do not find test to perform it. Can you suggest me a software. I uselly used JMP for my stats… but open to other softwares…

THank for your time G

November 16, 2020 at 5:42 am

Thank you very much for writing about this topic!

Your explanation made more sense to me about: Why we reject Null Hypothesis when p value < significance level

Kind greetings, Jalal

September 25, 2020 at 1:04 pm

Hi Jim, Your explanations are so helpful! Thank you. I wondered about your first graph. I see that the mean of the graph is 260 from the null hypothesis, and it looks like the standard deviation of the graph is about 31. Where did you get 31 from? Thank you

September 25, 2020 at 4:08 pm

Hi Michelle,

That is a great question. Very observant. And it gets to how these tests work. The hypothesis test that I’m illustrating here is the one-sample t-test. And this graph illustrates the sampling distribution for the t-test. T-tests use the t-distribution to determine the sampling distribution. For the t-distribution, you need to specify the degrees of freedom, which entirely defines the distribution (i.e., it’s the only parameter). For 1-sample t-tests, the degrees of freedom equal the number of observations minus 1. This dataset has 25 observations. Hence, the 24 DF you see in the graph.

Unlike the normal distribution, there is no standard deviation parameter. Instead, the degrees of freedom determines the spread of the curve. Typically, with t-tests, you’ll see results discussed in terms of t-values, both for your sample and for defining the critical regions. However, for this introductory example, I’ve converted the t-values into the raw data units (t-value * SE mean).

So, the standard deviation you’re seeing in the graph is a result of the spread of the underlying t-distribution that has 24 degrees of freedom and then applying the conversion from t-values to raw values.

September 10, 2020 at 8:19 am

Your blog is incredible.

I am having difficulty understanding why the phrase ‘as extreme as’ is required in the definition of p-value (“P values are the probability that a sample will have an effect at least as extreme as the effect observed in your sample if the null hypothesis is correct.”)

Why can’t P-Values simply be defined as “The probability of sample observation if the null hypothesis is correct?”

In your other blog titled ‘Interpreting P values’ you have explained p-values as “P-values indicate the believability of the devil’s advocate case that the null hypothesis is correct given the sample data”. I understand (or accept) this explanation. How does one move from this definition to one that contains the phrase ‘as extreme as’?

September 11, 2020 at 5:05 pm

Thanks so much for your kind words! I’m glad that my website has been helpful!

The key to understanding the “at least as extreme” wording lies in the probability plots for p-values. Using probability plots for continuous data, you can calculate probabilities, but only for ranges of values. I discuss this in my post about understanding probability distributions . In a nutshell, we need a range of values for these probabilities because the probabilities are derived from the area under a distribution curve. A single value just produces a line on these graphs rather than an area. Those ranges are the shaded regions in the probability plots. For p-values, the range corresponds to the “at least as extreme” wording. That’s where it comes from. We need a range to calculate a probability. We can’t use the single value of the observed effect because it doesn’t produce an area under the curve.

I hope that helps! I think this is a particularly confusing part of understanding p-values that most people don’t understand.

August 7, 2020 at 5:45 pm

Hi Jim, thanks for the post.

Could you please clarify the following excerpt from ‘Graphing Significance Levels as Critical Regions’:

“The percentage of the area under the curve that is shaded equals the probability that the sample value will fall in those regions if the null hypothesis is correct.”

I’m not sure if I understood this correctly. If the sample value fall in one of the shaded regions, doesn’t mean that the null hypothesis can be rejected, hence that is not correct?

August 7, 2020 at 10:23 pm

Think of it this way. There are two basic reasons for why a sample value could fall in a critical region:

• The null hypothesis is correct and random chance caused the sample value to be unusual.
• The null hypothesis is not correct.

You don’t know which one is true. Remember, just because you reject the null hypothesis it doesn’t mean the null is false. However, by using hypothesis tests to determine statistical significance, you control the chances of #1 occurring. The rate at which #1 occurs equals your significance level. On the hand, you don’t know the probability of the sample value falling in a critical region if the alternative hypothesis is correct (#2). It depends on the precise distribution for the alternative hypothesis and you usually don’t know that, which is why you’re testing the hypotheses in the first place!

I hope I answered the question you were asking. If not, feel free to ask follow up questions. Also, this ties into how to interpret p-values . It’s not exactly straightforward. Click the link to learn more.

June 4, 2020 at 6:17 am

Hi Jim, thank you very much for your answer. You helped me a lot!

June 3, 2020 at 5:23 pm

Hi, Thanks for this post. I’ve been learning a lot with you. My question is regarding to lack of fit. The p-value of my lack of fit is really low, making my lack of fit significant, meaning my model does not fit well. Is my case a “false negative”? given that my pure error is really low, making the computation of the lack of fit low. So it means my model is good. Below I show some information, that I hope helps to clarify my question.

SumSq DF MeanSq F pValue ________ __ ________ ______ __________

Total 1246.5 18 69.25 Model 1241.7 6 206.94 514.43 9.3841e-14 . Linear 1196.6 3 398.87 991.53 1.2318e-14 . Nonlinear 45.046 3 15.015 37.326 2.3092e-06 Residual 4.8274 12 0.40228 . Lack of fit 4.7388 7 0.67698 38.238 0.0004787 . Pure error 0.088521 5 0.017704

June 3, 2020 at 7:53 pm

As you say, a low p-value for a lack of fit test indicates that the model doesn’t fit your data adequately. This is a positive result for the test, which means it can’t be a “false negative.” At best, it could be a false positive, meaning that your data actually fit model well despite the low p-value.

I’d recommend graphing the residuals and looking for patterns . There is probably a relationship between variables that you’re not modeling correctly, such as curvature or interaction effects. There’s no way to diagnose the specific nature of the lack-of-fit problem by using the statistical output. You’ll need the graphs.

If there are no patterns in the residual plots, then your lack-of-fit results might be a false positive.

I hope this helps!

May 30, 2020 at 6:23 am

First of all, I have to say there are not many resources that explain a complicated topic in an easier manner.

My question is, how do we arrive at “if p value is less than alpha, we reject the null hypothesis.”

Is this covered in a separate article I could read?

Thanks Shekhar

May 25, 2020 at 12:21 pm

Hi Jim, terrific website, blog, and after this I’m ordering your book. One of my biggest challenges is nomenclature, definitions, context, and formulating the hypotheses. Here’s one I want to double-be-sure I understand: From above you write: ” These tools allow us to test these two hypotheses:

Null hypothesis: The population mean equals the null hypothesis mean (260). Alternative hypothesis: The population mean does not equal the null hypothesis mean (260). ” I keep thinking that 260 is the population mean mu, the underlying population (that we never really know exactly) and that the Null Hypothesis is comparing mu to x-bar (the sample mean of the 25 families randomly sampled w mean = sample mean = x-bar = 330.6).

So is the following incorrect, and if so, why? Null hypothesis: The population mean mu=260 equals the null hypothesis mean x-bar (330.6). Alternative hypothesis: The population mean mu=269 does not equal the null hypothesis mean x-bar (330.6).

And my thinking is that usually the formulation of null and alternative hypotheses is “test value” = “mu current of underlying population”, whereas I read the formulation on the webpage above to be the reverse.

Any comments appreciated. Many Thanks,

May 26, 2020 at 8:56 pm

The null hypothesis states that population value equals the null value. Now, I know that’s not particularly helpful! But, the null value varies based on test and context. So, in this example, we’re setting the null value aa $260, which was the mean from the previous year. So, our null hypothesis states:

Null: the population mean (mu) = 260. Alternative: the population mean ≠ 260.

These hypothesis statements are about the population parameter. For this type of one-sample analysis, the target or reference value you specify is the null hypothesis value. Additionally, you don’t include the sample estimate in these statements, which is the X-bar portion you tacked on at the end. It’s strictly about the value of the population parameter you’re testing. You don’t know the value of the underlying distribution. However, given the mutually exclusive nature of the null and alternative hypothesis, you know one or the other is correct. The null states that mu equals 260 while the alternative states that it doesn’t equal 260. The data help you decide, which brings us to . . .

However, the procedure does compare our sample data to the null hypothesis value, which is how it determines how strong our evidence is against the null hypothesis.

I hope I answered your question. If not, please let me know!

May 8, 2020 at 6:00 pm

Really using the interpretation “In other words, you will observe sample effects at least as large as 70.6 about 3.1% of the time if the null is true”, our head seems to tie a knot. However, doing the reverse interpretation, it is much more intuitive and easier. That is, we will observe the sample effect of at least 70.6 in about 96.9% of the time, if the null is false (that is, our hypothesis is true).

May 8, 2020 at 7:25 pm

Your phrasing really isn’t any simpler. And it has the additional misfortune of being incorrect.

What you’re essentially doing is creating a one-sided confidence interval by using the p-value from a two-sided test. That’s incorrect in two ways.

• Don’t mix and match one-sided and two-sided test results.
• Confidence levels are determine by the significance level, not p-values.

So, what you need is a two-sided 95% CI (1-alpha). You could then state the results are statistically significant and you have 95% confidence that the population effect is between X and Y. If you want a lower bound as you propose, then you’ll need to use a one-sided hypothesis test with a 95% Lower Bound. That’ll give you a different value for the lower bound than the one you use.

I like confidence intervals. As I write elsewhere, I think they’re easier to understand and provide more information than a binary test result. But, you need to use them correctly!

One other point. When you are talking about p-values, it’s always under the assumption that the null hypothesis is correct. You *never* state anything about the p-value in relation to the null being false (i.e. alternative is true). But, if you want to use the type of phrasing you suggest, use it in the context of CIs and incorporate the points I cover above.

February 10, 2020 at 11:13 am

Muchas gracias profesor por compartir sus conocimientos. Un saliud especial desde Colombia.

August 6, 2019 at 11:46 pm

i found this really helpful . also can you help me out ?

I’m a little confused Can you tell me if level of significance and pvalue are comparable or not and if they are what does it mean if pvalue < LS . Do we reject the null hypothesis or do we accept the null hypothesis ?

August 7, 2019 at 12:49 am

Hi Divyanshu,

Yes, you compare the p-value to the significance level. When the p-value is less than the significance level (alpha), your results are statistically significant and you reject the null hypothesis.

I’d suggest re-reading the “Using P values and Significance Levels Together” section near the end of this post more closely. That describes the process. The next section describes what it all means.

July 1, 2019 at 4:19 am

sure.. I will use only in my class rooms that too offline with due credits to your orginal page. I will encourage my students to visit your blog . I have purchased your eBook on Regressions….immensely useful.

July 1, 2019 at 9:52 am

Hi Narasimha, that sounds perfect. Thanks for buying my ebook as well. I’m thrilled to hear that you’ve found it to be helpful!

June 28, 2019 at 6:22 am

I have benefited a lot by your writings….Can I share the same with my students in the classroom?

June 30, 2019 at 8:44 pm

Hi Narasimha,

Yes, you can certainly share with your students. Please attribute my original page. And please don’t copy whole sections of my posts onto another webpage as that can be bad with Google! Thanks!

February 11, 2019 at 7:46 pm

Hello, great site and my apologies if the answer to the following question exists already.

I’ve always wondered why we put the sampling distribution about the null hypothesis rather than simply leave it about the observed mean. I can see mathematically we are measuring the same distance from the null and basically can draw the same conclusions.

For example we take a sample (say 50 people) we gather an observation (mean wage) estimate the standard error in that observation and so can build a sampling distribution about the observed mean. That sampling distribution contains a confidence interval, where say, i am 95% confident the true mean lies (i.e. in repeated sampling the true mean would reside within this interval 95% of the time).

When i use this for a hyp-test, am i right in saying that we place the sampling dist over the reference level simply because it’s mathematically equivalent and it just seems easier to gauge how far the observation is from 0 via t-stats or its likelihood via p-values?

It seems more natural to me to look at it the other way around. leave the sampling distribution on the observed value, and then look where the null sits…if it’s too far left or right then it is unlikely the true population parameter is what we believed it to be, because if the null were true it would only occur ~ 5% of the time in repeated samples…so perhaps we need to change our opinion.

Can i interpret a hyp-test that way? Or do i have a misconception?

February 12, 2019 at 8:25 pm

The short answer is that, yes, you can draw the interval around the sample mean instead. And, that is, in fact, how you construct confidence intervals. The distance around the null hypothesis for hypothesis tests and the distance around the sample for confidence intervals are the same distance, which is why the results will always agree as long as you use corresponding alpha levels and confidence levels (e.g., alpha 0.05 with a 95% confidence level). I write about how this works in a post about confidence intervals .

I prefer confidence intervals for a number of reasons. They’ll indicate whether you have significant results if they exclude the null value and they indicate the precision of the effect size estimate. Corresponding with what you’re saying, it’s easier to gauge how far a confidence interval is from the null value (often zero) whereas a p-value doesn’t provide that information. See Practical versus Statistical Significance .

So, you don’t have any misconception at all! Just refer to it as a confidence interval rather than a hypothesis test, but, of course, they are very closely related.

January 9, 2019 at 10:37 pm

Hi Jim, Nice Article.. I have a question… I read the Central limit theorem article before this article…

Coming to this article, During almost every hypothesis test, we draw a normal distribution curve assuming there is a sampling distribution (and then we go for test statistic, p value etc…). Do we draw a normal distribution curve for hypo tests because of the central limit theorem…

Thanks in advance, Surya

January 10, 2019 at 1:57 am

These distributions are actually the t-distribution which are different from the normal distribution. T-distributions only have one parameter–the degrees of freedom. As the DF of increases, the t-distribution tightens up. Around 25 degrees of freedom, the t-distribution approximates the normal distribution. Depending on the type of t-test, this corresponds to a sample size of 26 or 27. Similarly, the sampling distribution of the means also approximate the normal distribution at around these sample sizes. With a large enough sample size, both the t-distribution and the sample distribution converge to a normal distribution regardless (largely) of the underlying population distribution. So, yes, the central limit theorem plays a strong role in this.

It’s more accurate to say that central limit theorem causes the sampling distribution of the means to converge on the same distribution that the t-test uses, which allows you to assume that the test produces valid results. But, technically, the t-test is based on the t-distribution.

Problems can occur if the underlying distribution is non-normal and you have a small sample size. In that case, the sampling distribution of the means won’t approximate the t-distribution that the t-test uses. However, the test results will assume that it does and produce results based on that–which is why it causes problems!

November 19, 2018 at 9:15 am

Dear Jim! Thank you very much for your explanation. I need your help to understand my data. I have two samples (about 300 observations) with biased distributions. I did the ttest and obtained the p-value, which is quite small. Can I draw the conclusion that the effect size is small even when the distribution of my data is not normal? Thank you

November 19, 2018 at 9:34 am

Hi Tetyana,

First, when you say that your p-value is small and that you want to “draw the conclusion that the effect size is small,” I assume that you mean statistically significant. When the p-value is low, the null hypothesis must go! In other words, you reject the null and conclude that there is a statistically significant effect–not a small effect.

Now, back to the question at hand! Yes, When you have a sufficiently large sample-size, t-tests are robust to departures from normality. For a 2-sample t-test, you should have at least 15 samples per group, which you exceed by quite a bit. So, yes, you can reliably conclude that your results are statistically significant!

You can thank the central limit theorem! 🙂

September 10, 2018 at 12:18 am

Hello Jim, I am very sorry; I have very elementary of knowledge of stats. So, would you please explain how you got a p- value of 0.03112 in the above calculation/t-test? By looking at a chart? Would you also explain how you got the information that “you will observe sample effects at least as large as 70.6 about 3.1% of the time if the null is true”?

July 6, 2018 at 7:02 am

A quick question regarding your use of two-tailed critical regions in the article above: why? I mean, what is a real-world scenario that would warrant a two-tailed test of any kind (z, t, etc.)? And if there are none, why keep using the two-tailed scenario as an example, instead of the one-tailed which is both more intuitive and applicable to most if not all practical situations. Just curious, as one person attempting to educate people on stats to another (my take on the one vs. two-tailed tests can be seen here: http://blog.analytics-toolkit.com/2017/one-tailed-two-tailed-tests-significance-ab-testing/ )

Thanks, Georgi

July 6, 2018 at 12:05 pm

There’s the appropriate time and place for both one-tailed and two-tailed tests. I plan to write a post on this issue specifically, so I’ll keep my comments here brief.

So much of statistics is context sensitive. People often want concrete rules for how to do things in statistics but that’s often hard to provide because the answer depends on the context, goals, etc. The question of whether to use a one-tailed or two-tailed test falls firmly in this category of it depends.

I did read the article you wrote. I’ll say that I can see how in the context of A/B testing specifically there might be a propensity to use one-tailed tests. You only care about improvements. There’s probably not too much downside in only caring about one direction. In fact, in a post where I compare different tests and different options , I suggest using a one-tailed test for a similar type of casing involving defects. So, I’m onboard with the idea of using one-tailed tests when they’re appropriate. However, I do think that two-tailed tests should be considered the default choice and that you need good reasons to move to a one-tailed test. Again, your A/B testing area might supply those reasons on a regular basis, but I can’t make that a blanket statement for all research areas.

I think your article mischaracterizes some of the pros and cons of both types of tests. Just a couple of for instances. In a two-tailed test, you don’t have to take the same action regardless of which direction the results are significant (example below). And, yes, you can determine the direction of the effect in a two-tailed test. You simply look at the estimated effect. Is it positive or negative?

On the other hand, I do agree that one-tailed tests don’t increase the overall Type I error. However, there is a big caveat for that. In a two-tailed test, the Type I error rate is evenly split in both tails. For a one-tailed test, the overall Type I error rate does not change, but the Type I errors are redistributed so they all occur in the direction that you are interested in rather than being split between the positive and negative directions. In other words, you’ll have twice as many Type I errors in the specific direction that you’re interested in. That’s not good.

My big concerns with one-tailed tests are that it makes it easier to obtain the results that you want to obtain. And, all of the Type I errors (false positives) are in that direction too. It’s just not a good combination.

To answer your question about when you might want to use two-tailed tests, there are plenty of reasons. For one, you might want to avoid the situation I describe above. Additionally, in a lot of scientific research, the researchers truly are interested in detecting effects in either direction for the sake of science. Even in cases with a practical application, you might want to learn about effects in either direction.

For example, I was involved in a research study that looked at the effects of an exercise intervention on bone density. The idea was that it might be a good way to prevent osteoporosis. I used a two-tailed test. Obviously, we’re hoping that there was positive effect. However, we’d be very interested in knowing whether there was a negative effect too. And, this illustrates how you can have different actions based on both directions. If there was a positive effect, you can recommend that as a good approach and try to promote its use. If there’s a negative effect, you’d issue a warning to not do that intervention. You have the potential for learning both what is good and what is bad. The extra false-positives would’ve cause problems because we’d think that there’d be health benefits for participants when those benefits don’t actually exist. Also, if we had performed only a one-tailed test and didn’t obtain significant results, we’d learn that it wasn’t a positive effect, but we would not know whether it was actually detrimental or not.

