null hypothesis symbol in google docs

Lukas Kollmer

List of mathematical symbol shortcuts supported in google docs equations, greek letters, miscellaneous operations, punctuation, linear algebra, probability theory, trigonometry.

The Google Docs equation editor allows entering certain mathematical symbols and operations via a LaTeX \LaTeX L A T E ​ X -style command syntax.

In addition to the symbols listed in the various dropdown menus in the equation toolbar, there are also several other undocumented commands recognized by Google Docs.

This page is an attempt at compiling a complete list of all commands that can be used in an equation. As of August 2020, there's a total of 173 supported commands I know of.

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How to insert symbols into google docs and slides.

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You can insert special characters in your documents and presentations without having to remember all those Alt-codes by using Google Docs and Slides easy-to-use character insertion tool. It offers a myriad of symbols, characters, symbols, languages, and more. Here's how you can insert special characters into your documents.

Note:  You can't insert characters directly into Google Sheets, but you can copy and paste them into a cell on the spreadsheet.

Inserting symbols into your file is a straightforward process that you can perform in several ways. Whether you want some silly emojis, arrows, or a different language's scripts you can achieve this by manually selecting a category, typing in the search bar, or by drawing what you're looking for.

The first thing you'll need to do is open up a new   Google Docs  or Slides file to get started.

Alternatively, if you're using the latest version of Chrome, you can type "docs.new" or "slides.new" into a new tab's address bar.

In your document, open the "Insert" menu and then click the "Special Characters" command.

Manually Search for Symbols

If you don't have a particular character in mind (or you're not sure how to search for what you do have in mind), you can use the drop-down menus to browse through the plethora of available symbols.

Click the second drop-down menu to choose a category. You can choose from symbols, punctuation, emojis, different language's scripts, and even different whitespace characters. There are a lot, so be prepared to spend some time browsing.

Next, click on the other drop-down menu to refine the characters even further.

Once you've chosen the categories, all you need to do is click the character you want to insert it into your file.

Use the Search Bar

If you know what you're looking for you can use the search bar located to the right of the pop-up window. You can search by keyword, description, or by Unicode value--if you know it.

Using the search bar can prove a bit troublesome as searching for an emoji with a smile didn't produce the intended results. This is because it uses the word to match the description of the character.

If you search "Smiling" instead, you get more results.

Still, searching for a symbol is usually faster than browsing all the menus to find one manually.

Draw a Your Character to Search

Finally, if both your attempts to find the correct character or symbol have turned up dry, you can try the draw feature that lets you sketch whatever you want.

Start drawing/writing in the box to the right of the window, and similar characters will appear in the pane to the left. You don't have to draw it all in one stroke, and you can keep adding to your drawing if it requires multiple gestures.

Once you're done, click the arrow in the bottom right corner to reset the box and start drawing the next one.

If you regularly use any these characters, you'll find them first drop-down menu under "Recent Characters."

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Hypothesis Testing with One Sample

31 Null and Alternative Hypotheses

[latexpage]

The actual test begins by considering two hypotheses . They are called the null hypothesis and the alternative hypothesis . These hypotheses contain opposing viewpoints.

H 0 : The null hypothesis: It is a statement of no difference between the variables—they are not related. This can often be considered the status quo and as a result if you cannot accept the null it requires some action.

H 1 : The alternative hypothesis: It is a claim about the population that is contradictory to H 0 and what we conclude when we reject H 0 . This is usually what the researcher is trying to prove.

Since the null and alternative hypotheses are contradictory, you must examine evidence to decide if you have enough evidence to reject the null hypothesis or not. The evidence is in the form of sample data.

After you have determined which hypothesis the sample supports, you make a decision. There are two options for a decision. They are “reject H 0 ” if the sample information favors the alternative hypothesis or “do not reject H 0 ” or “fail to reject H 0 ” if the sample information is insufficient to reject the null hypothesis.

Mathematical Symbols Used in H 0 and H 1 :

H 0 always has an equal symbol in it. H 1 never has a symbol with an equal in it. The choice of symbol depends on the wording of the hypothesis test. This practice is acceptable because we only make the decision to reject or not reject the null hypothesis.

