zero null hypothesis

Have a language expert improve your writing

Run a free plagiarism check in 10 minutes, generate accurate citations for free.

• Knowledge Base
• 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.

Receive feedback on language, structure, and formatting

Professional editors proofread and edit your paper by focusing on:

• Academic style
• Vague sentences
• Style consistency

See an example

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.

Prevent plagiarism. Run a free check.

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.

Cite this Scribbr article

If you want to cite this source, you can copy and paste the citation or click the “Cite this Scribbr article” button to automatically add the citation to our free Citation Generator.

Turney, S. (2023, June 22). Null & Alternative Hypotheses | Definitions, Templates & Examples. Scribbr. Retrieved April 9, 2024, from https://www.scribbr.com/statistics/null-and-alternative-hypotheses/

Is this article helpful?

Shaun Turney

Other students also liked, inferential statistics | an easy introduction & examples, hypothesis testing | a step-by-step guide with easy examples, type i & type ii errors | differences, examples, visualizations, what is your plagiarism score.

What is The Null Hypothesis & When Do You Reject The Null Hypothesis

Julia Simkus

Editor at Simply Psychology

BA (Hons) Psychology, Princeton University

Julia Simkus is a graduate of Princeton University with a Bachelor of Arts in Psychology. She is currently studying for a Master's Degree in Counseling for Mental Health and Wellness in September 2023. Julia's research has been published in peer reviewed journals.

Learn about our Editorial Process

Saul Mcleod, PhD

Editor-in-Chief for Simply Psychology

BSc (Hons) Psychology, MRes, PhD, University of Manchester

Saul Mcleod, PhD., is a qualified psychology teacher with over 18 years of experience in further and higher education. He has been published in peer-reviewed journals, including the Journal of Clinical Psychology.

Olivia Guy-Evans, MSc

Associate Editor for Simply Psychology

BSc (Hons) Psychology, MSc Psychology of Education

Olivia Guy-Evans is a writer and associate editor for Simply Psychology. She has previously worked in healthcare and educational sectors.

On This Page:

A null hypothesis is a statistical concept suggesting no significant difference or relationship between measured variables. It’s the default assumption unless empirical evidence proves otherwise.

The null hypothesis states no relationship exists between the two variables being studied (i.e., one variable does not affect the other).

The null hypothesis is the statement that a researcher or an investigator wants to disprove.

Testing the null hypothesis can tell you whether your results are due to the effects of manipulating ​ the dependent variable or due to random chance. 

How to Write a Null Hypothesis

Null hypotheses (H0) start as research questions that the investigator rephrases as statements indicating no effect or relationship between the independent and dependent variables.

It is a default position that your research aims to challenge or confirm.

For example, if studying the impact of exercise on weight loss, your null hypothesis might be:

There is no significant difference in weight loss between individuals who exercise daily and those who do not.

Examples of Null Hypotheses

When do we reject the null hypothesis .

We reject the null hypothesis when the data provide strong enough evidence to conclude that it is likely incorrect. This often occurs when the p-value (probability of observing the data given the null hypothesis is true) is below a predetermined significance level.

If the collected data does not meet the expectation of the null hypothesis, a researcher can conclude that the data lacks sufficient evidence to back up the null hypothesis, and thus the null hypothesis is rejected. 

Rejecting the null hypothesis means that a relationship does exist between a set of variables and the effect is statistically significant ( p > 0.05).

If the data collected from the random sample is not statistically significance , then the null hypothesis will be accepted, and the researchers can conclude that there is no relationship between the variables. 

You need to perform a statistical test on your data in order to evaluate how consistent it is with the null hypothesis. A p-value is one statistical measurement used to validate a hypothesis against observed data.

Calculating the p-value is a critical part of null-hypothesis significance testing because it quantifies how strongly the sample data contradicts the null hypothesis.

The level of statistical significance is often expressed as a  p  -value between 0 and 1. The smaller the p-value, the stronger the evidence that you should reject the null hypothesis.