Here’s when I’d say it’s OK to use a one-tailed test. Consider a one-tailed test when you’re in situation where you truly only need to know whether an effect exists in one direction, and the extra Type I errors in that direction are an acceptable risk (false positives don’t cause problems), and there’s no benefit in determining whether an effect exists in the other direction. Those conditions really restrict when one-tailed tests are the best choice. Again, those restrictions might not be relevant for your specific field, but as for the usage of statistics as a whole, they’re absolutely crucial to consider.

On the other hand, according to this article, two-tailed tests might be important in A/B testing !

March 30, 2018 at 5:29 am

Dear Sir, please confirm if there is an inadvertent mistake in interpretation as, “We can conclude that mean fuel expenditures have increased since last year.” Our null hypothesis is =260. If found significant, it implies two possibilities – both increase and decrease. Please let us know if we are mistaken here. Many Thanks!

March 30, 2018 at 9:59 am

Hi Khalid, the null hypothesis as it is defined for this test represents the mean monthly expenditure for the previous year (260). The mean expenditure for the current year is 330.6 whereas it was 260 for the previous year. Consequently, the mean has increased from 260 to 330.7 over the course of a year. The p-value indicates that this increase is statistically significant. This finding does not suggest both an increase and a decrease–just an increase. Keep in mind that a significant result prompts us to reject the null hypothesis. So, we reject the null that the mean equals 260.

Let’s explore the other possible findings to be sure that this makes sense. Suppose the sample mean had been closer to 260 and the p-value was greater than the significance level, those results would indicate that the results were not statistically significant. The conclusion that we’d draw is that we have insufficient evidence to conclude that mean fuel expenditures have changed since the previous year.

If the sample mean was less than the null hypothesis (260) and if the p-value is statistically significant, we’d concluded that mean fuel expenditures have decreased and that this decrease is statistically significant.

When you interpret the results, you have to be sure to understand what the null hypothesis represents. In this case, it represents the mean monthly expenditure for the previous year and we’re comparing this year’s mean to it–hence our sample suggests an increase.

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Hypothesis Testing: Principles and Methods

Learn about hypothesis testing. The types of tests, common errors, best practices, and more. Perfect for all researchers.

Hypothesis testing is a fundamental tool used in scientific research to validate or reject hypotheses about population parameters based on sample data. It provides a structured framework for evaluating the statistical significance of a hypothesis and drawing conclusions about the true nature of a population. Hypothesis testing is widely used in fields such as biology, psychology, economics, and engineering to determine the effectiveness of new treatments, explore relationships between variables , and make data-driven decisions. However, despite its importance , hypothesis testing can be a challenging topic to understand and apply correctly.

In this article, we will provide an introduction to hypothesis testing, including its purpose, types of tests, steps involved, common errors, and best practices. Whether you are a beginner or an experienced researcher, this article will serve as a valuable guide to mastering hypothesis testing in your work.

Introduction to Hypothesis Testing

Hypothesis testing is a statistical tool that is commonly used in research to determine whether there is enough evidence to support or reject a hypothesis. It involves formulating a hypothesis about a population parameter, collecting data, and analyzing the data to determine the likelihood of the hypothesis being true. It is a critical component of the scientific method, and it is used in a wide range of fields.

The process of hypothesis testing typically involves two hypotheses: the null hypothesis and the alternative hypothesis. The null hypothesis is a statement that there is no significant difference between two variables or no relationship between them, while the alternative hypothesis suggests the presence of a relationship or difference. Researchers collect data and perform statistical analysis to determine if the null hypothesis can be rejected in favor of the alternative hypothesis.

Hypothesis testing is used to make decisions based on data, and it is important to understand the underlying assumptions and limitations of the process. It is crucial to choose appropriate statistical tests and sample sizes to ensure that the results are accurate and reliable, and it can be a powerful tool for researchers to validate their theories and make evidence-based decisions.

Types of Hypothesis Tests

Hypothesis testing can be broadly classified into two categories: one-sample hypothesis tests and two-sample hypothesis tests. Let’s take a closer look at each of these categories:

One Sample Hypothesis Tests

In a one-sample hypothesis test, a researcher collects data from a single population and compares it to a known value or hypothesis. The null hypothesis usually assumes that there is no significant difference between the population means and the known value or hypothesized value. The researcher then performs a statistical test to determine whether the observed difference is statistically significant. Some examples of one-sample hypothesis tests are:

One Sample t-test: This test is used to determine whether the sample mean is significantly different from the hypothesized mean of the population.

One Sample z-test: This test is used to determine whether the sample mean is significantly different from the hypothesized mean of the population when the population standard deviation is known.

Two Sample Hypothesis Tests

In a two-sample hypothesis test, a researcher collects data from two different populations and compares them to each other. The null hypothesis typically assumes that there is no significant difference between the two populations, and the researcher performs a statistical test to determine whether the observed difference is statistically significant. Some examples of two sample hypothesis tests are:

Independent Samples t-test: This test is used to compare the means of two independent samples to determine whether they are significantly different from each other.

Paired Samples t-test: This test is used to compare the means of two related samples, such as pre-test and post-test scores of the same group of subjects.

Figure: https://statstest.b-cdn.net/wp-content/uploads/2020/10/Paired-Samples-T-Test.jpg

In summary, one-sample hypothesis tests are used to test hypotheses about a single population, while two-sample hypothesis tests are used to compare two populations. The appropriate test to use depends on the nature of the data and the research question being investigated.

Steps of Hypothesis Testing

Hypothesis testing involves a series of steps that help researchers determine whether there is enough evidence to support or reject a hypothesis. These steps can be broadly classified into four categories:

Formulating the Hypothesis

The first step in hypothesis testing is to formulate the null hypothesis and alternative hypothesis. The null hypothesis usually assumes that there is no significant difference between two variables, while the alternative hypothesis suggests the presence of a relationship or difference. It is important to formulate clear and testable hypotheses before proceeding with data collection.

Collecting Data

The second step is to collect relevant data that can be used to test the hypotheses. The data collection process should be carefully designed to ensure that the sample is representative of the population of interest. The sample size should be large enough to produce statistically valid results.

Analyzing Data

The third step is to analyze the data using appropriate statistical tests. The choice of test depends on the nature of the data and the research question being investigated. The results of the statistical analysis will provide information on whether the null hypothesis can be rejected in favor of the alternative hypothesis.

Interpreting Results

The final step is to interpret the results of the statistical analysis. The researcher needs to determine whether the results are statistically significant and whether they support or reject the hypothesis. The researcher should also consider the limitations of the study and the potential implications of the results.

Common Errors in Hypothesis Testing

Hypothesis testing is a statistical method used to determine if there is enough evidence to support or reject a specific hypothesis about a population parameter based on a sample of data. The two types of errors that can occur in hypothesis testing are:

Type I error: This occurs when the researcher rejects the null hypothesis even though it is true. Type I error is also known as a false positive.

Type II error: This occurs when the researcher fails to reject the null hypothesis even though it is false. Type II error is also known as a false negative.

To minimize these errors, it is important to carefully design and conduct the study, choose appropriate statistical tests, and properly interpret the results. Researchers should also acknowledge the limitations of their study and consider the potential sources of error when drawing conclusions.

Null and Alternative Hypotheses

In hypothesis testing, there are two types of hypotheses: null hypothesis and alternative hypothesis.

The Null Hypothesis

The null hypothesis (H0) is a statement that assumes there is no significant difference or relationship between two variables. It is the default hypothesis that is assumed to be true until there is sufficient evidence to reject it. The null hypothesis is often written as a statement of equality, such as “the mean of Group A is equal to the mean of Group B.”

The Alternative Hypothesis

The alternative hypothesis (Ha) is a statement that suggests the presence of a significant difference or relationship between two variables. It is the hypothesis that the researcher is interested in testing. The alternative hypothesis is often written as a statement of inequality, such as “the mean of Group A is not equal to the mean of Group B.”

The null and alternative hypotheses are complementary and mutually exclusive. If the null hypothesis is rejected, the alternative hypothesis is accepted. If the null hypothesis cannot be rejected, the alternative hypothesis is not supported.

It is important to note that the null hypothesis is not necessarily true. It is simply a statement that assumes there is no significant difference or relationship between the variables being studied. The purpose of hypothesis testing is to determine whether there is sufficient evidence to reject the null hypothesis in favor of the alternative hypothesis.

Significance Level and P Value

In hypothesis testing, the significance level (alpha) is the probability of making a Type I error, which is rejecting the null hypothesis when it is actually true. The most commonly used significance level in scientific research is 0.05, meaning that there is a 5% chance of making a Type I error.

The p-value is a statistical measure that indicates the probability of obtaining the observed results or more extreme results if the null hypothesis is true. It is a measure of the strength of evidence against the null hypothesis. A small p-value (typically less than the chosen significance level of 0.05) suggests that there is strong evidence against the null hypothesis, while a large p-value suggests that there is not enough evidence to reject the null hypothesis.

If the p-value is less than the significance level (p < alpha), then the null hypothesis is rejected and the alternative hypothesis is accepted. This means that there is sufficient evidence to suggest that there is a significant difference or relationship between the variables being studied. On the other hand, if the p-value is greater than the significance level (p > alpha), then the null hypothesis is not rejected and the alternative hypothesis is not supported.

If you want an easy-to-understand summary of the significance level, you will find it in this article: An easy-to-understand summary of significance level .

It is important to note that statistical significance does not necessarily imply practical significance or importance. A small difference or relationship between variables may be statistically significant but may not be practically significant. Additionally, statistical significance depends on sample size and effect size, among other factors, and should be interpreted in the context of the study design and research question.

Power Analysis for Hypothesis Testing

Power analysis is a statistical method used in hypothesis testing to determine the sample size needed to detect a specific effect size with a certain level of confidence. The power of a statistical test is the probability of correctly rejecting the null hypothesis when it is false or the probability of avoiding a Type II error.

Power analysis is important because it helps researchers determine the appropriate sample size needed to achieve a desired level of power. A study with low power may fail to detect a true effect, leading to a Type II error, while a study with high power is more likely to detect a true effect, leading to more accurate and reliable results.

To conduct a power analysis, researchers need to specify the desired power level, significance level, effect size, and sample size. Effect size is a measure of the magnitude of the difference or relationship between variables being studied, and is typically estimated from previous research or pilot studies. The power analysis can then determine the necessary sample size needed to achieve the desired power level.

Power analysis can also be used retrospectively to determine the power of a completed study, based on the sample size, effect size, and significance level. This can help researchers evaluate the strength of their conclusions and determine whether additional research is needed.

Overall, power analysis is an important tool in hypothesis testing, as it helps researchers design studies that are adequately powered to detect true effects and avoid Type II errors

Bayesian Hypothesis Testing

Bayesian hypothesis testing is a statistical method that allows researchers to evaluate the evidence for and against competing hypotheses, based on the likelihood of the observed data under each hypothesis, as well as the prior probability of each hypothesis. Unlike classical hypothesis testing, which focuses on rejecting null hypotheses based on p-values, Bayesian hypothesis testing provides a more nuanced and informative approach to hypothesis testing, by allowing researchers to quantify the strength of evidence for and against each hypothesis.

In Bayesian hypothesis testing, researchers start with a prior probability distribution for each hypothesis, based on existing knowledge or beliefs. They then update the prior probability distribution based on the likelihood of the observed data under each hypothesis, using Bayes’ theorem. The resulting posterior probability distribution represents the probability of each hypothesis, given the observed data.

The strength of evidence for one hypothesis versus another can be quantified by calculating the Bayes factor, which is the ratio of the likelihood of the observed data under one hypothesis versus another, weighted by their prior probabilities. A Bayes factor greater than 1 indicates evidence in favor of one hypothesis, while a Bayes factor less than 1 indicates evidence in favor of the other hypothesis.

Bayesian hypothesis testing has several advantages over classical hypothesis testing. First, it allows researchers to update their prior beliefs based on observed data, which can lead to more accurate and reliable conclusions. Second, it provides a more informative measure of evidence than p-values, which only indicate whether the observed data is statistically significant at a predetermined level. Finally, it can accommodate complex models with multiple parameters and hypotheses, which may be difficult to analyze using classical methods.

Overall, Bayesian hypothesis testing is a powerful and flexible statistical method that can help researchers make more informed decisions and draw more accurate conclusions from their data.

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Guide: Hypothesis Testing

Author: Daniel Croft

Daniel Croft is an experienced continuous improvement manager with a Lean Six Sigma Black Belt and a Bachelor's degree in Business Management. With more than ten years of experience applying his skills across various industries, Daniel specializes in optimizing processes and improving efficiency. His approach combines practical experience with a deep understanding of business fundamentals to drive meaningful change.

In the world of data-driven decision-making, Hypothesis Testing stands as a cornerstone methodology. It serves as the statistical backbone for a multitude of sectors, from manufacturing and logistics to healthcare and finance. But what exactly is Hypothesis Testing, and why is it so indispensable? Simply put, it’s a technique that allows you to validate or invalidate claims about a population based on sample data. Whether you’re looking to streamline a manufacturing process, optimize logistics, or improve customer satisfaction, Hypothesis Testing offers a structured approach to reach conclusive, data-supported decisions.

The graphical example above provides a simplified snapshot of a hypothesis test. The bell curve represents a normal distribution, the green area is where you’d accept the null hypothesis ( H 0​), and the red area is the “rejection zone,” where you’d favor the alternative hypothesis ( Ha ​). The vertical blue line represents the threshold value or “critical value,” beyond which you’d reject H 0​.

Here’s a graphical example of a hypothesis test, which you can include in the introduction section of your guide. In this graph:

• The curve represents a standard normal distribution, often encountered in hypothesis tests.
• The green-shaded area signifies the “Acceptance Region,” where you would fail to reject the null hypothesis ( H 0​).
• The red-shaded areas are the “Rejection Regions,” where you would reject H 0​ in favor of the alternative hypothesis ( Ha ​).
• The blue dashed lines indicate the “Critical Values” (±1.96), which are the thresholds for rejecting H 0​.

This graphical representation serves as a conceptual foundation for understanding the mechanics of hypothesis testing. It visually illustrates what it means to accept or reject a hypothesis based on a predefined level of significance.

What is Hypothesis Testing?

Hypothesis testing is a structured procedure in statistics used for drawing conclusions about a larger population based on a subset of that population, known as a sample. The method is widely used across different industries and sectors for a variety of purposes. Below, we’ll dissect the key components of hypothesis testing to provide a more in-depth understanding.

The Hypotheses: H 0 and Ha

In every hypothesis test, there are two competing statements:

• Null Hypothesis ( H 0) : This is the “status quo” hypothesis that you are trying to test against. It is a statement that asserts that there is no effect or difference. For example, in a manufacturing setting, the null hypothesis might state that a new production process does not improve the average output quality.
• Alternative Hypothesis ( Ha or H 1) : This is what you aim to prove by conducting the hypothesis test. It is the statement that there is an effect or difference. Using the same manufacturing example, the alternative hypothesis might state that the new process does improve the average output quality.

Significance Level ( α )

Before conducting the test, you decide on a “Significance Level” ( α ), typically set at 0.05 or 5%. This level represents the probability of rejecting the null hypothesis when it is actually true. Lower α values make the test more stringent, reducing the chances of a ‘false positive’.

Data Collection

You then proceed to gather data, which is usually a sample from a larger population. The quality of your test heavily relies on how well this sample represents the population. The data can be collected through various means such as surveys, observations, or experiments.

Statistical Test

Depending on the nature of the data and what you’re trying to prove, different statistical tests can be applied (e.g., t-test, chi-square test , ANOVA , etc.). These tests will compute a test statistic (e.g., t , 2 χ 2, F , etc.) based on your sample data.

Here are graphical examples of the distributions commonly used in three different types of statistical tests: t-test, Chi-square test, and ANOVA (Analysis of Variance), displayed side by side for comparison.

• Graph 1 (Leftmost): This graph represents a t-distribution, often used in t-tests. The t-distribution is similar to the normal distribution but tends to have heavier tails. It is commonly used when the sample size is small or the population variance is unknown.

Chi-square Test

• Graph 2 (Middle): The Chi-square distribution is used in Chi-square tests, often for testing independence or goodness-of-fit. Unlike the t-distribution, the Chi-square distribution is not symmetrical and only takes on positive values.

ANOVA (F-distribution)

• Graph 3 (Rightmost): The F-distribution is used in Analysis of Variance (ANOVA), a statistical test used to analyze the differences between group means. Like the Chi-square distribution, the F-distribution is also not symmetrical and takes only positive values.

These visual representations provide an intuitive understanding of the different statistical tests and their underlying distributions. Knowing which test to use and when is crucial for conducting accurate and meaningful hypothesis tests.

Decision Making

The test statistic is then compared to a critical value determined by the significance level ( α ) and the sample size. This comparison will give you a p-value. If the p-value is less than α , you reject the null hypothesis in favor of the alternative hypothesis. Otherwise, you fail to reject the null hypothesis.

Interpretation

Finally, you interpret the results in the context of what you were investigating. Rejecting the null hypothesis might mean implementing a new process or strategy, while failing to reject it might lead to a continuation of current practices.

To sum it up, hypothesis testing is not just a set of formulas but a methodical approach to problem-solving and decision-making based on data. It’s a crucial tool for anyone interested in deriving meaningful insights from data to make informed decisions.

Why is Hypothesis Testing Important?

Hypothesis testing is a cornerstone of statistical and empirical research, serving multiple functions in various fields. Let’s delve into each of the key areas where hypothesis testing holds significant importance:

Data-Driven Decisions

In today’s complex business environment, making decisions based on gut feeling or intuition is not enough; you need data to back up your choices. Hypothesis testing serves as a rigorous methodology for making decisions based on data. By setting up a null hypothesis and an alternative hypothesis, you can use statistical methods to determine which is more likely to be true given a data sample. This structured approach eliminates guesswork and adds empirical weight to your decisions, thereby increasing their credibility and effectiveness.

Risk Management

Hypothesis testing allows you to assign a ‘p-value’ to your findings, which is essentially the probability of observing the given sample data if the null hypothesis is true. This p-value can be directly used to quantify risk. For instance, a p-value of 0.05 implies there’s a 5% risk of rejecting the null hypothesis when it’s actually true. This is invaluable in scenarios like product launches or changes in operational processes, where understanding the risk involved can be as crucial as the decision itself.