Example 8.1

Statement : No more than 30% of the registered voters in Santa Clara County voted in the primary election.

From the statement above, identify the Null and Alternative Hypothesis.

Solution 8 .1

Claim (in symbols): p ≤ 0.30

Opposite of Claim: p > 0.30

Looking at the Claim and the Opposite, we first find the one that has an equal sign. In this example, it is the Claim because that is where we see the “≤” (it is read “less than or equal to” which is why we say it has the “equal” in it). This is what we convert into our null hypothesis H 0 . So p ≤ 0.30 becomes p = 0.30.

H 0 : p = 0.30

Looking again at the Claim and Opposite, find the the one that does not have an equal sign. In this example, it is the Opposite. This becomes our Alternative Hypothesis.

H 1 : p > 0.30

It will become important later on to remember whether the Null Hypothesis or the Alternative Hypothesis came from the Claim, so it is a good practice to associate them somehow. In the table below, I put the Claim and Opposite in the first column. Then, in the second column, I put the Null Hypothesis H 0 in the first row across from the Claim, because H 0 “came from” the Claim (ie the claim had the equal sign in it).

A medical trial is conducted to test a drug company’s claim that a new medicine reduces cholesterol by 25%. State the null and alternative hypotheses.

Fill in the correct symbol (=, ≠, ≥, <, ≤, >) for the claim, opposite, null and alternative hypotheses. Also fill in the blanks indicating where the null and alternative hypotheses would go, indicating their correspondence with the claim and the opposite.

Example 8.2

We want to test whether the mean GPA of students in American colleges is different from 2.0 (out of 4.0).

Solution 8.2

Notice that this time, as opposed to Example 8.1, our null hypothesis corresponds to the Opposite. This is because the claim did not have an equal sign in it; in fact, it had a “not equal” sign because the original statement used the phrase “different from”, which suggests “not equal”.

We want to test whether the mean height of eighth graders is 66 inches. State the null and alternative hypotheses.

Example 8.3

We want to test if college students take less than five years to graduate from college, on the average. The null and alternative hypotheses are: H 0 : μ ≥ 5 H 1 : μ < 5

Solution 8.3

Notice that in this example, we had to convert the symbol in our “Opposite” statement from “≥” to an equal symbol in our Null Hypothesis.

We want to test if it takes fewer than 45 minutes to teach a lesson plan. State the null and alternative hypotheses.

Create the table identifying the claim, opposite, null hypothesis and alternative hypothesis. Put the null and alternative hypothesis in the correct row to show correspondence with the claim and opposite.

Example 8.4

In an issue of U. S. News and World Report , an article on school standards stated that about half of all students in France, Germany, and Israel take advanced placement exams and a third pass. The same article stated that 6.6% of U.S. students take advanced placement exams and 4.4% pass. Test if the percentage of U.S. students who take advanced placement exams is more than 6.6%. State the null and alternative hypotheses. H 0 : p ≤ 0.066 H 1 : p > 0.066

Solution 8.4

On a state driver’s test, about 40% pass the test on the first try. We want to test if more than 40% pass on the first try.

Introductory Statistics with Google Sheets by jkesler is licensed under a Creative Commons Attribution 4.0 International License , except where otherwise noted.

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• Null and Alternative Hypotheses | Definitions & Examples

Null & Alternative Hypotheses | Definitions, Templates & Examples

Published on May 6, 2022 by Shaun Turney . Revised on June 22, 2023.

The null and alternative hypotheses are two competing claims that researchers weigh evidence for and against using a statistical test :

• Null hypothesis ( H 0 ): There’s no effect in the population .
• Alternative hypothesis ( H a or H 1 ) : There’s an effect in the population.

Table of contents

Answering your research question with hypotheses, what is a null hypothesis, what is an alternative hypothesis, similarities and differences between null and alternative hypotheses, how to write null and alternative hypotheses, other interesting articles, frequently asked questions.