Usually, a researcher uses a confidence level of 95% or 99% (p-value of 0.05 or 0.01) as general guidelines to decide if you should reject or keep the null.

When your p-value is less than or equal to your significance level, you reject the null hypothesis.

In other words, smaller p-values are taken as stronger evidence against the null hypothesis. Conversely, when the p-value is greater than your significance level, you fail to reject the null hypothesis.

In this case, the sample data provides insufficient data to conclude that the effect exists in the population.

Because you can never know with complete certainty whether there is an effect in the population, your inferences about a population will sometimes 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 called a type II error.

Why Do We Never Accept The Null Hypothesis?

The reason we do not say “accept the null” is because we are always assuming the null hypothesis is true and then conducting a study to see if there is evidence against it. And, even if we don’t find evidence against it, a null hypothesis is not accepted.

A lack of evidence only means that you haven’t proven that something exists. It does not prove that something doesn’t exist. 

It is risky to conclude that the null hypothesis is true merely because we did not find evidence to reject it. It is always possible that researchers elsewhere have disproved the null hypothesis, so we cannot accept it as true, but instead, we state that we failed to reject the null. 

One can either reject the null hypothesis, or fail to reject it, but can never accept it.

Why Do We Use The Null Hypothesis?

We can never prove with 100% certainty that a hypothesis is true; We can only collect evidence that supports a theory. However, testing a hypothesis can set the stage for rejecting or accepting this hypothesis within a certain confidence level.

The null hypothesis is useful because it can tell us whether the results of our study are due to random chance or the manipulation of a variable (with a certain level of confidence).

A null hypothesis is rejected if the measured data is significantly unlikely to have occurred and a null hypothesis is accepted if the observed outcome is consistent with the position held by the null hypothesis.

Rejecting the null hypothesis sets the stage for further experimentation to see if a relationship between two variables exists. 

Hypothesis testing is a critical part of the scientific method as it helps decide whether the results of a research study support a particular theory about a given population. Hypothesis testing is a systematic way of backing up researchers’ predictions with statistical analysis.

It helps provide sufficient statistical evidence that either favors or rejects a certain hypothesis about the population parameter. 

Purpose of a Null Hypothesis 

• The primary purpose of the null hypothesis is to disprove an assumption. 
• Whether rejected or accepted, the null hypothesis can help further progress a theory in many scientific cases.
• A null hypothesis can be used to ascertain how consistent the outcomes of multiple studies are.

Do you always need both a Null Hypothesis and an Alternative Hypothesis?

The null (H0) and alternative (Ha or H1) hypotheses are two competing claims that describe the effect of the independent variable on the dependent variable. They are mutually exclusive, which means that only one of the two hypotheses can be true. 

While the null hypothesis states that there is no effect in the population, an alternative hypothesis states that there is statistical significance between two variables. 

The goal of hypothesis testing is to make inferences about a population based on a sample. In order to undertake hypothesis testing, you must express your research hypothesis as a null and alternative hypothesis. Both hypotheses are required to cover every possible outcome of the study. 

What is the difference between a null hypothesis and an alternative hypothesis?

The alternative hypothesis is the complement to the null hypothesis. The null hypothesis states that there is no effect or no relationship between variables, while the alternative hypothesis claims that there is an effect or relationship in the population.

It is the claim that you expect or hope will be true. The null hypothesis and the alternative hypothesis are always mutually exclusive, meaning that only one can be true at a time.

What are some problems with the null hypothesis?

One major problem with the null hypothesis is that researchers typically will assume that accepting the null is a failure of the experiment. However, accepting or rejecting any hypothesis is a positive result. Even if the null is not refuted, the researchers will still learn something new.

Why can a null hypothesis not be accepted?

We can either reject or fail to reject a null hypothesis, but never accept it. If your test fails to detect an effect, this is not proof that the effect doesn’t exist. It just means that your sample did not have enough evidence to conclude that it exists.