Here’s an example to help you understand the concept better.

The graph above serves as a graphical representation to help explain the concept of a ‘p-value’ and its role in quantifying risk in hypothesis testing. Here’s how to interpret the graph:

Elements of the Graph

• The curve represents a Standard Normal Distribution , which is often used to represent z-scores in hypothesis testing.
• The red-shaded area on the right represents the Rejection Region . It corresponds to a 5% risk ( α =0.05) of rejecting the null hypothesis when it is actually true. This is the area where, if your test statistic falls, you would reject the null hypothesis.
• The green-shaded area represents the Acceptance Region , with a 95% level of confidence. If your test statistic falls in this region, you would fail to reject the null hypothesis.
• The blue dashed line is the Critical Value (approximately 1.645 in this example). If your standardized test statistic (z-value) exceeds this point, you enter the rejection region, and your p-value becomes less than 0.05, leading you to reject the null hypothesis.

Relating to Risk Management

The p-value can be directly related to risk management. For example, if you’re considering implementing a new manufacturing process, the p-value quantifies the risk of that decision. A low p-value (less than α ) would mean that the risk of rejecting the null hypothesis (i.e., going ahead with the new process) when it’s actually true is low, thus indicating a lower risk in implementing the change.

Quality Control

In sectors like manufacturing, automotive, and logistics, maintaining a high level of quality is not just an option but a necessity. Hypothesis testing is often employed in quality assurance and control processes to test whether a certain process or product conforms to standards. For example, if a car manufacturing line claims its error rate is below 5%, hypothesis testing can confirm or disprove this claim based on a sample of products. This ensures that quality is not compromised and that stakeholders can trust the end product.

Resource Optimization

Resource allocation is a significant challenge for any organization. Hypothesis testing can be a valuable tool in determining where resources will be most effectively utilized. For instance, in a manufacturing setting, you might want to test whether a new piece of machinery significantly increases production speed. A hypothesis test could provide the statistical evidence needed to decide whether investing in more of such machinery would be a wise use of resources.

In the realm of research and development, hypothesis testing can be a game-changer. When developing a new product or process, you’ll likely have various theories or hypotheses. Hypothesis testing allows you to systematically test these, filtering out the less likely options and focusing on the most promising ones. This not only speeds up the innovation process but also makes it more cost-effective by reducing the likelihood of investing in ideas that are statistically unlikely to be successful.

In summary, hypothesis testing is a versatile tool that adds rigor, reduces risk, and enhances the decision-making and innovation processes across various sectors and functions.

This graphical representation makes it easier to grasp how the p-value is used to quantify the risk involved in making a decision based on a hypothesis test.

Step-by-Step Guide to Hypothesis Testing

To make this guide practical and helpful if you are new learning about the concept we will explain each step of the process and follow it up with an example of the method being applied to a manufacturing line, and you want to test if a new process reduces the average time it takes to assemble a product.

Step 1: State the Hypotheses

The first and foremost step in hypothesis testing is to clearly define your hypotheses. This sets the stage for your entire test and guides the subsequent steps, from data collection to decision-making. At this stage, you formulate two competing hypotheses:

Null Hypothesis ( H 0)

The null hypothesis is a statement that there is no effect or no difference, and it serves as the hypothesis that you are trying to test against. It’s the default assumption that any kind of effect or difference you suspect is not real, and is due to chance. Formulating a clear null hypothesis is crucial, as your statistical tests will be aimed at challenging this hypothesis.

In a manufacturing context, if you’re testing whether a new assembly line process has reduced the time it takes to produce an item, your null hypothesis ( H 0) could be:

H 0:”The new process does not reduce the average assembly time.”

Alternative Hypothesis ( Ha or H 1)

The alternative hypothesis is what you want to prove. It is a statement that there is an effect or difference. This hypothesis is considered only after you find enough evidence against the null hypothesis.

Continuing with the manufacturing example, the alternative hypothesis ( Ha ) could be:

Ha :”The new process reduces the average assembly time.”

Types of Alternative Hypothesis

Depending on what exactly you are trying to prove, the alternative hypothesis can be:

• Two-Sided : You’re interested in deviations in either direction (greater or smaller).
• One-Sided : You’re interested in deviations only in one direction (either greater or smaller).

Scenario: Reducing Assembly Time in a Car Manufacturing Plant

You are a continuous improvement manager at a car manufacturing plant. One of the assembly lines has been struggling with longer assembly times, affecting the overall production schedule. A new assembly process has been proposed, promising to reduce the assembly time per car. Before rolling it out on the entire line, you decide to conduct a hypothesis test to see if the new process actually makes a difference. Null Hypothesis ( H 0​) In this context, the null hypothesis would be the status quo, asserting that the new assembly process doesn’t reduce the assembly time per car. Mathematically, you could state it as: H 0:The average assembly time per car with the new process ≥ The average assembly time per car with the old process. Or simply: H 0​:”The new process does not reduce the average assembly time per car.” Alternative Hypothesis ( Ha ​ or H 1​) The alternative hypothesis is what you aim to prove — that the new process is more efficient. Mathematically, it could be stated as: Ha :The average assembly time per car with the new process < The average assembly time per car with the old process Or simply: Ha ​:”The new process reduces the average assembly time per car.” Types of Alternative Hypothesis In this example, you’re only interested in knowing if the new process reduces the time, making it a One-Sided Alternative Hypothesis .

Step 2: Determine the Significance Level ( α )

Once you’ve clearly stated your null and alternative hypotheses, the next step is to decide on the significance level, often denoted by α . The significance level is a threshold below which the null hypothesis will be rejected. It quantifies the level of risk you’re willing to accept when making a decision based on the hypothesis test.

What is a Significance Level?

The significance level, usually expressed as a percentage, represents the probability of rejecting the null hypothesis when it is actually true. Common choices for α are 0.05, 0.01, and 0.10, representing 5%, 1%, and 10% levels of significance, respectively.

• 5% Significance Level ( α =0.05) : This is the most commonly used level and implies that you are willing to accept a 5% chance of rejecting the null hypothesis when it is true.
• 1% Significance Level ( α =0.01) : This is a more stringent level, used when you want to be more sure of your decision. The risk of falsely rejecting the null hypothesis is reduced to 1%.
• 10% Significance Level ( α =0.10) : This is a more lenient level, used when you are willing to take a higher risk. Here, the chance of falsely rejecting the null hypothesis is 10%.

Continuing with the manufacturing example, let’s say you decide to set α =0.05, meaning you’re willing to take a 5% risk of concluding that the new process is effective when it might not be.

How to Choose the Right Significance Level?

Choosing the right significance level depends on the context and the consequences of making a wrong decision. Here are some factors to consider:

• Criticality of Decision : For highly critical decisions with severe consequences if wrong, a lower α like 0.01 may be appropriate.
• Resource Constraints : If the cost of collecting more data is high, you may choose a higher α to make a decision based on a smaller sample size.
• Industry Standards : Sometimes, the choice of α may be dictated by industry norms or regulatory guidelines.

By the end of Step 2, you should have a well-defined significance level that will guide the rest of your hypothesis testing process. This level serves as the cut-off for determining whether the observed effect or difference in your sample is statistically significant or not.

Continuing the Scenario: Reducing Assembly Time in a Car Manufacturing Plant

After formulating the hypotheses, the next step is to set the significance level ( α ) that will be used to interpret the results of the hypothesis test. This is a critical decision as it quantifies the level of risk you’re willing to accept when making a conclusion based on the test. Setting the Significance Level Given that assembly time is a critical factor affecting the production schedule, and ultimately, the company’s bottom line, you decide to be fairly stringent in your test. You opt for a commonly used significance level: α = 0.05 This means you are willing to accept a 5% chance of rejecting the null hypothesis when it is actually true. In practical terms, if you find that the p-value of the test is less than 0.05, you will conclude that the new process significantly reduces assembly time and consider implementing it across the entire line. Why α = 0.05 ? Industry Standard : A 5% significance level is widely accepted in many industries, including manufacturing, for hypothesis testing. Risk Management : By setting  α = 0.05 , you’re limiting the risk of concluding that the new process is effective when it may not be to just 5%. Balanced Approach : This level offers a balance between being too lenient (e.g., α=0.10) and too stringent (e.g., α=0.01), making it a reasonable choice for this scenario.

Step 3: Collect and Prepare the Data

After stating your hypotheses and setting the significance level, the next vital step is data collection. The data you collect serves as the basis for your hypothesis test, so it’s essential to gather accurate and relevant data.

Types of Data

Depending on your hypothesis, you’ll need to collect either:

• Quantitative Data : Numerical data that can be measured. Examples include height, weight, and temperature.
• Qualitative Data : Categorical data that represent characteristics. Examples include colors, gender, and material types.

Data Collection Methods

Various methods can be used to collect data, such as:

• Surveys and Questionnaires : Useful for collecting qualitative data and opinions.
• Observation : Collecting data through direct or participant observation.
• Experiments : Especially useful in scientific research where control over variables is possible.
• Existing Data : Utilizing databases, records, or any other data previously collected.

Sample Size

The sample size ( n ) is another crucial factor. A larger sample size generally gives more accurate results, but it’s often constrained by resources like time and money. The choice of sample size might also depend on the statistical test you plan to use.

Continuing with the manufacturing example, suppose you decide to collect data on the assembly time of 30 randomly chosen products, 15 made using the old process and 15 made using the new process. Here, your sample size n =30.

Data Preparation

Once data is collected, it often needs to be cleaned and prepared for analysis. This could involve:

• Removing Outliers : Outliers can skew the results and provide an inaccurate picture.
• Data Transformation : Converting data into a format suitable for statistical analysis.
• Data Coding : Categorizing or labeling data, necessary for qualitative data.

By the end of Step 3, you should have a dataset that is ready for statistical analysis. This dataset should be representative of the population you’re interested in and prepared in a way that makes it suitable for hypothesis testing.

With the hypotheses stated and the significance level set, you’re now ready to collect the data that will serve as the foundation for your hypothesis test. Given that you’re testing a change in a manufacturing process, the data will most likely be quantitative, representing the assembly time of cars produced on the line. Data Collection Plan You decide to use a Random Sampling Method for your data collection. For two weeks, assembly times for randomly selected cars will be recorded: one week using the old process and another week using the new process. Your aim is to collect data for 40 cars from each process, giving you a sample size ( n ) of 80 cars in total. Types of Data Quantitative Data : In this case, you’re collecting numerical data representing the assembly time in minutes for each car. Data Preparation Data Cleaning : Once the data is collected, you’ll need to inspect it for any anomalies or outliers that could skew your results. For example, if a significant machine breakdown happened during one of the weeks, you may need to adjust your data or collect more. Data Transformation : Given that you’re dealing with time, you may not need to transform your data, but it’s something to consider, depending on the statistical test you plan to use. Data Coding : Since you’re dealing with quantitative data in this scenario, coding is likely unnecessary unless you’re planning to categorize assembly times into bins (e.g., ‘fast’, ‘medium’, ‘slow’) for some reason. Example Data Points: Car_ID Process_Type Assembly_Time_Minutes 1 Old 38.53 2 Old 35.80 3 Old 36.96 4 Old 39.48 5 Old 38.74 6 Old 33.05 7 Old 36.90 8 Old 34.70 9 Old 34.79 … … … The complete dataset would contain 80 rows: 40 for the old process and 40 for the new process.

Step 4: Conduct the Statistical Test

After you have your hypotheses, significance level, and collected data, the next step is to actually perform the statistical test. This step involves calculations that will lead to a test statistic, which you’ll then use to make your decision regarding the null hypothesis.

Choose the Right Test

The first task is to decide which statistical test to use. The choice depends on several factors:

• Type of Data : Quantitative or Qualitative
• Sample Size : Large or Small
• Number of Groups or Categories : One-sample, Two-sample, or Multiple groups

For instance, you might choose a t-test for comparing means of two groups when you have a small sample size. Chi-square tests are often used for categorical data, and ANOVA is used for comparing means across more than two groups.

Calculation of Test Statistic

Once you’ve chosen the appropriate statistical test, the next step is to calculate the test statistic. This involves using the sample data in a specific formula for the chosen test.

Obtain the p-value

After calculating the test statistic, the next step is to find the p-value associated with it. The p-value represents the probability of observing the given test statistic if the null hypothesis is true.

• A small p-value (< α ) indicates strong evidence against the null hypothesis, so you reject the null hypothesis.
• A large p-value (> α ) indicates weak evidence against the null hypothesis, so you fail to reject the null hypothesis.

Make the Decision

You now compare the p-value to the predetermined significance level ( α ):

• If p < α , you reject the null hypothesis in favor of the alternative hypothesis.
• If p > α , you fail to reject the null hypothesis.

In the manufacturing case, if your calculated p-value is 0.03 and your α is 0.05, you would reject the null hypothesis, concluding that the new process effectively reduces the average assembly time.

By the end of Step 4, you will have either rejected or failed to reject the null hypothesis, providing a statistical basis for your decision-making process.

Now that you have collected and prepared your data, the next step is to conduct the actual statistical test to evaluate the null and alternative hypotheses. In this case, you’ll be comparing the mean assembly times between cars produced using the old and new processes to determine if the new process is statistically significantly faster. Choosing the Right Test Given that you have two sets of independent samples (old process and new process), a Two-sample t-test for Equality of Means seems appropriate for comparing the average assembly times. Preparing Data for Minitab Firstly, you would prepare your data in an Excel sheet or CSV file with one column for the assembly times using the old process and another column for the assembly times using the new process. Import this file into Minitab. Steps to Perform the Two-sample t-test in Minitab Open Minitab : Launch the Minitab software on your computer. Import Data : Navigate to File > Open and import your data file. Navigate to the t-test Menu : Go to Stat > Basic Statistics > 2-Sample t... . Select Columns : In the dialog box, specify the columns corresponding to the old and new process assembly times under “Sample 1” and “Sample 2.” Options : Click on Options and make sure that you set the confidence level to 95% (which corresponds to α = 0.05 ). Run the Test : Click OK to run the test. In this example output, the p-value is 0.0012, which is less than the significance level α = 0.05 . Hence, you would reject the null hypothesis. The t-statistic is -3.45, indicating that the mean of the new process is statistically significantly less than the mean of the old process, which aligns with your alternative hypothesis. Showing the data displayed as a Box plot in the below graphic it is easy to see the new process is statistically significantly better.

Why do a Hypothesis test?

You might ask, after all this why do a hypothesis test and not just look at the averages, which is a good question. While looking at average times might give you a general idea of which process is faster, hypothesis testing provides several advantages that a simple comparison of averages doesn’t offer:

Statistical Significance

Account for Random Variability : Hypothesis testing considers not just the averages, but also the variability within each group. This allows you to make more robust conclusions that account for random chance.

Quantify the Evidence : With hypothesis testing, you obtain a p-value that quantifies the strength of the evidence against the null hypothesis. A simple comparison of averages doesn’t provide this level of detail.

Control Type I Error : Hypothesis testing allows you to control the probability of making a Type I error (i.e., rejecting a true null hypothesis). This is particularly useful in settings where the consequences of such an error could be costly or risky.

Quantify Risk : Hypothesis testing provides a structured way to make decisions based on a predefined level of risk (the significance level, α ).

Decision-making Confidence

Objective Decision Making : The formal structure of hypothesis testing provides an objective framework for decision-making. This is especially useful in a business setting where decisions often have to be justified to stakeholders.

Replicability : The statistical rigor ensures that the results are replicable. Another team could perform the same test and expect to get similar results, which is not necessarily the case when comparing only averages.

Additional Insights

Understanding of Variability : Hypothesis testing often involves looking at measures of spread and distribution, not just the mean. This can offer additional insights into the processes you’re comparing.

Basis for Further Analysis : Once you’ve performed a hypothesis test, you can often follow it up with other analyses (like confidence intervals for the difference in means, or effect size calculations) that offer more detailed information.

In summary, while comparing averages is quicker and simpler, hypothesis testing provides a more reliable, nuanced, and objective basis for making data-driven decisions.

Step 5: Interpret the Results and Make Conclusions

Having conducted the statistical test and obtained the p-value, you’re now at a stage where you can interpret these results in the context of the problem you’re investigating. This step is crucial for transforming the statistical findings into actionable insights.

Interpret the p-value

The p-value you obtained tells you the significance of your results:

• Low p-value ( p < α ) : Indicates that the results are statistically significant, and it’s unlikely that the observed effects are due to random chance. In this case, you generally reject the null hypothesis.
• High p-value ( p > α ) : Indicates that the results are not statistically significant, and the observed effects could well be due to random chance. Here, you generally fail to reject the null hypothesis.

Relate to Real-world Context

You should then relate these statistical conclusions to the real-world context of your problem. This is where your expertise in your specific field comes into play.

In our manufacturing example, if you’ve found a statistically significant reduction in assembly time with a p-value of 0.03 (which is less than the α level of 0.05), you can confidently conclude that the new manufacturing process is more efficient. You might then consider implementing this new process across the entire assembly line.

Make Recommendations

Based on your conclusions, you can make recommendations for action or further study. For example:

• Implement Changes : If the test results are significant, consider making the changes on a larger scale.
• Further Research : If the test results are not clear or not significant, you may recommend further studies or data collection.
• Review Methodology : If you find that the results are not as expected, it might be useful to review the methodology and see if the test was conducted under the right conditions and with the right test parameters.

Document the Findings

Lastly, it’s essential to document all the steps taken, the methodology used, the data collected, and the conclusions drawn. This documentation is not only useful for any further studies but also for auditing purposes or for stakeholders who may need to understand the process and the findings.

By the end of Step 5, you’ll have turned the raw statistical findings into meaningful conclusions and actionable insights. This is the final step in the hypothesis testing process, making it a complete, robust method for informed decision-making.

You’ve successfully conducted the hypothesis test and found strong evidence to reject the null hypothesis in favor of the alternative: The new assembly process is statistically significantly faster than the old one. It’s now time to interpret these results in the context of your business operations and make actionable recommendations. Interpretation of Results Statistical Significance : The p-value of 0.0012 is well below the significance level of = 0.05   α = 0.05 , indicating that the results are statistically significant. Practical Significance : The boxplot and t-statistic (-3.45) suggest not just statistical, but also practical significance. The new process appears to be both consistently and substantially faster. Risk Assessment : The low p-value allows you to reject the null hypothesis with a high degree of confidence, meaning the risk of making a Type I error is minimal. Business Implications Increased Productivity : Implementing the new process could lead to an increase in the number of cars produced, thereby enhancing productivity. Cost Savings : Faster assembly time likely translates to lower labor costs. Quality Control : Consider monitoring the quality of cars produced under the new process closely to ensure that the speedier assembly does not compromise quality. Recommendations Implement New Process : Given the statistical and practical significance of the findings, recommend implementing the new process across the entire assembly line. Monitor and Adjust : Implement a control phase where the new process is monitored for both speed and quality. This could involve additional hypothesis tests or control charts. Communicate Findings : Share the results and recommendations with stakeholders through a formal presentation or report, emphasizing both the statistical rigor and the potential business benefits. Review Resource Allocation : Given the likely increase in productivity, assess if resources like labor and parts need to be reallocated to optimize the workflow further.