The null and alternative hypotheses offer competing answers to your research question . When the research question asks “Does the independent variable affect the dependent variable?”:

• The null hypothesis ( H 0 ) answers “No, there’s no effect in the population.”
• The alternative hypothesis ( H a ) answers “Yes, there is an effect in the population.”

The null and alternative are always claims about the population. That’s because the goal of hypothesis testing is to make inferences about a population based on a sample . Often, we infer whether there’s an effect in the population by looking at differences between groups or relationships between variables in the sample. It’s critical for your research to write strong hypotheses .

You can use a statistical test to decide whether the evidence favors the null or alternative hypothesis. Each type of statistical test comes with a specific way of phrasing the null and alternative hypothesis. However, the hypotheses can also be phrased in a general way that applies to any test.

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The null hypothesis is the claim that there’s no effect in the population.

If the sample provides enough evidence against the claim that there’s no effect in the population ( p ≤ α), then we can reject the null hypothesis . Otherwise, we fail to reject the null hypothesis.

Although “fail to reject” may sound awkward, it’s the only wording that statisticians accept . Be careful not to say you “prove” or “accept” the null hypothesis.

Null hypotheses often include phrases such as “no effect,” “no difference,” or “no relationship.” When written in mathematical terms, they always include an equality (usually =, but sometimes ≥ or ≤).

You can never know with complete certainty whether there is an effect in the population. Some percentage of the time, your inference about the population will be incorrect. When you incorrectly reject the null hypothesis, it’s called a type I error . When you incorrectly fail to reject it, it’s a type II error.

Examples of null hypotheses

The table below gives examples of research questions and null hypotheses. There’s always more than one way to answer a research question, but these null hypotheses can help you get started.

*Note that some researchers prefer to always write the null hypothesis in terms of “no effect” and “=”. It would be fine to say that daily meditation has no effect on the incidence of depression and p 1 = p 2 .

The alternative hypothesis ( H a ) is the other answer to your research question . It claims that there’s an effect in the population.

Often, your alternative hypothesis is the same as your research hypothesis. In other words, it’s the claim that you expect or hope will be true.

The alternative hypothesis is the complement to the null hypothesis. Null and alternative hypotheses are exhaustive, meaning that together they cover every possible outcome. They are also mutually exclusive, meaning that only one can be true at a time.

Alternative hypotheses often include phrases such as “an effect,” “a difference,” or “a relationship.” When alternative hypotheses are written in mathematical terms, they always include an inequality (usually ≠, but sometimes < or >). As with null hypotheses, there are many acceptable ways to phrase an alternative hypothesis.

Examples of alternative hypotheses

The table below gives examples of research questions and alternative hypotheses to help you get started with formulating your own.

Null and alternative hypotheses are similar in some ways:

• They’re both answers to the research question.
• They both make claims about the population.
• They’re both evaluated by statistical tests.

However, there are important differences between the two types of hypotheses, summarized in the following table.

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To help you write your hypotheses, you can use the template sentences below. If you know which statistical test you’re going to use, you can use the test-specific template sentences. Otherwise, you can use the general template sentences.

General template sentences

The only thing you need to know to use these general template sentences are your dependent and independent variables. To write your research question, null hypothesis, and alternative hypothesis, fill in the following sentences with your variables:

Does independent variable affect dependent variable ?

• Null hypothesis ( H 0 ): Independent variable does not affect dependent variable.
• Alternative hypothesis ( H a ): Independent variable affects dependent variable.

Test-specific template sentences

Once you know the statistical test you’ll be using, you can write your hypotheses in a more precise and mathematical way specific to the test you chose. The table below provides template sentences for common statistical tests.

Note: The template sentences above assume that you’re performing one-tailed tests . One-tailed tests are appropriate for most studies.

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.

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.

The null hypothesis is often abbreviated as H 0 . When the null hypothesis is written using mathematical symbols, it always includes an equality symbol (usually =, but sometimes ≥ or ≤).

The alternative hypothesis is often abbreviated as H a or H 1 . When the alternative hypothesis is written using mathematical symbols, it always includes an inequality symbol (usually ≠, but sometimes < or >).