We can’t accept a null hypothesis because a lack of evidence does not prove something that does not exist. Instead, we fail to reject it.

Failing to reject the null indicates that the sample did not provide sufficient enough evidence to conclude that an effect exists.

If the p-value is greater than the significance level, then you fail to reject the null hypothesis.

Is a null hypothesis directional or non-directional?

A hypothesis test can either contain an alternative directional hypothesis or a non-directional alternative hypothesis. A directional hypothesis is one that contains the less than (“<“) or greater than (“>”) sign.

A nondirectional hypothesis contains the not equal sign (“≠”).  However, a null hypothesis is neither directional nor non-directional.

A null hypothesis is a prediction that there will be no change, relationship, or difference between two variables.

The directional hypothesis or nondirectional hypothesis would then be considered alternative hypotheses to the null hypothesis.

Gill, J. (1999). The insignificance of null hypothesis significance testing.  Political research quarterly ,  52 (3), 647-674.

Krueger, J. (2001). Null hypothesis significance testing: On the survival of a flawed method.  American Psychologist ,  56 (1), 16.

Masson, M. E. (2011). A tutorial on a practical Bayesian alternative to null-hypothesis significance testing.  Behavior research methods ,  43 , 679-690.

Nickerson, R. S. (2000). Null hypothesis significance testing: a review of an old and continuing controversy.  Psychological methods ,  5 (2), 241.

Rozeboom, W. W. (1960). The fallacy of the null-hypothesis significance test.  Psychological bulletin ,  57 (5), 416.

• school Campus Bookshelves
• menu_book Bookshelves
• perm_media Learning Objects
• login Login
• how_to_reg Request Instructor Account
• hub Instructor Commons
• Download Page (PDF)
• Download Full Book (PDF)
• Periodic Table
• Physics Constants
• Scientific Calculator
• Reference & Cite
• Tools expand_more
• Readability

selected template will load here

This action is not available.

10.2: Null and Alternative Hypotheses

• Last updated
• Save as PDF
• Page ID 100392

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

• The null hypothesis (\(H_{0}\)) is a statement about the population that either is believed to be true or is used to put forth an argument unless it can be shown to be incorrect beyond a reasonable doubt.
• The alternative hypothesis (\(H_{a}\)) 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.

\(H_{0}\) 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 \(\PageIndex{1}\)

• \(H_{0}\): No more than 30% of the registered voters in Santa Clara County voted in the primary election. \(p \leq 30\)
• \(H_{a}\): More than 30% of the registered voters in Santa Clara County voted in the primary election. \(p > 30\)

Exercise \(\PageIndex{1}\)

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

• \(H_{0}\): The drug reduces cholesterol by 25%. \(p = 0.25\)
• \(H_{a}\): The drug does not reduce cholesterol by 25%. \(p \neq 0.25\)

Example \(\PageIndex{2}\)

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:

• \(H_{0}: \mu = 2.0\)
• \(H_{a}: \mu \neq 2.0\)

Exercise \(\PageIndex{2}\)

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 \((=, \neq, \geq, <, \leq, >)\) for the null and alternative hypotheses.

• \(H_{0}: \mu \  \_ \  66\)
• \(H_{a}: \mu \  \_ \  66\)
• \(H_{0}: \mu = 66\)
• \(H_{a}: \mu \neq 66\)

Example \(\PageIndex{3}\)

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:

• \(H_{0}: \mu \geq 5\)
• \(H_{a}: \mu < 5\)

Exercise \(\PageIndex{3}\)

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.

• \(H_{0}: \mu \  \_ \  45\)
• \(H_{a}: \mu \  \_ \  45\)
• \(H_{0}: \mu \geq 45\)
• \(H_{a}: \mu < 45\)

Example \(\PageIndex{4}\)

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.

• \(H_{0}: p \leq 0.066\)
• \(H_{a}: p > 0.066\)

Exercise \(\PageIndex{4}\)

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. Fill in the correct symbol (\(=, \neq, \geq, <, \leq, >\)) for the null and alternative hypotheses.