By following this step-by-step guide, you’ve journeyed through the rigorous yet enlightening process of hypothesis testing. From stating clear hypotheses to interpreting the results, each step has paved the way for making informed, data-driven decisions that can significantly impact your projects, business, or research.

Hypothesis testing is more than just a set of formulas or calculations; it’s a holistic approach to problem-solving that incorporates context, statistics, and strategic decision-making. While the process may seem daunting at first, each step serves a crucial role in ensuring that your conclusions are both statistically sound and practically relevant.

• McKenzie, C.R., 2004. Hypothesis testing and evaluation .  Blackwell handbook of judgment and decision making , pp.200-219.
• Park, H.M., 2015. Hypothesis testing and statistical power of a test.
• Eberhardt, L.L., 2003. What should we do about hypothesis testing? .  The Journal of wildlife management , pp.241-247.

Q: What is hypothesis testing in the context of Lean Six Sigma?

A: Hypothesis testing is a statistical method used in Lean Six Sigma to determine whether there is enough evidence in a sample of data to infer that a certain condition holds true for the entire population. In the Lean Six Sigma process, it’s commonly used to validate the effectiveness of process improvements by comparing performance metrics before and after changes are implemented. A null hypothesis ( H 0 ​ ) usually represents no change or effect, while the alternative hypothesis ( H 1 ​ ) indicates a significant change or effect.

Q: How do I determine which statistical test to use for my hypothesis?

A: The choice of statistical test for hypothesis testing depends on several factors, including the type of data (nominal, ordinal, interval, or ratio), the sample size, the number of samples (one sample, two samples, paired), and whether the data distribution is normal. For example, a t-test is used for comparing the means of two groups when the data is normally distributed, while a Chi-square test is suitable for categorical data to test the relationship between two variables. It’s important to choose the right test to ensure the validity of your hypothesis testing results.

Q: What is a p-value, and how does it relate to hypothesis testing?

A: A p-value is a probability value that helps you determine the significance of your results in hypothesis testing. It represents the likelihood of obtaining a result at least as extreme as the one observed during the test, assuming that the null hypothesis is true. In hypothesis testing, if the p-value is lower than the predetermined significance level (commonly α = 0.05 ), you reject the null hypothesis, suggesting that the observed effect is statistically significant. If the p-value is higher, you fail to reject the null hypothesis, indicating that there is not enough evidence to support the alternative hypothesis.

Q: Can you explain Type I and Type II errors in hypothesis testing?

A: Type I and Type II errors are potential errors that can occur in hypothesis testing. A Type I error, also known as a “false positive,” occurs when the null hypothesis is true, but it is incorrectly rejected. It is equivalent to a false alarm. On the other hand, a Type II error, or a “false negative,” happens when the null hypothesis is false, but it is erroneously failed to be rejected. This means a real effect or difference was missed. The risk of a Type I error is represented by the significance level ( α ), while the risk of a Type II error is denoted by β . Minimizing these errors is crucial for the reliability of hypothesis tests in continuous improvement projects.

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Hi im Daniel continuous improvement manager with a Black Belt in Lean Six Sigma and over 10 years of real-world experience across a range sectors, I have a passion for optimizing processes and creating a culture of efficiency. I wanted to create Learn Lean Siigma to be a platform dedicated to Lean Six Sigma and process improvement insights and provide all the guides, tools, techniques and templates I looked for in one place as someone new to the world of Lean Six Sigma and Continuous improvement.

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Hypothesis Testing for Means & Proportions

Lisa Sullivan, PhD

Professor of Biostatistics

Boston University School of Public Health

Introduction

This is the first of three modules that will addresses the second area of statistical inference, which is hypothesis testing, in which a specific statement or hypothesis is generated about a population parameter, and sample statistics are used to assess the likelihood that the hypothesis is true. The hypothesis is based on available information and the investigator's belief about the population parameters. The process of hypothesis testing involves setting up two competing hypotheses, the null hypothesis and the alternate hypothesis. One selects a random sample (or multiple samples when there are more comparison groups), computes summary statistics and then assesses the likelihood that the sample data support the research or alternative hypothesis. Similar to estimation, the process of hypothesis testing is based on probability theory and the Central Limit Theorem.  

This module will focus on hypothesis testing for means and proportions. The next two modules in this series will address analysis of variance and chi-squared tests. 

Learning Objectives

After completing this module, the student will be able to:

• Define null and research hypothesis, test statistic, level of significance and decision rule
• Distinguish between Type I and Type II errors and discuss the implications of each
• Explain the difference between one and two sided tests of hypothesis
• Estimate and interpret p-values
• Explain the relationship between confidence interval estimates and p-values in drawing inferences
• Differentiate hypothesis testing procedures based on type of outcome variable and number of sample

Introduction to Hypothesis Testing

Techniques for hypothesis testing  .

The techniques for hypothesis testing depend on

• the type of outcome variable being analyzed (continuous, dichotomous, discrete)
• the number of comparison groups in the investigation
• whether the comparison groups are independent (i.e., physically separate such as men versus women) or dependent (i.e., matched or paired such as pre- and post-assessments on the same participants).

In estimation we focused explicitly on techniques for one and two samples and discussed estimation for a specific parameter (e.g., the mean or proportion of a population), for differences (e.g., difference in means, the risk difference) and ratios (e.g., the relative risk and odds ratio). Here we will focus on procedures for one and two samples when the outcome is either continuous (and we focus on means) or dichotomous (and we focus on proportions).

General Approach: A Simple Example

The Centers for Disease Control (CDC) reported on trends in weight, height and body mass index from the 1960's through 2002. 1 The general trend was that Americans were much heavier and slightly taller in 2002 as compared to 1960; both men and women gained approximately 24 pounds, on average, between 1960 and 2002.   In 2002, the mean weight for men was reported at 191 pounds. Suppose that an investigator hypothesizes that weights are even higher in 2006 (i.e., that the trend continued over the subsequent 4 years). The research hypothesis is that the mean weight in men in 2006 is more than 191 pounds. The null hypothesis is that there is no change in weight, and therefore the mean weight is still 191 pounds in 2006.  

In order to test the hypotheses, we select a random sample of American males in 2006 and measure their weights. Suppose we have resources available to recruit n=100 men into our sample. We weigh each participant and compute summary statistics on the sample data. Suppose in the sample we determine the following:

Do the sample data support the null or research hypothesis? The sample mean of 197.1 is numerically higher than 191. However, is this difference more than would be expected by chance? In hypothesis testing, we assume that the null hypothesis holds until proven otherwise. We therefore need to determine the likelihood of observing a sample mean of 197.1 or higher when the true population mean is 191 (i.e., if the null hypothesis is true or under the null hypothesis). We can compute this probability using the Central Limit Theorem. Specifically,

(Notice that we use the sample standard deviation in computing the Z score. This is generally an appropriate substitution as long as the sample size is large, n > 30. Thus, there is less than a 1% probability of observing a sample mean as large as 197.1 when the true population mean is 191. Do you think that the null hypothesis is likely true? Based on how unlikely it is to observe a sample mean of 197.1 under the null hypothesis (i.e., <1% probability), we might infer, from our data, that the null hypothesis is probably not true.

Suppose that the sample data had turned out differently. Suppose that we instead observed the following in 2006:

How likely it is to observe a sample mean of 192.1 or higher when the true population mean is 191 (i.e., if the null hypothesis is true)? We can again compute this probability using the Central Limit Theorem. Specifically,

There is a 33.4% probability of observing a sample mean as large as 192.1 when the true population mean is 191. Do you think that the null hypothesis is likely true?  

Neither of the sample means that we obtained allows us to know with certainty whether the null hypothesis is true or not. However, our computations suggest that, if the null hypothesis were true, the probability of observing a sample mean >197.1 is less than 1%. In contrast, if the null hypothesis were true, the probability of observing a sample mean >192.1 is about 33%. We can't know whether the null hypothesis is true, but the sample that provided a mean value of 197.1 provides much stronger evidence in favor of rejecting the null hypothesis, than the sample that provided a mean value of 192.1. Note that this does not mean that a sample mean of 192.1 indicates that the null hypothesis is true; it just doesn't provide compelling evidence to reject it.

In essence, hypothesis testing is a procedure to compute a probability that reflects the strength of the evidence (based on a given sample) for rejecting the null hypothesis. In hypothesis testing, we determine a threshold or cut-off point (called the critical value) to decide when to believe the null hypothesis and when to believe the research hypothesis. It is important to note that it is possible to observe any sample mean when the true population mean is true (in this example equal to 191), but some sample means are very unlikely. Based on the two samples above it would seem reasonable to believe the research hypothesis when x̄ = 197.1, but to believe the null hypothesis when x̄ =192.1. What we need is a threshold value such that if x̄ is above that threshold then we believe that H 1 is true and if x̄ is below that threshold then we believe that H 0 is true. The difficulty in determining a threshold for x̄ is that it depends on the scale of measurement. In this example, the threshold, sometimes called the critical value, might be 195 (i.e., if the sample mean is 195 or more then we believe that H 1 is true and if the sample mean is less than 195 then we believe that H 0 is true). Suppose we are interested in assessing an increase in blood pressure over time, the critical value will be different because blood pressures are measured in millimeters of mercury (mmHg) as opposed to in pounds. In the following we will explain how the critical value is determined and how we handle the issue of scale.

First, to address the issue of scale in determining the critical value, we convert our sample data (in particular the sample mean) into a Z score. We know from the module on probability that the center of the Z distribution is zero and extreme values are those that exceed 2 or fall below -2. Z scores above 2 and below -2 represent approximately 5% of all Z values. If the observed sample mean is close to the mean specified in H 0 (here m =191), then Z will be close to zero. If the observed sample mean is much larger than the mean specified in H 0 , then Z will be large.  

In hypothesis testing, we select a critical value from the Z distribution. This is done by first determining what is called the level of significance, denoted α ("alpha"). What we are doing here is drawing a line at extreme values. The level of significance is the probability that we reject the null hypothesis (in favor of the alternative) when it is actually true and is also called the Type I error rate.

α = Level of significance = P(Type I error) = P(Reject H 0 | H 0 is true).

Because α is a probability, it ranges between 0 and 1. The most commonly used value in the medical literature for α is 0.05, or 5%. Thus, if an investigator selects α=0.05, then they are allowing a 5% probability of incorrectly rejecting the null hypothesis in favor of the alternative when the null is in fact true. Depending on the circumstances, one might choose to use a level of significance of 1% or 10%. For example, if an investigator wanted to reject the null only if there were even stronger evidence than that ensured with α=0.05, they could choose a =0.01as their level of significance. The typical values for α are 0.01, 0.05 and 0.10, with α=0.05 the most commonly used value.  

Suppose in our weight study we select α=0.05. We need to determine the value of Z that holds 5% of the values above it (see below).

The critical value of Z for α =0.05 is Z = 1.645 (i.e., 5% of the distribution is above Z=1.645). With this value we can set up what is called our decision rule for the test. The rule is to reject H 0 if the Z score is 1.645 or more.  

With the first sample we have

Because 2.38 > 1.645, we reject the null hypothesis. (The same conclusion can be drawn by comparing the 0.0087 probability of observing a sample mean as extreme as 197.1 to the level of significance of 0.05. If the observed probability is smaller than the level of significance we reject H 0 ). Because the Z score exceeds the critical value, we conclude that the mean weight for men in 2006 is more than 191 pounds, the value reported in 2002. If we observed the second sample (i.e., sample mean =192.1), we would not be able to reject the null hypothesis because the Z score is 0.43 which is not in the rejection region (i.e., the region in the tail end of the curve above 1.645). With the second sample we do not have sufficient evidence (because we set our level of significance at 5%) to conclude that weights have increased. Again, the same conclusion can be reached by comparing probabilities. The probability of observing a sample mean as extreme as 192.1 is 33.4% which is not below our 5% level of significance.

Hypothesis Testing: Upper-, Lower, and Two Tailed Tests

The procedure for hypothesis testing is based on the ideas described above. Specifically, we set up competing hypotheses, select a random sample from the population of interest and compute summary statistics. We then determine whether the sample data supports the null or alternative hypotheses. The procedure can be broken down into the following five steps.  

• Step 1. Set up hypotheses and select the level of significance α.

H 0 : Null hypothesis (no change, no difference);  

H 1 : Research hypothesis (investigator's belief); α =0.05

• Step 2. Select the appropriate test statistic.  

The test statistic is a single number that summarizes the sample information.   An example of a test statistic is the Z statistic computed as follows:

When the sample size is small, we will use t statistics (just as we did when constructing confidence intervals for small samples). As we present each scenario, alternative test statistics are provided along with conditions for their appropriate use.

• Step 3.  Set up decision rule.  

The decision rule is a statement that tells under what circumstances to reject the null hypothesis. The decision rule is based on specific values of the test statistic (e.g., reject H 0 if Z > 1.645). The decision rule for a specific test depends on 3 factors: the research or alternative hypothesis, the test statistic and the level of significance. Each is discussed below.

• The decision rule depends on whether an upper-tailed, lower-tailed, or two-tailed test is proposed. In an upper-tailed test the decision rule has investigators reject H 0 if the test statistic is larger than the critical value. In a lower-tailed test the decision rule has investigators reject H 0 if the test statistic is smaller than the critical value.  In a two-tailed test the decision rule has investigators reject H 0 if the test statistic is extreme, either larger than an upper critical value or smaller than a lower critical value.
• The exact form of the test statistic is also important in determining the decision rule. If the test statistic follows the standard normal distribution (Z), then the decision rule will be based on the standard normal distribution. If the test statistic follows the t distribution, then the decision rule will be based on the t distribution. The appropriate critical value will be selected from the t distribution again depending on the specific alternative hypothesis and the level of significance.  
• The third factor is the level of significance. The level of significance which is selected in Step 1 (e.g., α =0.05) dictates the critical value.   For example, in an upper tailed Z test, if α =0.05 then the critical value is Z=1.645.  

The following figures illustrate the rejection regions defined by the decision rule for upper-, lower- and two-tailed Z tests with α=0.05. Notice that the rejection regions are in the upper, lower and both tails of the curves, respectively. The decision rules are written below each figure.

Rejection Region for Upper-Tailed Z Test (H : μ > μ ) with α=0.05 The decision rule is: Reject H if Z 1.645.