A research hypothesis is your proposed answer to your research question. The research hypothesis usually includes an explanation (“ x affects y because …”).

A statistical hypothesis, on the other hand, is a mathematical statement about a population parameter. Statistical hypotheses always come in pairs: the null and alternative hypotheses . In a well-designed study , the statistical hypotheses correspond logically to the research hypothesis.

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9.1 Null and Alternative Hypotheses

The actual test begins by considering two hypotheses . They are called the null hypothesis and the alternative hypothesis . These hypotheses contain opposing viewpoints.

H 0 : The null hypothesis: It is a statement of no difference between the variables—they are not related. This can often be considered the status quo and as a result if you cannot accept the null it requires some action.

H a : The alternative hypothesis: It is a claim about the population that is contradictory to H 0 and what we conclude when we reject H 0 . This is usually what the researcher is trying to prove.

Since the null and alternative hypotheses are contradictory, you must examine evidence to decide if you have enough evidence to reject the null hypothesis or not. The evidence is in the form of sample data.

After you have determined which hypothesis the sample supports, you make a decision. There are two options for a decision. They are "reject H 0 " if the sample information favors the alternative hypothesis or "do not reject H 0 " or "decline to reject H 0 " if the sample information is insufficient to reject the null hypothesis.

Mathematical Symbols Used in H 0 and H a :

H 0 always has a symbol with an equal in it. H a never has a symbol with an equal in it. The choice of symbol depends on the wording of the hypothesis test. However, be aware that many researchers (including one of the co-authors in research work) use = in the null hypothesis, even with > or < as the symbol in the alternative hypothesis. This practice is acceptable because we only make the decision to reject or not reject the null hypothesis.

Example 9.1

H 0 : No more than 30% of the registered voters in Santa Clara County voted in the primary election. p ≤ .30 H a : More than 30% of the registered voters in Santa Clara County voted in the primary election. p > 30

A medical trial is conducted to test whether or not a new medicine reduces cholesterol by 25%. State the null and alternative hypotheses.

Example 9.2

We want to test whether the mean GPA of students in American colleges is different from 2.0 (out of 4.0). The null and alternative hypotheses are: H 0 : μ = 2.0 H a : μ ≠ 2.0

We want to test whether the mean height of eighth graders is 66 inches. State the null and alternative hypotheses. Fill in the correct symbol (=, ≠, ≥, <, ≤, >) for the null and alternative hypotheses.

• H 0 : μ __ 66
• H a : μ __ 66

Example 9.3

We want to test if college students take less than five years to graduate from college, on the average. The null and alternative hypotheses are: H 0 : μ ≥ 5 H a : μ < 5

We want to test if it takes fewer than 45 minutes to teach a lesson plan. State the null and alternative hypotheses. Fill in the correct symbol ( =, ≠, ≥, <, ≤, >) for the null and alternative hypotheses.

• H 0 : μ __ 45
• H a : μ __ 45

Example 9.4

In an issue of U. S. News and World Report , an article on school standards stated that about half of all students in France, Germany, and Israel take advanced placement exams and a third pass. The same article stated that 6.6% of U.S. students take advanced placement exams and 4.4% pass. Test if the percentage of U.S. students who take advanced placement exams is more than 6.6%. State the null and alternative hypotheses. H 0 : p ≤ 0.066 H a : p > 0.066

On a state driver’s test, about 40% pass the test on the first try. We want to test if more than 40% pass on the first try. Fill in the correct symbol (=, ≠, ≥, <, ≤, >) for the null and alternative hypotheses.

• H 0 : p __ 0.40
• H a : p __ 0.40

Collaborative Exercise

Bring to class a newspaper, some news magazines, and some Internet articles . In groups, find articles from which your group can write null and alternative hypotheses. Discuss your hypotheses with the rest of the class.

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Want to cite, share, or modify this book? This book uses the Creative Commons Attribution License and you must attribute OpenStax.