• \(H_{0}: p \  \_ \  0.40\)
• \(H_{a}: p \  \_ \  0.40\)
• \(H_{0}: p = 0.40\)
• \(H_{a}: p > 0.40\)

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.

Chapter Review

In a hypothesis test , sample data is evaluated in order to arrive at a decision about some type of claim. If certain conditions about the sample are satisfied, then the claim can be evaluated for a population. In a hypothesis test, we:

• Evaluate the null hypothesis , typically denoted with \(H_{0}\). The null is not rejected unless the hypothesis test shows otherwise. The null statement must always contain some form of equality \((=, \leq \text{or} \geq)\)
• Always write the alternative hypothesis , typically denoted with \(H_{a}\) or \(H_{1}\), using less than, greater than, or not equals symbols, i.e., \((\neq, >, \text{or} <)\).
• If we reject the null hypothesis, then we can assume there is enough evidence to support the alternative hypothesis.
• Never state that a claim is proven true or false. Keep in mind the underlying fact that hypothesis testing is based on probability laws; therefore, we can talk only in terms of non-absolute certainties.

Formula Review

\(H_{0}\) and \(H_{a}\) are contradictory.

• If \(\alpha \leq p\)-value, then do not reject \(H_{0}\).
• If\(\alpha > p\)-value, then reject \(H_{0}\).

\(\alpha\) is preconceived. Its value is set before the hypothesis test starts. The \(p\)-value is calculated from the data.References

Data from the National Institute of Mental Health. Available online at http://www.nimh.nih.gov/publicat/depression.cfm .

Contributors

Barbara Illowsky and Susan Dean (De Anza College) with many other contributing authors. Content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at http://cnx.org/contents/[email protected] .

Want to create or adapt books like this? Learn more about how Pressbooks supports open publishing practices.

5.5 Introduction to Hypothesis Tests

One job of a statistician is to make statistical inferences about populations based on samples taken from the population. Confidence intervals are one way to estimate a population parameter.

Another way to make a statistical inference is to make a decision about a parameter. For instance, a car dealer advertises that its new small truck gets 35 miles per gallon, on average. A tutoring service claims that its method of tutoring helps 90% of its students get an A or a B. A company says that women managers in their company earn an average of $60,000 per year.  A statistician may want to make a decision about or evaluate these claims. A hypothesis test can be used to do this.

A hypothesis test involves collecting data from a sample and evaluating the data. Then, the statistician makes a decision as to whether or not there is sufficient evidence, based upon analyses of the data, to reject the null hypothesis.

In this section you will conduct hypothesis tests on single means when the population standard deviation is known.

Hypothesis testing consists of two contradictory hypotheses or statements, a decision based on the data, and a conclusion. To perform a hypothesis test, a statistician will perform some variation of these steps:

• Define hypotheses.
• Collect and/OR use the sample data to determine the correct distribution to use.
• Calculate Test Statistic.
• Make a decision
• Write a conclusion.

Defining your hypotheses

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

The null hypothesis ( H 0 ): It is often a statement of the accepted historical value or norm. This is your starting point that you must assume from the beginning in order to show an effect exists.

The alternative hypothesis ( H a ) : 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 :

Note: 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.

We want to test whether the mean GPA of students in American colleges is different from 2.0 (out of 4.0). The null hypothesis is: H 0 : μ = 2.0. What is the alternative hypothesis?

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

Using the Sample to Test the Null Hypothesis

Once you have defined your hypotheses the next step in the process, is to collect sample data.  In a classroom context most of the time the data or summary statistics will be given to you.

Then you will have to determine the correct distribution to perform the hypothesis test, given the assumptions you are able to make about the situation.  Right now we are demonstrating these ideas in a test for a mean when the population standard deviation is known using the Z distribution.  We will see other scenarios in the future.