α Z 0.10 1.282 0.05 1.645 0.025 1.960 0.010 2.326 0.005 2.576 0.001 3.090 0.0001 3.719 Rejection Region for Lower-Tailed Z Test (H 1 : μ < μ 0 ) with α =0.05 The decision rule is: Reject H 0 if Z < 1.645. a Z 0.10 -1.282 0.05 -1.645 0.025 -1.960 0.010 -2.326 0.005 -2.576 0.001 -3.090 0.0001 -3.719 Rejection Region for Two-Tailed Z Test (H 1 : μ ≠ μ 0 ) with α =0.05 The decision rule is: Reject H 0 if Z < -1.960 or if Z > 1.960. 0.20 1.282 0.10 1.645 0.05 1.960 0.010 2.576 0.001 3.291 0.0001 3.819 The complete table of critical values of Z for upper, lower and two-tailed tests can be found in the table of Z values to the right in "Other Resources." Critical values of t for upper, lower and two-tailed tests can be found in the table of t values in "Other Resources." Step 4. Compute the test statistic.   Here we compute the test statistic by substituting the observed sample data into the test statistic identified in Step 2. Step 5. Conclusion.   The final conclusion is made by comparing the test statistic (which is a summary of the information observed in the sample) to the decision rule. The final conclusion will be either to reject the null hypothesis (because the sample data are very unlikely if the null hypothesis is true) or not to reject the null hypothesis (because the sample data are not very unlikely).   If the null hypothesis is rejected, then an exact significance level is computed to describe the likelihood of observing the sample data assuming that the null hypothesis is true. The exact level of significance is called the p-value and it will be less than the chosen level of significance if we reject H 0 . Statistical computing packages provide exact p-values as part of their standard output for hypothesis tests. In fact, when using a statistical computing package, the steps outlined about can be abbreviated. The hypotheses (step 1) should always be set up in advance of any analysis and the significance criterion should also be determined (e.g., α =0.05). Statistical computing packages will produce the test statistic (usually reporting the test statistic as t) and a p-value. The investigator can then determine statistical significance using the following: If p < α then reject H 0 .   Step 1. Set up hypotheses and determine level of significance H 0 : μ = 191 H 1 : μ > 191                 α =0.05 The research hypothesis is that weights have increased, and therefore an upper tailed test is used. Step 2. Select the appropriate test statistic. Because the sample size is large (n > 30) the appropriate test statistic is Step 3. Set up decision rule.   In this example, we are performing an upper tailed test (H 1 : μ> 191), with a Z test statistic and selected α =0.05.   Reject H 0 if Z > 1.645. We now substitute the sample data into the formula for the test statistic identified in Step 2.   We reject H 0 because 2.38 > 1.645. We have statistically significant evidence at a =0.05, to show that the mean weight in men in 2006 is more than 191 pounds. Because we rejected the null hypothesis, we now approximate the p-value which is the likelihood of observing the sample data if the null hypothesis is true. An alternative definition of the p-value is the smallest level of significance where we can still reject H 0 . In this example, we observed Z=2.38 and for α=0.05, the critical value was 1.645. Because 2.38 exceeded 1.645 we rejected H 0 . In our conclusion we reported a statistically significant increase in mean weight at a 5% level of significance. Using the table of critical values for upper tailed tests, we can approximate the p-value. If we select α=0.025, the critical value is 1.96, and we still reject H 0 because 2.38 > 1.960. If we select α=0.010 the critical value is 2.326, and we still reject H 0 because 2.38 > 2.326. However, if we select α=0.005, the critical value is 2.576, and we cannot reject H 0 because 2.38 < 2.576. Therefore, the smallest α where we still reject H 0 is 0.010. This is the p-value. A statistical computing package would produce a more precise p-value which would be in between 0.005 and 0.010. Here we are approximating the p-value and would report p < 0.010.                   Type I and Type II Errors In all tests of hypothesis, there are two types of errors that can be committed. The first is called a Type I error and refers to the situation where we incorrectly reject H 0 when in fact it is true. This is also called a false positive result (as we incorrectly conclude that the research hypothesis is true when in fact it is not). When we run a test of hypothesis and decide to reject H 0 (e.g., because the test statistic exceeds the critical value in an upper tailed test) then either we make a correct decision because the research hypothesis is true or we commit a Type I error. The different conclusions are summarized in the table below. Note that we will never know whether the null hypothesis is really true or false (i.e., we will never know which row of the following table reflects reality). Table - Conclusions in Test of Hypothesis   is True Correct Decision Type I Error is False Type II Error Correct Decision In the first step of the hypothesis test, we select a level of significance, α, and α= P(Type I error). Because we purposely select a small value for α, we control the probability of committing a Type I error. For example, if we select α=0.05, and our test tells us to reject H 0 , then there is a 5% probability that we commit a Type I error. Most investigators are very comfortable with this and are confident when rejecting H 0 that the research hypothesis is true (as it is the more likely scenario when we reject H 0 ). When we run a test of hypothesis and decide not to reject H 0 (e.g., because the test statistic is below the critical value in an upper tailed test) then either we make a correct decision because the null hypothesis is true or we commit a Type II error. Beta (β) represents the probability of a Type II error and is defined as follows: β=P(Type II error) = P(Do not Reject H 0 | H 0 is false). Unfortunately, we cannot choose β to be small (e.g., 0.05) to control the probability of committing a Type II error because β depends on several factors including the sample size, α, and the research hypothesis. When we do not reject H 0 , it may be very likely that we are committing a Type II error (i.e., failing to reject H 0 when in fact it is false). Therefore, when tests are run and the null hypothesis is not rejected we often make a weak concluding statement allowing for the possibility that we might be committing a Type II error. If we do not reject H 0 , we conclude that we do not have significant evidence to show that H 1 is true. We do not conclude that H 0 is true.  The most common reason for a Type II error is a small sample size. Tests with One Sample, Continuous Outcome Hypothesis testing applications with a continuous outcome variable in a single population are performed according to the five-step procedure outlined above. A key component is setting up the null and research hypotheses. The objective is to compare the mean in a single population to known mean (μ 0 ). The known value is generally derived from another study or report, for example a study in a similar, but not identical, population or a study performed some years ago. The latter is called a historical control. It is important in setting up the hypotheses in a one sample test that the mean specified in the null hypothesis is a fair and reasonable comparator. This will be discussed in the examples that follow. Test Statistics for Testing H 0 : μ= μ 0 if n > 30 if n < 30 Note that statistical computing packages will use the t statistic exclusively and make the necessary adjustments for comparing the test statistic to appropriate values from probability tables to produce a p-value.  The National Center for Health Statistics (NCHS) published a report in 2005 entitled Health, United States, containing extensive information on major trends in the health of Americans. Data are provided for the US population as a whole and for specific ages, sexes and races.  The NCHS report indicated that in 2002 Americans paid an average of $3,302 per year on health care and prescription drugs. An investigator hypothesizes that in 2005 expenditures have decreased primarily due to the availability of generic drugs. To test the hypothesis, a sample of 100 Americans are selected and their expenditures on health care and prescription drugs in 2005 are measured.   The sample data are summarized as follows: n=100, x̄ =$3,190 and s=$890. Is there statistical evidence of a reduction in expenditures on health care and prescription drugs in 2005? Is the sample mean of $3,190 evidence of a true reduction in the mean or is it within chance fluctuation? We will run the test using the five-step approach.  Step 1.  Set up hypotheses and determine level of significance H 0 : μ = 3,302 H 1 : μ < 3,302           α =0.05 The research hypothesis is that expenditures have decreased, and therefore a lower-tailed test is used. This is a lower tailed test, using a Z statistic and a 5% level of significance.   Reject H 0 if Z < -1.645.   Step 4. Compute the test statistic.   We do not reject H 0 because -1.26 > -1.645. We do not have statistically significant evidence at α=0.05 to show that the mean expenditures on health care and prescription drugs are lower in 2005 than the mean of $3,302 reported in 2002.   Recall that when we fail to reject H 0 in a test of hypothesis that either the null hypothesis is true (here the mean expenditures in 2005 are the same as those in 2002 and equal to $3,302) or we committed a Type II error (i.e., we failed to reject H 0 when in fact it is false). In summarizing this test, we conclude that we do not have sufficient evidence to reject H 0 . We do not conclude that H 0 is true, because there may be a moderate to high probability that we committed a Type II error. It is possible that the sample size is not large enough to detect a difference in mean expenditures.       The NCHS reported that the mean total cholesterol level in 2002 for all adults was 203. Total cholesterol levels in participants who attended the seventh examination of the Offspring in the Framingham Heart Study are summarized as follows: n=3,310, x̄ =200.3, and s=36.8. Is there statistical evidence of a difference in mean cholesterol levels in the Framingham Offspring? Here we want to assess whether the sample mean of 200.3 in the Framingham sample is statistically significantly different from 203 (i.e., beyond what we would expect by chance). We will run the test using the five-step approach. H 0 : μ= 203 H 1 : μ≠ 203                       α=0.05 The research hypothesis is that cholesterol levels are different in the Framingham Offspring, and therefore a two-tailed test is used.   Step 3. Set up decision rule.   This is a two-tailed test, using a Z statistic and a 5% level of significance. Reject H 0 if Z < -1.960 or is Z > 1.960. We reject H 0 because -4.22 ≤ -1. .960. We have statistically significant evidence at α=0.05 to show that the mean total cholesterol level in the Framingham Offspring is different from the national average of 203 reported in 2002.   Because we reject H 0 , we also approximate a p-value. Using the two-sided significance levels, p < 0.0001.   Statistical Significance versus Clinical (Practical) Significance This example raises an important concept of statistical versus clinical or practical significance. From a statistical standpoint, the total cholesterol levels in the Framingham sample are highly statistically significantly different from the national average with p < 0.0001 (i.e., there is less than a 0.01% chance that we are incorrectly rejecting the null hypothesis). However, the sample mean in the Framingham Offspring study is 200.3, less than 3 units different from the national mean of 203. The reason that the data are so highly statistically significant is due to the very large sample size. It is always important to assess both statistical and clinical significance of data. This is particularly relevant when the sample size is large. Is a 3 unit difference in total cholesterol a meaningful difference?   Consider again the NCHS-reported mean total cholesterol level in 2002 for all adults of 203. Suppose a new drug is proposed to lower total cholesterol. A study is designed to evaluate the efficacy of the drug in lowering cholesterol.   Fifteen patients are enrolled in the study and asked to take the new drug for 6 weeks. At the end of 6 weeks, each patient's total cholesterol level is measured and the sample statistics are as follows:   n=15, x̄ =195.9 and s=28.7. Is there statistical evidence of a reduction in mean total cholesterol in patients after using the new drug for 6 weeks? We will run the test using the five-step approach.  H 0 : μ= 203 H 1 : μ< 203                   α=0.05  Step 2. Select the appropriate test statistic.   Because the sample size is small (n<30) the appropriate test statistic is This is a lower tailed test, using a t statistic and a 5% level of significance. In order to determine the critical value of t, we need degrees of freedom, df, defined as df=n-1. In this example df=15-1=14. The critical value for a lower tailed test with df=14 and a =0.05 is -2.145 and the decision rule is as follows:   Reject H 0 if t < -2.145. We do not reject H 0 because -0.96 > -2.145. We do not have statistically significant evidence at α=0.05 to show that the mean total cholesterol level is lower than the national mean in patients taking the new drug for 6 weeks. Again, because we failed to reject the null hypothesis we make a weaker concluding statement allowing for the possibility that we may have committed a Type II error (i.e., failed to reject H 0 when in fact the drug is efficacious). This example raises an important issue in terms of study design. In this example we assume in the null hypothesis that the mean cholesterol level is 203. This is taken to be the mean cholesterol level in patients without treatment. Is this an appropriate comparator? Alternative and potentially more efficient study designs to evaluate the effect of the new drug could involve two treatment groups, where one group receives the new drug and the other does not, or we could measure each patient's baseline or pre-treatment cholesterol level and then assess changes from baseline to 6 weeks post-treatment. These designs are also discussed here. Video - Comparing a Sample Mean to Known Population Mean (8:20) Link to transcript of the video Tests with One Sample, Dichotomous Outcome Hypothesis testing applications with a dichotomous outcome variable in a single population are also performed according to the five-step procedure. Similar to tests for means, a key component is setting up the null and research hypotheses. The objective is to compare the proportion of successes in a single population to a known proportion (p 0 ). That known proportion is generally derived from another study or report and is sometimes called a historical control. It is important in setting up the hypotheses in a one sample test that the proportion specified in the null hypothesis is a fair and reasonable comparator.     In one sample tests for a dichotomous outcome, we set up our hypotheses against an appropriate comparator. We select a sample and compute descriptive statistics on the sample data. Specifically, we compute the sample size (n) and the sample proportion which is computed by taking the ratio of the number of successes to the sample size, We then determine the appropriate test statistic (Step 2) for the hypothesis test. The formula for the test statistic is given below. Test Statistic for Testing H 0 : p = p 0 if min(np 0 , n(1-p 0 )) > 5 The formula above is appropriate for large samples, defined when the smaller of np 0 and n(1-p 0 ) is at least 5. This is similar, but not identical, to the condition required for appropriate use of the confidence interval formula for a population proportion, i.e., Here we use the proportion specified in the null hypothesis as the true proportion of successes rather than the sample proportion. If we fail to satisfy the condition, then alternative procedures, called exact methods must be used to test the hypothesis about the population proportion. Example:   The NCHS report indicated that in 2002 the prevalence of cigarette smoking among American adults was 21.1%.  Data on prevalent smoking in n=3,536 participants who attended the seventh examination of the Offspring in the Framingham Heart Study indicated that 482/3,536 = 13.6% of the respondents were currently smoking at the time of the exam. Suppose we want to assess whether the prevalence of smoking is lower in the Framingham Offspring sample given the focus on cardiovascular health in that community. Is there evidence of a statistically lower prevalence of smoking in the Framingham Offspring study as compared to the prevalence among all Americans? H 0 : p = 0.211 H 1 : p < 0.211                     α=0.05 We must first check that the sample size is adequate.   Specifically, we need to check min(np 0 , n(1-p 0 )) = min( 3,536(0.211), 3,536(1-0.211))=min(746, 2790)=746. The sample size is more than adequate so the following formula can be used: This is a lower tailed test, using a Z statistic and a 5% level of significance. Reject H 0 if Z < -1.645. We reject H 0 because -10.93 < -1.645. We have statistically significant evidence at α=0.05 to show that the prevalence of smoking in the Framingham Offspring is lower than the prevalence nationally (21.1%). Here, p < 0.0001.   The NCHS report indicated that in 2002, 75% of children aged 2 to 17 saw a dentist in the past year. An investigator wants to assess whether use of dental services is similar in children living in the city of Boston. A sample of 125 children aged 2 to 17 living in Boston are surveyed and 64 reported seeing a dentist over the past 12 months. Is there a significant difference in use of dental services between children living in Boston and the national data? Calculate this on your own before checking the answer. Video - Hypothesis Test for One Sample and a Dichotomous Outcome (3:55) Tests with Two Independent Samples, Continuous Outcome There are many applications where it is of interest to compare two independent groups with respect to their mean scores on a continuous outcome. Here we compare means between groups, but rather than generating an estimate of the difference, we will test whether the observed difference (increase, decrease or difference) is statistically significant or not. Remember, that hypothesis testing gives an assessment of statistical significance, whereas estimation gives an estimate of effect and both are important. Here we discuss the comparison of means when the two comparison groups are independent or physically separate. The two groups might be determined by a particular attribute (e.g., sex, diagnosis of cardiovascular disease) or might be set up by the investigator (e.g., participants assigned to receive an experimental treatment or placebo). The first step in the analysis involves computing descriptive statistics on each of the two samples. Specifically, we compute the sample size, mean and standard deviation in each sample and we denote these summary statistics as follows: for sample 1: for sample 2: The designation of sample 1 and sample 2 is arbitrary. In a clinical trial setting the convention is to call the treatment group 1 and the control group 2. However, when comparing men and women, for example, either group can be 1 or 2.   In the two independent samples application with a continuous outcome, the parameter of interest in the test of hypothesis is the difference in population means, μ 1 -μ 2 . The null hypothesis is always that there is no difference between groups with respect to means, i.e., The null hypothesis can also be written as follows: H 0 : μ 1 = μ 2 . In the research hypothesis, an investigator can hypothesize that the first mean is larger than the second (H 1 : μ 1 > μ 2 ), that the first mean is smaller than the second (H 1 : μ 1 < μ 2 ), or that the means are different (H 1 : μ 1 ≠ μ 2 ). The three different alternatives represent upper-, lower-, and two-tailed tests, respectively. The following test statistics are used to test these hypotheses. Test Statistics for Testing H 0 : μ 1 = μ 2 if n 1 > 30 and n 2 > 30 if n 1 < 30 or n 2 < 30 NOTE: The formulas above assume equal variability in the two populations (i.e., the population variances are equal, or s 1 2 = s 2 2 ). This means that the outcome is equally variable in each of the comparison populations. For analysis, we have samples from each of the comparison populations. If the sample variances are similar, then the assumption about variability in the populations is probably reasonable. As a guideline, if the ratio of the sample variances, s 1 2 /s 2 2 is between 0.5 and 2 (i.e., if one variance is no more than double the other), then the formulas above are appropriate. If the ratio of the sample variances is greater than 2 or less than 0.5 then alternative formulas must be used to account for the heterogeneity in variances.     The test statistics include Sp, which is the pooled estimate of the common standard deviation (again assuming that the variances in the populations are similar) computed as the weighted average of the standard deviations in the samples as follows: Because we are assuming equal variances between groups, we pool the information on variability (sample variances) to generate an estimate of the variability in the population. Note: Because Sp is a weighted average of the standard deviations in the sample, Sp will always be in between s 1 and s 2 .) Data measured on n=3,539 participants who attended the seventh examination of the Offspring in the Framingham Heart Study are shown below.     Characteristic n S n s Systolic Blood Pressure 1,623 128.2 17.5 1,911 126.5 20.1 Diastolic Blood Pressure 1,622 75.6 9.8 1,910 72.6 9.7 Total Serum Cholesterol 1,544 192.4 35.2 1,766 207.1 36.7 Weight 1,612 194.0 33.8 1,894 157.7 34.6 Height 1,545 68.9 2.7 1,781 63.4 2.5 Body Mass Index 1,545 28.8 4.6 1,781 27.6 5.9 Suppose we now wish to assess whether there is a statistically significant difference in mean systolic blood pressures between men and women using a 5% level of significance.   H 0 : μ 1 = μ 2 H 1 : μ 1 ≠ μ 2                       α=0.05 Because both samples are large ( > 30), we can use the Z test statistic as opposed to t. Note that statistical computing packages use t throughout. Before implementing the formula, we first check whether the assumption of equality of population variances is reasonable. The guideline suggests investigating the ratio of the sample variances, s 1 2 /s 2 2 . Suppose we call the men group 1 and the women group 2. Again, this is arbitrary; it only needs to be noted when interpreting the results. The ratio of the sample variances is 17.5 2 /20.1 2 = 0.76, which falls between 0.5 and 2 suggesting that the assumption of equality of population variances is reasonable. The appropriate test statistic is We now substitute the sample data into the formula for the test statistic identified in Step 2. Before substituting, we will first compute Sp, the pooled estimate of the common standard deviation. Notice that the pooled estimate of the common standard deviation, Sp, falls in between the standard deviations in the comparison groups (i.e., 17.5 and 20.1). Sp is slightly closer in value to the standard deviation in the women (20.1) as there were slightly more women in the sample.   Recall, Sp is a weight average of the standard deviations in the comparison groups, weighted by the respective sample sizes.   Now the test statistic: We reject H 0 because 2.66 > 1.960. We have statistically significant evidence at α=0.05 to show that there is a difference in mean systolic blood pressures between men and women. The p-value is p < 0.010.   