Access for free at https://openstax.org/books/introductory-statistics-2e/pages/1-introduction

• Authors: Barbara Illowsky, Susan Dean
• Publisher/website: OpenStax
• Book title: Introductory Statistics 2e
• Publication date: Dec 13, 2023
• Location: Houston, Texas
• Book URL: https://openstax.org/books/introductory-statistics-2e/pages/1-introduction
• Section URL: https://openstax.org/books/introductory-statistics-2e/pages/9-1-null-and-alternative-hypotheses

© Jul 18, 2024 OpenStax. Textbook content produced by OpenStax is licensed under a Creative Commons Attribution License . The OpenStax name, OpenStax logo, OpenStax book covers, OpenStax CNX name, and OpenStax CNX logo are not subject to the Creative Commons license and may not be reproduced without the prior and express written consent of Rice University.

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Writing Null Hypotheses in Research and Statistics

Last Updated: September 2, 2024 Fact Checked

This article was co-authored by Joseph Quinones and by wikiHow staff writer, Jennifer Mueller, JD . Joseph Quinones is a Physics Teacher working at South Bronx Community Charter High School. Joseph specializes in astronomy and astrophysics and is interested in science education and science outreach, currently practicing ways to make physics accessible to more students with the goal of bringing more students of color into the STEM fields. He has experience working on Astrophysics research projects at the Museum of Natural History (AMNH). Joseph recieved his Bachelor's degree in Physics from Lehman College and his Masters in Physics Education from City College of New York (CCNY). He is also a member of a network called New York City Men Teach. There are 7 references cited in this article, which can be found at the bottom of the page. This article has been fact-checked, ensuring the accuracy of any cited facts and confirming the authority of its sources. This article has been viewed 29,437 times.

Are you working on a research project and struggling with how to write a null hypothesis? Well, you've come to the right place! Keep reading to learn everything you need to know about the null hypothesis, including a review of what it is, how it relates to your research question and your alternative hypothesis, as well as how to use it in different types of studies.

Things You Should Know

• Write a research null hypothesis as a statement that the studied variables have no relationship to each other, or that there's no difference between 2 groups.

• Adjust the format of your null hypothesis to match the statistical method you used to test it, such as using "mean" if you're comparing the mean between 2 groups.

What is a null hypothesis?

• Research hypothesis: States in plain language that there's no relationship between the 2 variables or there's no difference between the 2 groups being studied.
• Statistical hypothesis: States the predicted outcome of statistical analysis through a mathematical equation related to the statistical method you're using.

Examples of Null Hypotheses

Null Hypothesis vs. Alternative Hypothesis

• For example, your alternative hypothesis could state a positive correlation between 2 variables while your null hypothesis states there's no relationship. If there's a negative correlation, then both hypotheses are false.

• You need additional data or evidence to show that your alternative hypothesis is correct—proving the null hypothesis false is just the first step.
• In smaller studies, sometimes it's enough to show that there's some relationship and your hypothesis could be correct—you can leave the additional proof as an open question for other researchers to tackle.

How do I test a null hypothesis?

• Group means: Compare the mean of the variable in your sample with the mean of the variable in the general population. [6] X Research source
• Group proportions: Compare the proportion of the variable in your sample with the proportion of the variable in the general population. [7] X Research source
• Correlation: Correlation analysis looks at the relationship between 2 variables—specifically, whether they tend to happen together. [8] X Research source
• Regression: Regression analysis reveals the correlation between 2 variables while also controlling for the effect of other, interrelated variables. [9] X Research source

Templates for Null Hypotheses

• Research null hypothesis: There is no difference in the mean [dependent variable] between [group 1] and [group 2].

• Research null hypothesis: The proportion of [dependent variable] in [group 1] and [group 2] is the same.

• Research null hypothesis: There is no correlation between [independent variable] and [dependent variable] in the population.

• Research null hypothesis: There is no relationship between [independent variable] and [dependent variable] in the population.

Expert Q&A

You Might Also Like

Expert Interview

Thanks for reading our article! If you’d like to learn more about physics, check out our in-depth interview with Joseph Quinones .