Calculating a Test Statistic

Next, you will start evaluating the data.  This begins with calculating your test statistic , which is a measure of how far what you observed is from what you are assuming to he true.  In this context, your test statistic, z ο , quantifies the number of standard deviations between the sample mean x and the population mean µ.  Calculating the test statistic is analogous to standardizing observations with Z-scores as discussed previously:

where µ o   is the value assumed to be true in the null hypothesis.

Making a Decision

Once you have your test statistic there are two methods to use it to make your decision:

• Critical value method—This is one way you can make a decision, but will not be discussed in detail at this time.
• P-Value method—This is the preferred method we will focus on.

P-Value Method

To find a p-value we use the test statistic to calculate the actual probability of getting the test result. Formally, the p -value is the probability that, if the null hypothesis is true, the results from another randomly selected sample will be as extreme or more extreme as the results obtained from the given sample.

A large p -value calculated from the data indicates that we should not reject the null hypothesis. The smaller the p -value, the more unlikely the outcome, and the stronger the evidence is against the null hypothesis. We would reject the null hypothesis if the evidence is strongly against it.

Draw a graph that shows the p -value. The hypothesis test is easier to perform if you use a graph because you see the problem more clearly.

Suppose a baker claims that his bread height is more than 15 cm, on average. Several of his customers do not believe him. To persuade his customers that he is right, the baker decides to do a hypothesis test. He bakes 10 loaves of bread. The mean height of the sample loaves is 17 cm. The baker knows from baking hundreds of loaves of bread that the standard deviation for the height is 0.5 cm. and the distribution of heights is normal.

The null hypothesis could be H 0 : μ ≤ 15

The alternate hypothesis is H a : μ > 15

The words “is more than” translates as a “>” so “ μ > 15″ goes into the alternate hypothesis. The null hypothesis must contradict the alternate hypothesis.

Suppose the null hypothesis is true (the mean height of the loaves is no more than 15 cm). Then is the mean height (17 cm) calculated from the sample unexpectedly large? The hypothesis test works by asking the question how unlikely the sample mean would be if the null hypothesis were true. The graph shows how far out the sample mean is on the normal curve. The p -value is the probability that, if we were to take other samples, any other sample mean would fall at least as far out as 17 cm.

The p -value, then, is the probability that a sample mean is the same or greater than 17 cm. when the population mean is, in fact, 15 cm. We can calculate this probability using the normal distribution for means.

A p -value of approximately zero tells us that it is highly unlikely that a loaf of bread rises no more than 15 cm, on average. That is, almost 0% of all loaves of bread would be at least as high as 17 cm. purely by CHANCE had the population mean height really been 15 cm. Because the outcome of 17 cm. is so unlikely (meaning it is happening NOT by chance alone), we conclude that the evidence is strongly against the null hypothesis (the mean height is at most 15 cm.). There is sufficient evidence that the true mean height for the population of the baker’s loaves of bread is greater than 15 cm.

A normal distribution has a standard deviation of 1. We want to verify a claim that the mean is greater than 12. A sample of 36 is taken with a sample mean of 12.5.

Find The P-value:

Decision and Conclusion

A systematic way to make a decision of whether to reject or not reject the null hypothesis is to compare the p -value and a preset or preconceived α (also called a significance level ). A preset α is the probability of a Type I error (rejecting the null hypothesis when the null hypothesis is true). It may or may not be given to you at the beginning of the problem.  If there is no given preconceived α , then use α = 0.05.

When you make a decision to reject or not reject H 0 , do as follows:

• If α > p -value, reject H 0 . The results of the sample data are statistically significant . You can say there is sufficient evidence to conclude that H 0 is an incorrect belief and that the alternative hypothesis, H a , may be correct.
• If α ≤ p -value, fail to reject H 0 . The results of the sample data are not significant. There is not sufficient evidence to conclude that the alternative hypothesis, H a , may be correct.

After you make your decision, write a thoughtful conclusion in the context of the scenario incorporating the hypotheses.