Here again we find that there is a statistically significant difference in mean systolic blood pressures between men and women at p < 0.010. Notice that there is a very small difference in the sample means (128.2-126.5 = 1.7 units), but this difference is beyond what would be expected by chance. Is this a clinically meaningful difference? The large sample size in this example is driving the statistical significance. A 95% confidence interval for the difference in mean systolic blood pressures is: 1.7 + 1.26 or (0.44, 2.96). The confidence interval provides an assessment of the magnitude of the difference between means whereas the test of hypothesis and p-value provide an assessment of the statistical significance of the difference.   Above we performed a study to evaluate a new drug designed to lower total cholesterol. The study involved one sample of patients, each patient took the new drug for 6 weeks and had their cholesterol measured. As a means of evaluating the efficacy of the new drug, the mean total cholesterol following 6 weeks of treatment was compared to the NCHS-reported mean total cholesterol level in 2002 for all adults of 203. At the end of the example, we discussed the appropriateness of the fixed comparator as well as an alternative study design to evaluate the effect of the new drug involving two treatment groups, where one group receives the new drug and the other does not. Here, we revisit the example with a concurrent or parallel control group, which is very typical in randomized controlled trials or clinical trials (refer to the EP713 module on Clinical Trials).   A new drug is proposed to lower total cholesterol. A randomized controlled trial is designed to evaluate the efficacy of the medication in lowering cholesterol. Thirty participants are enrolled in the trial and are randomly assigned to receive either the new drug or a placebo. The participants do not know which treatment they are assigned. Each participant is asked to take the assigned treatment for 6 weeks. At the end of 6 weeks, each patient's total cholesterol level is measured and the sample statistics are as follows. Treatment New Drug 15 195.9 28.7 Placebo 15 227.4 30.3 Is there statistical evidence of a reduction in mean total cholesterol in patients taking the new drug for 6 weeks as compared to participants taking placebo? We will run the test using the five-step approach. H 0 : μ 1 = μ 2 H 1 : μ 1 < μ 2                         α=0.05 Because both samples are small (< 30), we use the t test statistic. Before implementing the formula, we first check whether the assumption of equality of population variances is reasonable. The ratio of the sample variances, s 1 2 /s 2 2 =28.7 2 /30.3 2 = 0.90, which falls between 0.5 and 2, suggesting that the assumption of equality of population variances is reasonable. The appropriate test statistic is: This is a lower-tailed test, using a t statistic and a 5% level of significance. The appropriate critical value can be found in the t Table (in More Resources to the right). In order to determine the critical value of t we need degrees of freedom, df, defined as df=n 1 +n 2 -2 = 15+15-2=28. The critical value for a lower tailed test with df=28 and α=0.05 is -1.701 and the decision rule is: Reject H 0 if t < -1.701. Now the test statistic, We reject H 0 because -2.92 < -1.701. We have statistically significant evidence at α=0.05 to show that the mean total cholesterol level is lower in patients taking the new drug for 6 weeks as compared to patients taking placebo, p < 0.005. The clinical trial in this example finds a statistically significant reduction in total cholesterol, whereas in the previous example where we had a historical control (as opposed to a parallel control group) we did not demonstrate efficacy of the new drug. Notice that the mean total cholesterol level in patients taking placebo is 217.4 which is very different from the mean cholesterol reported among all Americans in 2002 of 203 and used as the comparator in the prior example. The historical control value may not have been the most appropriate comparator as cholesterol levels have been increasing over time. In the next section, we present another design that can be used to assess the efficacy of the new drug. Video - Comparison of Two Independent Samples With a Continuous Outcome (8:02) Tests with Matched Samples, Continuous Outcome In the previous section we compared two groups with respect to their mean scores on a continuous outcome. An alternative study design is to compare matched or paired samples. The two comparison groups are said to be dependent, and the data can arise from a single sample of participants where each participant is measured twice (possibly before and after an intervention) or from two samples that are matched on specific characteristics (e.g., siblings). When the samples are dependent, we focus on difference scores in each participant or between members of a pair and the test of hypothesis is based on the mean difference, μ d . The null hypothesis again reflects "no difference" and is stated as H 0 : μ d =0 . Note that there are some instances where it is of interest to test whether there is a difference of a particular magnitude (e.g., μ d =5) but in most instances the null hypothesis reflects no difference (i.e., μ d =0).   The appropriate formula for the test of hypothesis depends on the sample size. The formulas are shown below and are identical to those we presented for estimating the mean of a single sample presented (e.g., when comparing against an external or historical control), except here we focus on difference scores. Test Statistics for Testing H 0 : μ d =0 A new drug is proposed to lower total cholesterol and a study is designed to evaluate the efficacy of the drug in lowering cholesterol. Fifteen patients agree to participate in the study and each is asked to take the new drug for 6 weeks. However, before starting the treatment, each patient's total cholesterol level is measured. The initial measurement is a pre-treatment or baseline value. After taking the drug for 6 weeks, each patient's total cholesterol level is measured again and the data are shown below. The rightmost column contains difference scores for each patient, computed by subtracting the 6 week cholesterol level from the baseline level. The differences represent the reduction in total cholesterol over 4 weeks. (The differences could have been computed by subtracting the baseline total cholesterol level from the level measured at 6 weeks. The way in which the differences are computed does not affect the outcome of the analysis only the interpretation.) 1 215 205 10 2 190 156 34 3 230 190 40 4 220 180 40 5 214 201 13 6 240 227 13 7 210 197 13 8 193 173 20 9 210 204 6 10 230 217 13 11 180 142 38 12 260 262 -2 13 210 207 3 14 190 184 6 15 200 193 7 Because the differences are computed by subtracting the cholesterols measured at 6 weeks from the baseline values, positive differences indicate reductions and negative differences indicate increases (e.g., participant 12 increases by 2 units over 6 weeks). The goal here is to test whether there is a statistically significant reduction in cholesterol. Because of the way in which we computed the differences, we want to look for an increase in the mean difference (i.e., a positive reduction). In order to conduct the test, we need to summarize the differences. In this sample, we have The calculations are shown below.   1 10 100 2 34 1156 3 40 1600 4 40 1600 5 13 169 6 13 169 7 13 169 8 20 400 9 6 36 10 13 169 11 38 1444 12 -2 4 13 3 9 14 6 36 15 7 49 Is there statistical evidence of a reduction in mean total cholesterol in patients after using the new medication for 6 weeks? We will run the test using the five-step approach. H 0 : μ d = 0 H 1 : μ d > 0                 α=0.05 NOTE: If we had computed differences by subtracting the baseline level from the level measured at 6 weeks then negative differences would have reflected reductions and the research hypothesis would have been H 1 : μ d < 0.  Step 2 . Select the appropriate test statistic. This is an upper-tailed test, using a t statistic and a 5% level of significance. The appropriate critical value can be found in the t Table at the right, with df=15-1=14. The critical value for an upper-tailed test with df=14 and α=0.05 is 2.145 and the decision rule is Reject H 0 if t > 2.145. We now substitute the sample data into the formula for the test statistic identified in Step 2. We reject H 0 because 4.61 > 2.145. We have statistically significant evidence at α=0.05 to show that there is a reduction in cholesterol levels over 6 weeks.   Here we illustrate the use of a matched design to test the efficacy of a new drug to lower total cholesterol. We also considered a parallel design (randomized clinical trial) and a study using a historical comparator. It is extremely important to design studies that are best suited to detect a meaningful difference when one exists. There are often several alternatives and investigators work with biostatisticians to determine the best design for each application. It is worth noting that the matched design used here can be problematic in that observed differences may only reflect a "placebo" effect. All participants took the assigned medication, but is the observed reduction attributable to the medication or a result of these participation in a study. Video - Hypothesis Testing With a Matched Sample and a Continuous Outcome (3:11) Tests with Two Independent Samples, Dichotomous Outcome There are several approaches that can be used to test hypotheses concerning two independent proportions. Here we present one approach - the chi-square test of independence is an alternative, equivalent, and perhaps more popular approach to the same analysis. Hypothesis testing with the chi-square test is addressed in the third module in this series: BS704_HypothesisTesting-ChiSquare. In tests of hypothesis comparing proportions between two independent groups, one test is performed and results can be interpreted to apply to a risk difference, relative risk or odds ratio. As a reminder, the risk difference is computed by taking the difference in proportions between comparison groups, the risk ratio is computed by taking the ratio of proportions, and the odds ratio is computed by taking the ratio of the odds of success in the comparison groups. Because the null values for the risk difference, the risk ratio and the odds ratio are different, the hypotheses in tests of hypothesis look slightly different depending on which measure is used. When performing tests of hypothesis for the risk difference, relative risk or odds ratio, the convention is to label the exposed or treated group 1 and the unexposed or control group 2.       For example, suppose a study is designed to assess whether there is a significant difference in proportions in two independent comparison groups. The test of interest is as follows: H 0 : p 1 = p 2 versus H 1 : p 1 ≠ p 2 .   The following are the hypothesis for testing for a difference in proportions using the risk difference, the risk ratio and the odds ratio. First, the hypotheses above are equivalent to the following: For the risk difference, H 0 : p 1 - p 2 = 0 versus H 1 : p 1 - p 2 ≠ 0 which are, by definition, equal to H 0 : RD = 0 versus H 1 : RD ≠ 0. If an investigator wants to focus on the risk ratio, the equivalent hypotheses are H 0 : RR = 1 versus H 1 : RR ≠ 1. If the investigator wants to focus on the odds ratio, the equivalent hypotheses are H 0 : OR = 1 versus H 1 : OR ≠ 1.   Suppose a test is performed to test H 0 : RD = 0 versus H 1 : RD ≠ 0 and the test rejects H 0 at α=0.05. Based on this test we can conclude that there is significant evidence, α=0.05, of a difference in proportions, significant evidence that the risk difference is not zero, significant evidence that the risk ratio and odds ratio are not one. The risk difference is analogous to the difference in means when the outcome is continuous. Here the parameter of interest is the difference in proportions in the population, RD = p 1 -p 2 and the null value for the risk difference is zero. In a test of hypothesis for the risk difference, the null hypothesis is always H 0 : RD = 0. This is equivalent to H 0 : RR = 1 and H 0 : OR = 1. In the research hypothesis, an investigator can hypothesize that the first proportion is larger than the second (H 1 : p 1 > p 2 , which is equivalent to H 1 : RD > 0, H 1 : RR > 1 and H 1 : OR > 1), that the first proportion is smaller than the second (H 1 : p 1 < p 2 , which is equivalent to H 1 : RD < 0, H 1 : RR < 1 and H 1 : OR < 1), or that the proportions are different (H 1 : p 1 ≠ p 2 , which is equivalent to H 1 : RD ≠ 0, H 1 : RR ≠ 1 and H 1 : OR ≠ 1). The three different alternatives represent upper-, lower- and two-tailed tests, respectively.   The formula for the test of hypothesis for the difference in proportions is given below. Test Statistics for Testing H 0 : p 1 = p                                       The formula above is appropriate for large samples, defined as at least 5 successes (np > 5) and at least 5 failures (n(1-p > 5)) in each of the two samples. If there are fewer than 5 successes or failures in either comparison group, then alternative procedures, called exact methods must be used to estimate the difference in population proportions. The following table summarizes data from n=3,799 participants who attended the fifth examination of the Offspring in the Framingham Heart Study. The outcome of interest is prevalent CVD and we want to test whether the prevalence of CVD is significantly higher in smokers as compared to non-smokers.   Free of CVD History of CVD Total Non-Smoker 2,757 298 3,055 Current Smoker 663 81 744 Total 3,420 379 3,799 The prevalence of CVD (or proportion of participants with prevalent CVD) among non-smokers is 298/3,055 = 0.0975 and the prevalence of CVD among current smokers is 81/744 = 0.1089. Here smoking status defines the comparison groups and we will call the current smokers group 1 (exposed) and the non-smokers (unexposed) group 2. The test of hypothesis is conducted below using the five step approach. H 0 : p 1 = p 2     H 1 : p 1 ≠ p 2                 α=0.05 Step 2.  Select the appropriate test statistic.   We must first check that the sample size is adequate. Specifically, we need to ensure that we have at least 5 successes and 5 failures in each comparison group. In this example, we have more than enough successes (cases of prevalent CVD) and failures (persons free of CVD) in each comparison group. The sample size is more than adequate so the following formula can be used: Reject H 0 if Z < -1.960 or if Z > 1.960. We now substitute the sample data into the formula for the test statistic identified in Step 2. We first compute the overall proportion of successes: We now substitute to compute the test statistic. Step 5. Conclusion. We do not reject H 0 because -1.960 < 0.927 < 1.960. We do not have statistically significant evidence at α=0.05 to show that there is a difference in prevalent CVD between smokers and non-smokers.   A 95% confidence interval for the difference in prevalent CVD (or risk difference) between smokers and non-smokers as 0.0114 + 0.0247, or between -0.0133 and 0.0361. Because the 95% confidence interval for the risk difference includes zero we again conclude that there is no statistically significant difference in prevalent CVD between smokers and non-smokers.     Smoking has been shown over and over to be a risk factor for cardiovascular disease. What might explain the fact that we did not observe a statistically significant difference using data from the Framingham Heart Study? HINT: Here we consider prevalent CVD, would the results have been different if we considered incident CVD? A randomized trial is designed to evaluate the effectiveness of a newly developed pain reliever designed to reduce pain in patients following joint replacement surgery. The trial compares the new pain reliever to the pain reliever currently in use (called the standard of care). A total of 100 patients undergoing joint replacement surgery agreed to participate in the trial. Patients were randomly assigned to receive either the new pain reliever or the standard pain reliever following surgery and were blind to the treatment assignment. Before receiving the assigned treatment, patients were asked to rate their pain on a scale of 0-10 with higher scores indicative of more pain. Each patient was then given the assigned treatment and after 30 minutes was again asked to rate their pain on the same scale. The primary outcome was a reduction in pain of 3 or more scale points (defined by clinicians as a clinically meaningful reduction). The following data were observed in the trial. New Pain Reliever 50 23 0.46 Standard Pain Reliever 50 11 0.22 We now test whether there is a statistically significant difference in the proportions of patients reporting a meaningful reduction (i.e., a reduction of 3 or more scale points) using the five step approach.   H 0 : p 1 = p 2     H 1 : p 1 ≠ p 2              α=0.05 Here the new or experimental pain reliever is group 1 and the standard pain reliever is group 2. We must first check that the sample size is adequate. Specifically, we need to ensure that we have at least 5 successes and 5 failures in each comparison group, i.e., In this example, we have min(50(0.46), 50(1-0.46), 50(0.22), 50(1-0.22)) = min(23, 27, 11, 39) = 11. The sample size is adequate so the following formula can be used We reject H 0 because 2.526 > 1960. We have statistically significant evidence at a =0.05 to show that there is a difference in the proportions of patients on the new pain reliever reporting a meaningful reduction (i.e., a reduction of 3 or more scale points) as compared to patients on the standard pain reliever. A 95% confidence interval for the difference in proportions of patients on the new pain reliever reporting a meaningful reduction (i.e., a reduction of 3 or more scale points) as compared to patients on the standard pain reliever is 0.24 + 0.18 or between 0.06 and 0.42. Because the 95% confidence interval does not include zero we concluded that there was a statistically significant difference in proportions which is consistent with the test of hypothesis result.  Again, the procedures discussed here apply to applications where there are two independent comparison groups and a dichotomous outcome. There are other applications in which it is of interest to compare a dichotomous outcome in matched or paired samples. For example, in a clinical trial we might wish to test the effectiveness of a new antibiotic eye drop for the treatment of bacterial conjunctivitis. Participants use the new antibiotic eye drop in one eye and a comparator (placebo or active control treatment) in the other. The success of the treatment (yes/no) is recorded for each participant for each eye. Because the two assessments (success or failure) are paired, we cannot use the procedures discussed here. The appropriate test is called McNemar's test (sometimes called McNemar's test for dependent proportions).   Vide0 - Hypothesis Testing With Two Independent Samples and a Dichotomous Outcome (2:55) Here we presented hypothesis testing techniques for means and proportions in one and two sample situations. Tests of hypothesis involve several steps, including specifying the null and alternative or research hypothesis, selecting and computing an appropriate test statistic, setting up a decision rule and drawing a conclusion. There are many details to consider in hypothesis testing. The first is to determine the appropriate test. We discussed Z and t tests here for different applications. The appropriate test depends on the distribution of the outcome variable (continuous or dichotomous), the number of comparison groups (one, two) and whether the comparison groups are independent or dependent. The following table summarizes the different tests of hypothesis discussed here. Continuous Outcome, One Sample: H0: μ = μ0 Continuous Outcome, Two Independent Samples: H0: μ1 = μ2 Continuous Outcome, Two Matched Samples: H0: μd = 0 Dichotomous Outcome, One Sample: H0: p = p 0 Dichotomous Outcome, Two Independent Samples: H0: p1 = p2, RD=0, RR=1, OR=1 Once the type of test is determined, the details of the test must be specified. Specifically, the null and alternative hypotheses must be clearly stated. The null hypothesis always reflects the "no change" or "no difference" situation. The alternative or research hypothesis reflects the investigator's belief. The investigator might hypothesize that a parameter (e.g., a mean, proportion, difference in means or proportions) will increase, will decrease or will be different under specific conditions (sometimes the conditions are different experimental conditions and other times the conditions are simply different groups of participants). Once the hypotheses are specified, data are collected and summarized. The appropriate test is then conducted according to the five step approach. If the test leads to rejection of the null hypothesis, an approximate p-value is computed to summarize the significance of the findings. When tests of hypothesis are conducted using statistical computing packages, exact p-values are computed. Because the statistical tables in this textbook are limited, we can only approximate p-values. If the test fails to reject the null hypothesis, then a weaker concluding statement is made for the following reason. In hypothesis testing, there are two types of errors that can be committed. A Type I error occurs when a test incorrectly rejects the null hypothesis. This is referred to as a false positive result, and the probability that this occurs is equal to the level of significance, α. The investigator chooses the level of significance in Step 1, and purposely chooses a small value such as α=0.05 to control the probability of committing a Type I error. A Type II error occurs when a test fails to reject the null hypothesis when in fact it is false. The probability that this occurs is equal to β. Unfortunately, the investigator cannot specify β at the outset because it depends on several factors including the sample size (smaller samples have higher b), the level of significance (β decreases as a increases), and the difference in the parameter under the null and alternative hypothesis.     