• ↑ https://online.stat.psu.edu/stat100/lesson/10/10.1
• ↑ https://online.stat.psu.edu/stat501/lesson/2/2.12
• ↑ https://support.minitab.com/en-us/minitab/21/help-and-how-to/statistics/basic-statistics/supporting-topics/basics/null-and-alternative-hypotheses/
• ↑ https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5635437/
• ↑ https://online.stat.psu.edu/statprogram/reviews/statistical-concepts/hypothesis-testing
• ↑ https://education.arcus.chop.edu/null-hypothesis-testing/
• ↑ https://sphweb.bumc.bu.edu/otlt/mph-modules/bs/bs704_hypothesistest-means-proportions/bs704_hypothesistest-means-proportions_print.html

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Returns the probability associated with a Pearson’s chi-squared test on the two ranges of data. Determines the likelihood that the observed categorical data is drawn from an expected distribution.

Sample Usage

CHITEST(A1:A5, B1:B5)

CHITEST(A1:D3, A5:D7)

CHITEST(observed_range, expected_range)

observed_range - The counts associated with each category of data.

expected_range - The expected counts for each category under the null hypothesis.

observed_range and expected_range must both be ranges with the same number of rows and columns.

If any cell in either range is non-numeric, it and the corresponding cell in the other range do not count toward the calculation.

CHIDIST : Calculates the right-tailed chi-squared distribution, often used in hypothesis testing.

CHIINV : Calculates the inverse of the right-tailed chi-squared distribution.

CHISQ.DIST : Calculates the left-tailed chi-squared distribution, often used in hypothesis testing.

CHISQ.DIST.RT : Calculates the right-tailed chi-squared distribution, which is commonly used in hypothesis testing.

FTEST : Returns the probability associated with an F-test for equality of variances. Determines whether two samples are likely to have come from populations with the same variance.

TTEST : Returns the probability associated with t-test. Determines whether two samples are likely to have come from the same two underlying populations that have the same mean.

Suppose you want to test the fairness of a 6-sided die. You count the number of times each side is rolled for 60 trials, and compare it to an expected distribution where each side is rolled 10 times. There is only a 5.1% chance that the die is actually fair.

9.1 Null and Alternative Hypotheses

Introduction.

The actual test begins by considering two hypotheses . They are called the null hypothesis and the alternative hypothesis . These hypotheses contain opposing viewpoints.

H 0 — The null hypothesis: It is a statement of no difference between sample means or proportions or no difference between a sample mean or proportion and a population mean or proportion. In other words, the difference equals 0.

H a — The alternative hypothesis: It is a claim about the population that is contradictory to H 0 and what we conclude when we reject H 0 .

Since the null and alternative hypotheses are contradictory, you must examine evidence to decide if you have enough evidence to reject the null hypothesis or not. The evidence is in the form of sample data.

After you have determined which hypothesis the sample supports, you make a decision. There are two options for a decision. They are reject H 0 if the sample information favors the alternative hypothesis or do not reject H 0 or decline to reject H 0 if the sample information is insufficient to reject the null hypothesis.

Mathematical Symbols Used in H 0 and H a :

H 0 always has a symbol with an equal in it. H a never has a symbol with an equal in it. The choice of symbol depends on the wording of the hypothesis test. However, be aware that many researchers use = in the null hypothesis, even with > or < as the symbol in the alternative hypothesis. This practice is acceptable because we only make the decision to reject or not reject the null hypothesis.

Example 9.1

H 0 : No more than 30 percent of the registered voters in Santa Clara County voted in the primary election. p ≤ 30

A medical trial is conducted to test whether or not a new medicine reduces cholesterol by 25 percent. State the null and alternative hypotheses.

Example 9.2

We want to test whether the mean GPA of students in American colleges is different from 2.0 (out of 4.0). The null and alternative hypotheses are the following:

We want to test whether the mean height of eighth graders is 66 inches. State the null and alternative hypotheses. Fill in the correct symbol (=, ≠, ≥, <, ≤, >) for the null and alternative hypotheses.

• H 0 : μ __ 66
• H a : μ __ 66

Example 9.3

We want to test if college students take fewer than five years to graduate from college, on the average. The null and alternative hypotheses are the following:

We want to test if it takes fewer than 45 minutes to teach a lesson plan. State the null and alternative hypotheses. Fill in the correct symbol ( =, ≠, ≥, <, ≤, >) for the null and alternative hypotheses.

• H 0 : μ __ 45
• H a : μ __ 45

Example 9.4

An article on school standards stated that about half of all students in France, Germany, and Israel take advanced placement exams and a third of the students pass. The same article stated that 6.6 percent of U.S. students take advanced placement exams and 4.4 percent pass. Test if the percentage of U.S. students who take advanced placement exams is more than 6.6 percent. State the null and alternative hypotheses.

On a state driver’s test, about 40 percent pass the test on the first try. We want to test if more than 40 percent pass on the first try. Fill in the correct symbol (=, ≠, ≥, <, ≤, >) for the null and alternative hypotheses.

• H 0 : p __ 0.40
• H a : p __ 0.40

Collaborative Exercise

Bring to class a newspaper, some news magazines, and some internet articles. In groups, find articles from which your group can write null and alternative hypotheses. Discuss your hypotheses with the rest of the class.

Copy and paste the link code above.

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How to Insert the Null Hypothesis & Alternate Hypothesis Symbols in Microsoft Word

Although the symbols for the null hypothesis and alternative hypothesis -- sometimes called the alternate hypothesis -- do not exist as special characters in Microsoft Word, they are easily created with subscripts.The alternate hypothesis is symbolically represented by a capitalized "H," followed by a subscript "1," although some researchers prefer an "a." The null hypothesis is represented by a capitalized "H," followed by a subscript "0" or "o." The accepted practice in the scientific community is to use two hypotheses when testing the relationship between two events. The alternative hypothesis states that the two events are related. However, scientists have found that testing for a direct correlation can cause bias in the testing procedure. To avoid this bias, scientists test a null hypothesis that states there is no correlation. By disproving the null hypothesis, you imply a correlation in the alternate hypothesis. A similar system is used in the United States legal system where a defendant is found "not guilty," rather than being found "innocent."

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Open your document in Microsoft Word and click wherever you want the hypothesis symbols to appear.

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Type a capital "H" on your keyboard.

Click the subscript button, located in the "Font" group of the "Home" tab. This button's icon looks like an "x" with a subscript "2." Alternatively, hold the "Ctrl" key and press "=".

Type a "0" to create a null hypothesis symbol or "1" to create an alternative hypothesis symbol. Alternatively, type an "o" or "a" to represent the null and alternative hypotheses, respectively, although these symbols are not as frequently used.

Press the subscript button again to exit this formatting mode.

• University of New England: Null and Alternative Hypothesis

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How do I express this hypothesis using symbols?

I'm just making this up to understand the rules...

Claim: People wearing roller blades can get to places in lesser time than those who don't. (Assuming on foot)

Now, I'd like to express this using symbols (that's what I call them, unless there's another name for this).

I'm guessing: If there's no problem with the symbol to be used, I'd like to use t as the time taken to walk. So my $H_0$ would be that even with roller blades on, they would need the same amount of time.

Based on this, is it correct to express the following?

$H_0$ : $\mu$ = t

Or is there a better way to express this?

Please feel free to correct me - I really want to learn this!

• hypothesis-testing
• association-rules

For simplicity, let us fix definite start and end points to the hypothesis, so in words the hypothesis might be

"People traveling from point A to point B by roller blade arrive sooner than those who walk."

Probably the simplest way to express this hypothesis is to add the qualifier "... on average".

Then the hypothesis becomes a definite statement about the conditional expectation of the travel time, given the mode of transportation.

To proceed you would then introduce a symbol for the travel time, say $T$, and a symbol for the mode of travel. If we are assuming that all travelers either use roller blades or they walk, then we can represent the travel mode by a dummy variable such as $$R=\begin{cases}1 & \text{if roller blade} \\ 0 & \text{if walk} \end{cases}$$

Then the hypothesis becomes

$$\mathbb{E}[\,T\mid R=1\,] < \mathbb{E}[\,T\mid R=0\,]$$

where $\mathbb{E}[\,x\mid a\,]$ should be read as "the expected value of $x$, given $a$".