We noted in several examples in this chapter, the relationship between confidence intervals and tests of hypothesis. The approaches are different, yet related. It is possible to draw a conclusion about statistical significance by examining a confidence interval. For example, if a 95% confidence interval does not contain the null value (e.g., zero when analyzing a mean difference or risk difference, one when analyzing relative risks or odds ratios), then one can conclude that a two-sided test of hypothesis would reject the null at α=0.05. It is important to note that the correspondence between a confidence interval and test of hypothesis relates to a two-sided test and that the confidence level corresponds to a specific level of significance (e.g., 95% to α=0.05, 90% to α=0.10 and so on). The exact significance of the test, the p-value, can only be determined using the hypothesis testing approach and the p-value provides an assessment of the strength of the evidence and not an estimate of the effect. Answers to Selected Problems Dental services problem - bottom of page 5. Step 1: Set up hypotheses and determine the level of significance. α=0.05 Step 2: Select the appropriate test statistic. First, determine whether the sample size is adequate. Therefore the sample size is adequate, and we can use the following formula: Step 3: Set up the decision rule. Reject H0 if Z is less than or equal to -1.96 or if Z is greater than or equal to 1.96. Step 4: Compute the test statistic Step 5: Conclusion. We reject the null hypothesis because -6.15<-1.96. Therefore there is a statistically significant difference in the proportion of children in Boston using dental services compated to the national proportion. Tutorial Playlist Statistics tutorial, everything you need to know about the probability density function in statistics, the best guide to understand central limit theorem, an in-depth guide to measures of central tendency : mean, median and mode, the ultimate guide to understand conditional probability. A Comprehensive Look at Percentile in Statistics The Best Guide to Understand Bayes Theorem Everything you need to know about the normal distribution, an in-depth explanation of cumulative distribution function, a complete guide to chi-square test, what is hypothesis testing in statistics types and examples, understanding the fundamentals of arithmetic and geometric progression, the definitive guide to understand spearman’s rank correlation, mean squared error: overview, examples, concepts and more, all you need to know about the empirical rule in statistics, the complete guide to skewness and kurtosis, a holistic look at bernoulli distribution. All You Need to Know About Bias in Statistics A Complete Guide to Get a Grasp of Time Series Analysis The Key Differences Between Z-Test Vs. T-Test The Complete Guide to Understand Pearson's Correlation A complete guide on the types of statistical studies, everything you need to know about poisson distribution, your best guide to understand correlation vs. regression, the most comprehensive guide for beginners on what is correlation, hypothesis testing in statistics - types | examples. Lesson 10 of 24 By Avijeet Biswal Table of Contents In today’s data-driven world, decisions are based on data all the time. Hypothesis plays a crucial role in that process, whether it may be making business decisions, in the health sector, academia, or in quality improvement. Without hypothesis & hypothesis tests, you risk drawing the wrong conclusions and making bad decisions. In this tutorial, you will look at Hypothesis Testing in Statistics. The Ultimate Ticket to Top Data Science Job Roles What Is Hypothesis Testing in Statistics? Hypothesis Testing is a type of statistical analysis in which you put your assumptions about a population parameter to the test. It is used to estimate the relationship between 2 statistical variables. Let's discuss few examples of statistical hypothesis from real-life -  A teacher assumes that 60% of his college's students come from lower-middle-class families. A doctor believes that 3D (Diet, Dose, and Discipline) is 90% effective for diabetic patients. Now that you know about hypothesis testing, look at the two types of hypothesis testing in statistics. Hypothesis Testing Formula Z = ( x̅ – μ0 ) / (σ /√n) Here, x̅ is the sample mean, μ0 is the population mean, σ is the standard deviation, n is the sample size. How Hypothesis Testing Works? An analyst performs hypothesis testing on a statistical sample to present evidence of the plausibility of the null hypothesis. Measurements and analyses are conducted on a random sample of the population to test a theory. Analysts use a random population sample to test two hypotheses: the null and alternative hypotheses. The null hypothesis is typically an equality hypothesis between population parameters; for example, a null hypothesis may claim that the population means return equals zero. The alternate hypothesis is essentially the inverse of the null hypothesis (e.g., the population means the return is not equal to zero). As a result, they are mutually exclusive, and only one can be correct. One of the two possibilities, however, will always be correct. Your Dream Career is Just Around The Corner! Null Hypothesis and Alternative Hypothesis The Null Hypothesis is the assumption that the event will not occur. A null hypothesis has no bearing on the study's outcome unless it is rejected. H0 is the symbol for it, and it is pronounced H-naught. The Alternate Hypothesis is the logical opposite of the null hypothesis. The acceptance of the alternative hypothesis follows the rejection of the null hypothesis. H1 is the symbol for it. Let's understand this with an example. A sanitizer manufacturer claims that its product kills 95 percent of germs on average.  To put this company's claim to the test, create a null and alternate hypothesis. H0 (Null Hypothesis): Average = 95%. Alternative Hypothesis (H1): The average is less than 95%. Another straightforward example to understand this concept is determining whether or not a coin is fair and balanced. The null hypothesis states that the probability of a show of heads is equal to the likelihood of a show of tails. In contrast, the alternate theory states that the probability of a show of heads and tails would be very different. Become a Data Scientist with Hands-on Training! Hypothesis Testing Calculation With Examples Let's consider a hypothesis test for the average height of women in the United States. Suppose our null hypothesis is that the average height is 5'4". We gather a sample of 100 women and determine that their average height is 5'5". The standard deviation of population is 2. To calculate the z-score, we would use the following formula: z = ( x̅ – μ0 ) / (σ /√n) z = (5'5" - 5'4") / (2" / √100) z = 0.5 / (0.045) We will reject the null hypothesis as the z-score of 11.11 is very large and conclude that there is evidence to suggest that the average height of women in the US is greater than 5'4". Steps in Hypothesis Testing Hypothesis testing is a statistical method to determine if there is enough evidence in a sample of data to infer that a certain condition is true for the entire population. Here’s a breakdown of the typical steps involved in hypothesis testing: Formulate Hypotheses Null Hypothesis (H0): This hypothesis states that there is no effect or difference, and it is the hypothesis you attempt to reject with your test. Alternative Hypothesis (H1 or Ha): This hypothesis is what you might believe to be true or hope to prove true. It is usually considered the opposite of the null hypothesis. Choose the Significance Level (α) The significance level, often denoted by alpha (α), is the probability of rejecting the null hypothesis when it is true. Common choices for α are 0.05 (5%), 0.01 (1%), and 0.10 (10%). Select the Appropriate Test Choose a statistical test based on the type of data and the hypothesis. Common tests include t-tests, chi-square tests, ANOVA, and regression analysis. The selection depends on data type, distribution, sample size, and whether the hypothesis is one-tailed or two-tailed. Collect Data Gather the data that will be analyzed in the test. This data should be representative of the population to infer conclusions accurately. Calculate the Test Statistic Based on the collected data and the chosen test, calculate a test statistic that reflects how much the observed data deviates from the null hypothesis. Determine the p-value The p-value is the probability of observing test results at least as extreme as the results observed, assuming the null hypothesis is correct. It helps determine the strength of the evidence against the null hypothesis. Make a Decision Compare the p-value to the chosen significance level: If the p-value ≤ α: Reject the null hypothesis, suggesting sufficient evidence in the data supports the alternative hypothesis. If the p-value > α: Do not reject the null hypothesis, suggesting insufficient evidence to support the alternative hypothesis. Report the Results Present the findings from the hypothesis test, including the test statistic, p-value, and the conclusion about the hypotheses. Perform Post-hoc Analysis (if necessary) Depending on the results and the study design, further analysis may be needed to explore the data more deeply or to address multiple comparisons if several hypotheses were tested simultaneously. Types of Hypothesis Testing To determine whether a discovery or relationship is statistically significant, hypothesis testing uses a z-test. It usually checks to see if two means are the same (the null hypothesis). Only when the population standard deviation is known and the sample size is 30 data points or more, can a z-test be applied. A statistical test called a t-test is employed to compare the means of two groups. To determine whether two groups differ or if a procedure or treatment affects the population of interest, it is frequently used in hypothesis testing. Chi-Square  You utilize a Chi-square test for hypothesis testing concerning whether your data is as predicted. To determine if the expected and observed results are well-fitted, the Chi-square test analyzes the differences between categorical variables from a random sample. The test's fundamental premise is that the observed values in your data should be compared to the predicted values that would be present if the null hypothesis were true. Hypothesis Testing and Confidence Intervals Both confidence intervals and hypothesis tests are inferential techniques that depend on approximating the sample distribution. Data from a sample is used to estimate a population parameter using confidence intervals. Data from a sample is used in hypothesis testing to examine a given hypothesis. We must have a postulated parameter to conduct hypothesis testing. Bootstrap distributions and randomization distributions are created using comparable simulation techniques. The observed sample statistic is the focal point of a bootstrap distribution, whereas the null hypothesis value is the focal point of a randomization distribution. A variety of feasible population parameter estimates are included in confidence ranges. In this lesson, we created just two-tailed confidence intervals. There is a direct connection between these two-tail confidence intervals and these two-tail hypothesis tests. The results of a two-tailed hypothesis test and two-tailed confidence intervals typically provide the same results. In other words, a hypothesis test at the 0.05 level will virtually always fail to reject the null hypothesis if the 95% confidence interval contains the predicted value. A hypothesis test at the 0.05 level will nearly certainly reject the null hypothesis if the 95% confidence interval does not include the hypothesized parameter. Become a Data Scientist through hands-on learning with hackathons, masterclasses, webinars, and Ask-Me-Anything! Start learning now! Simple and Composite Hypothesis Testing Depending on the population distribution, you can classify the statistical hypothesis into two types. Simple Hypothesis: A simple hypothesis specifies an exact value for the parameter. Composite Hypothesis: A composite hypothesis specifies a range of values. A company is claiming that their average sales for this quarter are 1000 units. This is an example of a simple hypothesis. Suppose the company claims that the sales are in the range of 900 to 1000 units. Then this is a case of a composite hypothesis. One-Tailed and Two-Tailed Hypothesis Testing The One-Tailed test, also called a directional test, considers a critical region of data that would result in the null hypothesis being rejected if the test sample falls into it, inevitably meaning the acceptance of the alternate hypothesis. In a one-tailed test, the critical distribution area is one-sided, meaning the test sample is either greater or lesser than a specific value. In two tails, the test sample is checked to be greater or less than a range of values in a Two-Tailed test, implying that the critical distribution area is two-sided. If the sample falls within this range, the alternate hypothesis will be accepted, and the null hypothesis will be rejected. Become a Data Scientist With Real-World Experience Right Tailed Hypothesis Testing If the larger than (>) sign appears in your hypothesis statement, you are using a right-tailed test, also known as an upper test. Or, to put it another way, the disparity is to the right. For instance, you can contrast the battery life before and after a change in production. Your hypothesis statements can be the following if you want to know if the battery life is longer than the original (let's say 90 hours): The null hypothesis is (H0 <= 90) or less change. A possibility is that battery life has risen (H1) > 90. The crucial point in this situation is that the alternate hypothesis (H1), not the null hypothesis, decides whether you get a right-tailed test. Left Tailed Hypothesis Testing Alternative hypotheses that assert the true value of a parameter is lower than the null hypothesis are tested with a left-tailed test; they are indicated by the asterisk "<". Suppose H0: mean = 50 and H1: mean not equal to 50 According to the H1, the mean can be greater than or less than 50. This is an example of a Two-tailed test. In a similar manner, if H0: mean >=50, then H1: mean <50 Here the mean is less than 50. It is called a One-tailed test. Type 1 and Type 2 Error A hypothesis test can result in two types of errors. Type 1 Error: A Type-I error occurs when sample results reject the null hypothesis despite being true. Type 2 Error: A Type-II error occurs when the null hypothesis is not rejected when it is false, unlike a Type-I error. Suppose a teacher evaluates the examination paper to decide whether a student passes or fails. H0: Student has passed H1: Student has failed Type I error will be the teacher failing the student [rejects H0] although the student scored the passing marks [H0 was true].  Type II error will be the case where the teacher passes the student [do not reject H0] although the student did not score the passing marks [H1 is true]. Our Data Scientist Master's Program covers core topics such as R, Python, Machine Learning, Tableau, Hadoop, and Spark. Get started on your journey today! Limitations of Hypothesis Testing Hypothesis testing has some limitations that researchers should be aware of: It cannot prove or establish the truth: Hypothesis testing provides evidence to support or reject a hypothesis, but it cannot confirm the absolute truth of the research question. Results are sample-specific: Hypothesis testing is based on analyzing a sample from a population, and the conclusions drawn are specific to that particular sample. Possible errors: During hypothesis testing, there is a chance of committing type I error (rejecting a true null hypothesis) or type II error (failing to reject a false null hypothesis). Assumptions and requirements: Different tests have specific assumptions and requirements that must be met to accurately interpret results. Learn All The Tricks Of The BI Trade After reading this tutorial, you would have a much better understanding of hypothesis testing, one of the most important concepts in the field of Data Science . The majority of hypotheses are based on speculation about observed behavior, natural phenomena, or established theories. If you are interested in statistics of data science and skills needed for such a career, you ought to explore the Post Graduate Program in Data Science. If you have any questions regarding this ‘Hypothesis Testing In Statistics’ tutorial, do share them in the comment section. Our subject matter expert will respond to your queries. Happy learning! 1. What is hypothesis testing in statistics with example? Hypothesis testing is a statistical method used to determine if there is enough evidence in a sample data to draw conclusions about a population. It involves formulating two competing hypotheses, the null hypothesis (H0) and the alternative hypothesis (Ha), and then collecting data to assess the evidence. An example: testing if a new drug improves patient recovery (Ha) compared to the standard treatment (H0) based on collected patient data. 2. What is H0 and H1 in statistics? In statistics, H0​ and H1​ represent the null and alternative hypotheses. The null hypothesis, H0​, is the default assumption that no effect or difference exists between groups or conditions. The alternative hypothesis, H1​, is the competing claim suggesting an effect or a difference. Statistical tests determine whether to reject the null hypothesis in favor of the alternative hypothesis based on the data. 3. What is a simple hypothesis with an example? A simple hypothesis is a specific statement predicting a single relationship between two variables. It posits a direct and uncomplicated outcome. For example, a simple hypothesis might state, "Increased sunlight exposure increases the growth rate of sunflowers." Here, the hypothesis suggests a direct relationship between the amount of sunlight (independent variable) and the growth rate of sunflowers (dependent variable), with no additional variables considered. 4. What are the 3 major types of hypothesis? The three major types of hypotheses are: Null Hypothesis (H0): Represents the default assumption, stating that there is no significant effect or relationship in the data. Alternative Hypothesis (Ha): Contradicts the null hypothesis and proposes a specific effect or relationship that researchers want to investigate. Nondirectional Hypothesis: An alternative hypothesis that doesn't specify the direction of the effect, leaving it open for both positive and negative possibilities. Find our PL-300 Microsoft Power BI Certification Training Online Classroom training classes in top cities: Name Date Place 7 Sep -22 Sep 2024, Weekend batch Your City 21 Sep -6 Oct 2024, Weekend batch Your City 12 Oct -27 Oct 2024, Weekend batch Your City About the Author Avijeet is a Senior Research Analyst at Simplilearn. Passionate about Data Analytics, Machine Learning, and Deep Learning, Avijeet is also interested in politics, cricket, and football. Recommended Resources Free eBook: Top Programming Languages For A Data Scientist Normality Test in Minitab: Minitab with Statistics Machine Learning Career Guide: A Playbook to Becoming a Machine Learning Engineer PMP, PMI, PMBOK, CAPM, PgMP, PfMP, ACP, PBA, RMP, SP, and OPM3 are registered marks of the Project Management Institute, Inc. Search Search Please fill out this field. What Is Hypothesis Testing? How It Works 4 Step Process The bottom line. Fundamental Analysis Hypothesis Testing: 4 Steps and Example Hypothesis testing, sometimes called significance testing, is an act in statistics whereby an analyst tests an assumption regarding a population parameter. The methodology employed by the analyst depends on the nature of the data used and the reason for the analysis. Hypothesis testing is used to assess the plausibility of a hypothesis by using sample data. Such data may come from a larger population or a data-generating process. The word "population" will be used for both of these cases in the following descriptions. Key Takeaways Hypothesis testing is used to assess the plausibility of a hypothesis by using sample data. The test provides evidence concerning the plausibility of the hypothesis, given the data. Statistical analysts test a hypothesis by measuring and examining a random sample of the population being analyzed. The four steps of hypothesis testing include stating the hypotheses, formulating an analysis plan, analyzing the sample data, and analyzing the result. How Hypothesis Testing Works In hypothesis testing, an  analyst  tests a statistical sample, intending to provide evidence on the plausibility of the null hypothesis. Statistical analysts measure and examine a random sample of the population being analyzed. All analysts use a random population sample to test two different hypotheses: the null hypothesis and the alternative hypothesis. The null hypothesis is usually a hypothesis of equality between population parameters; e.g., a null hypothesis may state that the population mean return is equal to zero. The alternative hypothesis is effectively the opposite of a null hypothesis. Thus, they are mutually exclusive , and only one can be true. However, one of the two hypotheses will always be true. The null hypothesis is a statement about a population parameter, such as the population mean, that is assumed to be true. State the hypotheses. Formulate an analysis plan, which outlines how the data will be evaluated. Carry out the plan and analyze the sample data. Analyze the results and either reject the null hypothesis, or state that the null hypothesis is plausible, given the data. Example of Hypothesis Testing If an individual wants to test that a penny has exactly a 50% chance of landing on heads, the null hypothesis would be that 50% is correct, and the alternative hypothesis would be that 50% is not correct. Mathematically, the null hypothesis is represented as Ho: P = 0.5. The alternative hypothesis is shown as "Ha" and is identical to the null hypothesis, except with the equal sign struck-through, meaning that it does not equal 50%. A random sample of 100 coin flips is taken, and the null hypothesis is tested. If it is found that the 100 coin flips were distributed as 40 heads and 60 tails, the analyst would assume that a penny does not have a 50% chance of landing on heads and would reject the null hypothesis and accept the alternative hypothesis. If there were 48 heads and 52 tails, then it is plausible that the coin could be fair and still produce such a result. In cases such as this where the null hypothesis is "accepted," the analyst states that the difference between the expected results (50 heads and 50 tails) and the observed results (48 heads and 52 tails) is "explainable by chance alone." When Did Hypothesis Testing Begin? Some statisticians attribute the first hypothesis tests to satirical writer John Arbuthnot in 1710, who studied male and female births in England after observing that in nearly every year, male births exceeded female births by a slight proportion. Arbuthnot calculated that the probability of this happening by chance was small, and therefore it was due to “divine providence.” What are the Benefits of Hypothesis Testing? Hypothesis testing helps assess the accuracy of new ideas or theories by testing them against data. This allows researchers to determine whether the evidence supports their hypothesis, helping to avoid false claims and conclusions. Hypothesis testing also provides a framework for decision-making based on data rather than personal opinions or biases. By relying on statistical analysis, hypothesis testing helps to reduce the effects of chance and confounding variables, providing a robust framework for making informed conclusions. What are the Limitations of Hypothesis Testing? Hypothesis testing relies exclusively on data and doesn’t provide a comprehensive understanding of the subject being studied. Additionally, the accuracy of the results depends on the quality of the available data and the statistical methods used. Inaccurate data or inappropriate hypothesis formulation may lead to incorrect conclusions or failed tests. Hypothesis testing can also lead to errors, such as analysts either accepting or rejecting a null hypothesis when they shouldn’t have. These errors may result in false conclusions or missed opportunities to identify significant patterns or relationships in the data. Hypothesis testing refers to a statistical process that helps researchers determine the reliability of a study. By using a well-formulated hypothesis and set of statistical tests, individuals or businesses can make inferences about the population that they are studying and draw conclusions based on the data presented. All hypothesis testing methods have the same four-step process, which includes stating the hypotheses, formulating an analysis plan, analyzing the sample data, and analyzing the result. Sage. " Introduction to Hypothesis Testing ," Page 4. Elder Research. " Who Invented the Null Hypothesis? " Formplus. " Hypothesis Testing: Definition, Uses, Limitations and Examples ." Terms of Service Editorial Policy Privacy Policy Pardon Our Interruption As you were browsing something about your browser made us think you were a bot. There are a few reasons this might happen: You've disabled JavaScript in your web browser. You're a power user moving through this website with super-human speed. You've disabled cookies in your web browser. A third-party browser plugin, such as Ghostery or NoScript, is preventing JavaScript from running. Additional information is available in this support article . To regain access, please make sure that cookies and JavaScript are enabled before reloading the page. Browser not supported This probably isn't the experience you were expecting. Internet Explorer isn't supported on Uber.com. Try switching to a different browser to view our site. Come reimagine with us Shifting E2E Testing Left at Uber A few years ago, Uber primarily relied on incremental rollout and production probing/alerts to catch regressions. While this approach is sound, it became very operationally expensive and we also experienced many leakages. Detecting issues late requires developers to bisect the exact bad change and then go back through the whole process again. Many of our outage postmortems indicated little to no testing beyond a few basic unit tests, which were often so dependent on mocks that it was difficult to understand how much protection they actually offered. Uber is well-known for having fully embraced microservice-based architecture. Our core business logic is encapsulated across large groups of microservices such that it is difficult to validate functionality within a single service without over-mocking. Because of our architecture, the only sane way to test has been to perform E2E testing. At the same time, our internal NPS surveys consistently highlighted that E2E testing was the hardest part of the job for developers – it’s no surprise it was often skipped.  