Usually the symbol $H_0$ that you use is reserved for the null hypothesis , which is typically the negation of the hypothesis of interest. So for example in this case, $H_0$ might be "roller blading is no faster than walking".

One last point: The hypothesis as expressed in the equation above is not actually empirically testable , because the theoretical expectations cannot be literally computed. So usually a next step is to use the theoretical hypothesis to generate predictions about what would be observed in a data-set ( sample ) of actually observed $(T,R)$ pairs.

• $\begingroup$ That is one of the most beautiful explanations I've seen in a long time on the entire stackexchange. Thanks! I do have 2 questions on the what you've mentioned. First, how do I express the statement roller blading is no faster than walking as a symbols. Second, I thought that T is reserved for other things like the T-Score. So are we really free to use that symbol? Thanks a again! $\endgroup$ –  itsols Commented Sep 29, 2016 at 1:19
• 1 $\begingroup$ For the first: In the hypothesis (gray box), change $<$ to $geq$ (this is literally the "negation of the hypothesis of interest"). For the second: I am not really a statistician, so I tend to use whatever symbols are convenient! (I mentioned $H_0$ because that one is pretty standard in this context; AFAIK for the various "t" statistics the lower-case $t$ is used ... but that is also used for "time" series of course!) $\endgroup$ –  GeoMatt22 Commented Sep 29, 2016 at 1:27

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How to type degree symbol in Google docs?

How to Type Degree Symbol in Google Docs?

Are you struggling to find the degree symbol (°) in Google Docs? You’re not alone! Many users face this issue, but worry not, as we’re here to help you out. In this article, we’ll provide you with a step-by-step guide on how to type the degree symbol in Google Docs.

Method 1: Using the Keyboard Shortcut

One of the easiest ways to type the degree symbol in Google Docs is by using the keyboard shortcut. Here’s how:

• Windows: Press Alt + 0176 on your keyboard.
• Mac: Press Option + Shift + 8 on your keyboard.

This will insert the degree symbol (°) in your Google Doc. Make sure to release the keys before typing the next character , as holding the keys down will insert a different symbol.

Method 2: Using the Character Map

Another way to insert the degree symbol is by using the Character Map. Here’s how:

• Windows: Go to Start > All Programs > Accessories > System Tools > Character Map.
• Mac: Go to Applications > Utilities > Character Viewer.
• In the Character Map window, scroll down and find the "Symbol" section.
• Click on the "Degree" symbol (°) to select it.
• Click on the "Copy" button to copy the symbol to your clipboard.
• Go back to your Google Doc and right-click where you want to insert the symbol.
• Select "Paste" to insert the degree symbol.

Method 3: Using the Google Docs Menu

You can also find the degree symbol in the Google Docs menu. Here’s how:

• Go to your Google Doc and click on the "Insert" menu.
• Click on "Special characters" from the drop-down menu.
• In the "Special characters" window, scroll down and find the "Degree" symbol (°) under the "Symbols" section.
• Click on the "Insert" button to insert the degree symbol into your document.

Method 4: Using HTML Code

If you’re familiar with HTML code, you can also insert the degree symbol using the following code:

Simply type the HTML code in your Google Doc, and it will be converted to the degree symbol.

Tips and Tricks

Here are some additional tips and tricks to help you insert the degree symbol in Google Docs:

• Copy and paste: You can also copy the degree symbol from a website or document and paste it into your Google Doc.
• AutoCorrect: If you frequently need to insert the degree symbol, you can enable the AutoCorrect feature in Google Docs. This will automatically replace the text with the symbol as you type.
• Templates: If you’re using a template, you can also find the degree symbol in the template’s formatting options.

In conclusion, there are several ways to type the degree symbol in Google Docs, including using keyboard shortcuts, the Character Map, the Google Docs menu, and HTML code. By following these methods, you should be able to insert the degree symbol into your Google Doc with ease. Remember to try out different methods to see which one works best for you.

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Re: How do I make a "null sign?"

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