A well-known testing blog from 2015, Just Say No To More End-To-End Tests , calls out E2E tests as difficult to maintain, expensive to write, flaky, slow, and difficult to debug.  Uber’s journey into testing has run into all of the above problems and we’ve had to come up with creative ways to solve them. In this blog, we describe how we built a system that gates every code and configuration change to our core backend systems (1,000+ services). We have several thousand E2E tests that have an average pass rate of 90%+ per attempt. Imagine each of these tests going through a real E2E user flow, like going through an Uber Eats group order. We do all this fast enough to run on every diff before it gets landed. Prior Approaches Docker compose. In the past, some teams at Uber have tried to start up all relevant services and datastores locally to run a suite of integration tests against the stack. At a certain point of microservice complexity, this becomes prohibitively expensive with the number of containers that need to be started.  Deploy to staging At Uber, shared staging became too difficult to keep reliably usable for testing, because a single developer deploying a bad change could break the whole thing for everyone else. This approach only ever worked in small domains where developers could manually coordinate rollouts between a few teams. Could E2E tests really work well at Uber scale? This is the story of how we made it happen. Shift Left with BITS – Uber’s Current Testing Strategy We use bits to slice production. In order to shift testing left, we need to enable the ability to test changes without first deploying to production. To solve this, we launched a company-wide initiative called BITS (Backend Integration Testing Strategy) that enables on-demand deployment and routing to test sandboxes. Practically, this means individual commits can be tested in parallel before landing, without interfering with anyone else! Data Isolation To shift testing left safely , we need to provide isolation between production and testing traffic. At Uber, production services are expected to receive both testing and production traffic. User Context Most APIs are entity-scoped, meaning that access is scoped to a particular entity (i.e. user account). Using test users automatically limits the scope of side effects to that account. A small handful of cross-entity interactions (think algorithmic matching or Eats shopping) use the below strategies for isolation. Architecture BITS’ architecture can be expressed as a series of workflows that orchestrate between different infrastructure components to define a meaningful experience for the developer. We use Cadence , an open-source workflow engine developed at Uber that gives us a programming model for expressing retries, workflow state, and timers for teardown of resources. Our CI workflows run test/service selection algorithms, build and deploy test sandboxes, schedule tests, analyze results, and report results. Our provisioning workflows control container resource utilization and scheduling of workloads across different CI queues optimized for particular workload types (i.e., high network usage vs. CPU usage). *We’ve previously discussed SLATE , a sister project built on top of BITS infrastructure primitives for performing manual sanity testing. Check out SLATE if you haven’t already seen it! Infrastructure Isolation We’ve done a lot of work to properly isolate BITS’ development containers against production side effects. BITS test sandboxes do not receive any production traffic. The native Apache Kafka™ protocol at Uber is abstracted by a consumer proxy that remaps messages into a push-based forwarding model over gRPC. This design allows for intelligent routing of messages to BITS test sandboxes and also prevents test sandboxes from inadvertently polling production topics. Cadence workflows created from testing are isolated to the test sandbox. Metrics and logs are tagged with tenancy, so that they can be filtered out. Test sandboxes do not participate in production leader election protocols. SPIFFE ™ and SPIRE™ provide strongly attested identities to BITS sandboxes and test traffic. Uber’s forwarding proxy inspects requests and performs a conditional P2P redirect if routing overrides are detected. Testing Configuration Changes At Uber, configuration changes cause up to 30% of incidents. BITS also provides test coverage for configuration rollouts made in our largest backend config management system, Flipr. Test Management  Developers are able to manage their test suites in real time from our UI to perform actions like downtiming tests, tracking their health, and checking endpoint coverage of their services. Trace Indexing Every test execution is force-sampled with Jaeger ™. After each test execution, we capture the “trace” and use it to build several indexes: What services and endpoints were covered by the test flow What tests cover each service and endpoint These indexes are used to surface endpoint coverage metrics for teams, but more importantly, they allow us to intelligently determine which tests need to run when a given service is changed. Getting the infrastructure right has always been the easier part of the problem of testing. Machine problems can be solved, but developers at Uber are very sensitive to degradations in productivity and experience. The below captures some of our strategies  around mitigating problems with test maintenance, reliability, and signal. Challenge #1: Maintenance, state, and fixture management  Let’s say that you want to test forward dispatch, where a driver gets a new pickup request before they complete their current dropoff. Or shared rides, where you may or may not pick up another rider on the way to your destination. How do you represent the state of the trip in each situation? Composable Testing Framework (CTF) A foundational building block we built is CTF, a code-level DSL that provides a programming model for composing individual actions and flows and modeling their side effects on centralized trip state. CTF was originally developed to validate Uber’s Fulfillment Re-architecture and has since been adopted across the company. Test Cataloging Every test case is automatically registered in our datastore on land. Analytics on historical pass rate, common failure reasons, and ownership information are all available to developers who can track the stability of their tests over time. Common failures are aggregated so that they can be investigated more easily. Challenge #2: Reliability and Speed Test authors get their tests to pass at ≥90% per attempt (against main). Our platform then boosts this to 99.9% with retries. Our largest service has 300 tests. If tests individually pass at 90% per attempt, there is a 26% chance of one test failing due to non-determinism. In this case, we got the 300 tests to pass at 95% and with retries boost the signal up to ≥99%. The above is repeated thousands of times a day, debunking the myth that E2E tests can’t be stable. Our tests directly hit Uber APIs, so the majority of them run in sub-minute time. Challenge #3: Debuggability and Actionability Placebo Executions On every test run, we kick off a “placebo” parallel execution against a sandbox running the latest main branch version of code. The combination of these two executions allows us to move beyond a binary signal into a truth table: With retries, non-determinism is rare. Either production or your test is broken, or you’ve introduced a real defect. No matter what, the developer has a clear action. Automated quarantining of tests The pass rate of our tests follows a bimodal distribution–the vast majority of tests are very reliable. A small subset of tests are broken (consistently fail). An even smaller subset of tests pass non-deterministically. A single broken test can block CI/CD, so we take special care to guard against this. Once a test drops below 90% placebo pass rate, it is automatically marked non-blocking and a ticket and alert are automatically filed against the owning team. Our core flow and compliance tests are explicitly opted out of this behavior, given their importance to the business. At the end of the day, testing requires a balance between quality and velocity. When tuned well, developers spend less time rolling back deployments and mitigating incidents. Both velocity and quality go up. As part of our testing efforts, we tracked incidents per 1,000 diffs and we reduced this by 71% in 2023. Testing is a people problem as much as it is a technical problem Microservice architecture was born out of a desire to empower independent teams in a growing engineering organization. In a perfect world, we would have clean API abstractions everywhere to reinforce this model. Teams would be able to build AND test without impacting other teams. In reality, shipping most features at Uber requires heavy collaboration–E2E testing is a contract that enforces communication across independent teams, which inevitably breaks down because we’re human! Testing and Architecture are intrinsically linked No one at Uber intentionally planned for heavy reliance on E2E testing, but our microservice-heavy architecture and historical architectural investments in multi-tenancy paved the way to make this the best possible solution. Trying to fit our testing strategy to the classic “Testing Pyramid” didn’t work for us at Uber. Good testing requires you to think critically about what you’re actually trying to validate. You should try to test the smallest set of modules that serve some meaningful user functionality. At Uber, that just happens to often involve dozens of services.  Acknowledgments We are thankful to a dozen+ teams who have collaborated with us to make testing a first-class citizen in Uber Infrastructure, as well as all the various leaders who believed in the BITS initiative. Apache®, Apache Kafka, Kafka, and the star logo are either registered trademarks or trademarks of the Apache Software Foundation in the United States and/or other countries. No endorsement by The Apache Software Foundation is implied by the use of these marks. SPIFFE, OpenTelemetry and its logo are registered trademarks of The Cloud Native Computing Foundation® in the United States and other countries. No endorsement by The Cloud Native Computing Foundation is implied by the use of these marks. Jaeger and its logo are registered trademarks of The Linux Foundation® in the United States and other countries. No endorsement by The Linux Foundation is implied by the use of these marks. Quess Liu is a Senior Staff Engineer on the Developer Platform team. Over the past 3 years, he has led efforts to redefine microservice development and testing, with a particular focus on evolving Uber’s infrastructure and architecture to be more testable.. Daniel Tsui Daniel Tsui is a Senior Engineering Manager on the Developer Platform team. Over the past 9 years, He has built Uber’s testing infrastructure from scratch and has led his teams to shift testing left at scale. 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Preguntas frecuentes: estado del retorno de la prueba de vuelo tripulado boeing de la nasa. Astronauta de la NASA Frank Rubio Diez maneras en que los estudiantes pueden prepararse para ser astronautas Leadership to discuss nasa’s boeing crew flight test. Abbey A. Donaldson Nasa headquarters. NASA Administrator Bill Nelson and leadership will hold an internal Agency Test Flight Readiness Review on Saturday, Aug. 24, for NASA’s Boeing Crew Flight Test. About an hour later, NASA will host a live news conference at 1 p.m. EDT from the agency’s Johnson Space Center in Houston. Watch the media event on NASA+ , NASA Television, the NASA app , YouTube , and the agency’s website . Learn how to stream NASA content through a variety of platforms, including social media. Media interested in attending the news conference must contact the newsroom at NASA Johnson no later than 1 p.m., Friday, Aug. 23, at 281-483-5111 or [email protected] . Media participating by phone must RSVP no later than one hour prior to the start of the event. A copy of NASA’s media accreditation policy is online. NASA and Boeing have gathered data, both in space and on the ground, regarding the Starliner spacecraft’s propulsion and helium systems to better understand the ongoing technical challenges. The review will include a mission status update, review of technical data and closeout actions, as well as certify flight rationale to proceed with undocking and return from the space station. NASA’s Boeing Crew Flight Test launched on June 5 on a ULA (United Launch Alliance) Atlas V rocket from Space Launch Complex-41 at Cape Canaveral Space Force Station in Florida. It is an end-to-end test of the Starliner system as part of the agency’s Commercial Crew Program. Through partnership with American private industry, NASA is opening access to low Earth orbit and the space station to more people, science, and commercial opportunities. For NASA’s blog and more information about the mission, visit: https://www.nasa.gov/commercialcrew Meira Bernstein / Josh Finch Headquarters, Washington 202-358-1100 [email protected] / [email protected] Leah Cheshier / Sandra Jones Johnson Space Center, Houston 281-483-5111 [email protected] / [email protected] Steve Siceloff / Danielle Sempsrott / Stephanie Plucinsky Kennedy Space Center, Florida 321-867-2468 [email protected] / [email protected] / [email protected] 387 IMAGES PPT Hypothesis Testing Solved Examples(Questions and Solutions) Hypothesis Testing : Infographics PPT PPT PPT COMMENTS How to Write Hypothesis Test Conclusions (With Examples) When writing the conclusion of a hypothesis test, we typically include: Whether we reject or fail to reject the null hypothesis. The significance level. A short explanation in the context of the hypothesis test. For example, we would write: We reject the null hypothesis at the 5% significance level. Hypothesis Testing Present the findings in your results and discussion section. Though the specific details might vary, the procedure you will use when testing a hypothesis will always follow some version of these steps. Table of contents. Step 1: State your null and alternate hypothesis. Step 2: Collect data. Step 3: Perform a statistical test. How to State the Conclusion about a Hypothesis Test The best way to state the conclusion is to include the significance level of the test and a bit about the claim itself. For example, if the claim was the alternative that the mean score on a test was greater than 85, and your decision was to Reject then Null, then you could conclude: " At the 5% significance level, there is sufficient ... How to Write Hypothesis Test Conclusions (With Examples) A hypothesis test is used to test whether or not some hypothesis about a population parameter is true.. To perform a hypothesis test in the real world, researchers obtain a random sample from the population and perform a hypothesis test on the sample data, using a null and alternative hypothesis:. Null Hypothesis (H 0): The sample data occurs purely from chance. 6a.2 Below these are summarized into six such steps to conducting a test of a hypothesis. Set up the hypotheses and check conditions: Each hypothesis test includes two hypotheses about the population. One is the null hypothesis, notated as H 0, which is a statement of a particular parameter value. This hypothesis is assumed to be true until there is ... Hypothesis Testing Hypothesis Testing Step 1: State the Hypotheses. In all three examples, our aim is to decide between two opposing points of view, Claim 1 and Claim 2. ... One can actually draw conclusions in hypothesis testing just using the test statistic, and as we'll see the p-value is, in a sense, just another way of looking at the test statistic. ... Statistical Hypothesis Testing Overview Hypothesis testing is a crucial procedure to perform when you want to make inferences about a population using a random sample. These inferences include estimating population properties such as the mean, differences between means, proportions, and the relationships between variables. This post provides an overview of statistical hypothesis testing. Hypothesis Testing: Uses, Steps & Example The researchers write their hypotheses. These statements apply to the population, so they use the mu (μ) symbol for the population mean parameter.. Null Hypothesis (H 0): The population means of the test scores for the two groups are equal (μ 1 = μ 2).; Alternative Hypothesis (H A): The population means of the test scores for the two groups are unequal (μ 1 ≠ μ 2). 1.2 Step 7: Based on Steps 5 and 6, draw a conclusion about H 0. If F calculated is larger than F α, then you are in the rejection region and you can reject the null hypothesis with ( 1 − α) level of confidence. Note that modern statistical software condenses Steps 6 and 7 by providing a p -value. The p -value here is the probability of getting ... 9.1: Introduction to Hypothesis Testing In hypothesis testing, the goal is to see if there is sufficient statistical evidence to reject a presumed null hypothesis in favor of a conjectured alternative hypothesis.The null hypothesis is usually denoted \(H_0\) while the alternative hypothesis is usually denoted \(H_1\). An hypothesis test is a statistical decision; the conclusion will either be to reject the null hypothesis in favor ... Chapter 3: Hypothesis Testing Components of a Formal Hypothesis Test. The null hypothesis is a statement about the value of a population parameter, such as the population mean (µ) or the population proportion (p).It contains the condition of equality and is denoted as H 0 (H-naught).. H 0: µ = 157 or H 0: p = 0.37. The alternative hypothesis is the claim to be tested, the opposite of the null hypothesis. Statistical hypothesis test A statistical hypothesis test is a method of statistical inference used to decide whether the data sufficiently supports a particular hypothesis. ... This is equally true of hypothesis testing which can justify conclusions even when no scientific theory exists. ... Statistical hypothesis: A statement about the parameters describing a population ... S.3 Hypothesis Testing S.3 Hypothesis Testing. In reviewing hypothesis tests, we start first with the general idea. Then, we keep returning to the basic procedures of hypothesis testing, each time adding a little more detail. The general idea of hypothesis testing involves: Making an initial assumption. Collecting evidence (data). How Hypothesis Tests Work: Significance Levels (Alpha) and P values Using P values and Significance Levels Together. If your P value is less than or equal to your alpha level, reject the null hypothesis. The P value results are consistent with our graphical representation. The P value of 0.03112 is significant at the alpha level of 0.05 but not 0.01. Hypothesis Testing: Principles and Methods Hypothesis testing is a fundamental tool used in scientific research to validate or reject hypotheses about population parameters based on sample data. It provides a structured framework for evaluating the statistical significance of a hypothesis and drawing conclusions about the true nature of a population. Hypothesis testing is widely used in fields such as biology, psychology, economics ... Guide: Hypothesis Testing Hypothesis testing is a structured procedure in statistics used for drawing conclusions about a larger population based on a subset of that population, known as a sample. ... By following this step-by-step guide, you've journeyed through the rigorous yet enlightening process of hypothesis testing. From stating clear hypotheses to interpreting ... 7.1: Basics of Hypothesis Testing Test Statistic: z = ¯ x − μo σ / √n since it is calculated as part of the testing of the hypothesis. Definition 7.1.4. p - value: probability that the test statistic will take on more extreme values than the observed test statistic, given that the null hypothesis is true. Everything You Need To Know about Hypothesis Testing It is the interpretation of the data that we are interested in. Using Hypothesis Testing, we try to interpret or draw conclusions about the population using sample data. A Hypothesis Test evaluates two mutually exclusive statements about a population to determine which statement is best supported by the sample data. T-test and Hypothesis Testing (Explained Simply) Aug 5, 2022. 6. Photo by Andrew George on Unsplash. Student's t-tests are commonly used in inferential statistics for testing a hypothesis on the basis of a difference between sample means. However, people often misinterpret the results of t-tests, which leads to false research findings and a lack of reproducibility of studies. Hypothesis Testing for Means & Proportions We then determine the appropriate test statistic (Step 2) for the hypothesis test. The formula for the test statistic is given below. Test Statistic for Testing H0: p = p 0. if min (np 0 , n (1-p 0 )) > 5. The formula above is appropriate for large samples, defined when the smaller of np 0 and n (1-p 0) is at least 5. Hypothesis Testing in Statistics In other words, a hypothesis test at the 0.05 level will virtually always fail to reject the null hypothesis if the 95% confidence interval contains the predicted value. A hypothesis test at the 0.05 level will nearly certainly reject the null hypothesis if the 95% confidence interval does not include the hypothesized parameter. Hypothesis Testing: 4 Steps and Example Hypothesis testing is an act in statistics whereby an analyst tests an assumption regarding a population parameter. The methodology employed by the analyst depends on the nature of the data used ... Making Valid Conclusions about a Hypothesis Test for a Mean Make a valid conclusion based on this p-value. Step 1: State the null and alternative hypotheses for the mean. The null hypothesis is that the mean vehicle price is equal to $15,000. The ... Mastering Hypothesis Testing: 8 Steps Decoded 6. Calculate the p-value for this hypothesis test, and state the hypothesis conclusion based on the p-value. Does this match your results from the critical value method? Answer and Explanation: Enter your step-by-step answer and explanations here. To determine if I should reject or not reject the null hypothesis, I will need to compare my test statistic and my critical value. Shifting E2E Testing Left at Uber Our CI workflows run test/service selection algorithms, build and deploy test sandboxes, schedule tests, analyze results, and report results. Our provisioning workflows control container resource utilization and scheduling of workloads across different CI queues optimized for particular workload types (i.e., high network usage vs. CPU usage). Leadership to Discuss NASA's Boeing Crew Flight Test NASA's Boeing Crew Flight Test launched on June 5 on a ULA (United Launch Alliance) Atlas V rocket from Space Launch Complex-41 at Cape Canaveral Space Force Station in Florida. It is an end-to-end test of the Starliner system as part of the agency's Commercial Crew Program. Through partnership with American private industry, NASA is ... essay homework paper phd project resume review study thesis