NOTE: When you “do not reject H 0 “, it does not mean that you should believe that H 0 is true. It simply means that the sample data have failed to provide sufficient evidence to cast serious doubt about the truthfulness of H o .

When using the p -value to evaluate a hypothesis test, it is sometimes useful to use the following memory device

If the p -value is low, the null must go.

If the p -value is high, the null must fly.

This memory aid relates a p -value less than the established alpha (the p is low) as rejecting the null hypothesis and, likewise, relates a p -value higher than the established alpha (the p is high) as not rejecting the null hypothesis.

Fill in the blanks:

• Reject the null hypothesis when _____.
• The results of the sample data _____.
• Do not reject the null when hypothesis when _____.

It’s a Boy Genetics Labs claim their procedures improve the chances of a boy being born. The results for a test of a single population proportion are as follows:

H 0 : p = 0.50, H a : p > 0.50

p -value = 0.025

Interpret the results and state a conclusion in simple, non-technical terms.

MORE RESOURCES

Figure References

Figure 5.11: Alora Griffiths. 2019. Public domain. https://unsplash.com/photos/7aRQZtLsvqw

Figure 5.13: Kindred Grey. 2020. CC BY-SA 4.0.

A decision making procedure for determining whether sample evidence supports a hypothesis

The claim that is assumed to be true and is tested in a hypothesis test

A working hypothesis that is contradictory to the null hypothesis

A measure of how far what you observed is from the hypothesized (or claimed) value

The probability that an event will occur, assuming the null hypothesis is true

Probability that a true null hypothesis will be rejected, also known as Type I error and denoted by α

Finding sufficient evidence that the effect we see is not just due to variability, often from rejecting the null hypothesis

Significant Statistics Copyright © 2023 by John Morgan Russell, OpenStaxCollege, OpenIntro is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License , except where otherwise noted.

Share This Book

User Preferences

Content preview.

Arcu felis bibendum ut tristique et egestas quis:

• Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris
• Duis aute irure dolor in reprehenderit in voluptate
• Excepteur sint occaecat cupidatat non proident

Keyboard Shortcuts

5.2 - writing hypotheses.

The first step in conducting a hypothesis test is to write the hypothesis statements that are going to be tested. For each test you will have a null hypothesis (\(H_0\)) and an alternative hypothesis (\(H_a\)).

When writing hypotheses there are three things that we need to know: (1) the parameter that we are testing (2) the direction of the test (non-directional, right-tailed or left-tailed), and (3) the value of the hypothesized parameter.

• At this point we can write hypotheses for a single mean (\(\mu\)), paired means(\(\mu_d\)), a single proportion (\(p\)), the difference between two independent means (\(\mu_1-\mu_2\)), the difference between two proportions (\(p_1-p_2\)), a simple linear regression slope (\(\beta\)), and a correlation (\(\rho\)). 
• The research question will give us the information necessary to determine if the test is two-tailed (e.g., "different from," "not equal to"), right-tailed (e.g., "greater than," "more than"), or left-tailed (e.g., "less than," "fewer than").
• The research question will also give us the hypothesized parameter value. This is the number that goes in the hypothesis statements (i.e., \(\mu_0\) and \(p_0\)). For the difference between two groups, regression, and correlation, this value is typically 0.

Hypotheses are always written in terms of population parameters (e.g., \(p\) and \(\mu\)).  The tables below display all of the possible hypotheses for the parameters that we have learned thus far. Note that the null hypothesis always includes the equality (i.e., =).

• Math Article

Null Hypothesis

In mathematics, Statistics deals with the study of research and surveys on the numerical data. For taking surveys, we have to define the hypothesis. Generally, there are two types of hypothesis. One is a null hypothesis, and another is an alternative hypothesis .

In probability and statistics, the null hypothesis is a comprehensive statement or default status that there is zero happening or nothing happening. For example, there is no connection among groups or no association between two measured events. It is generally assumed here that the hypothesis is true until any other proof has been brought into the light to deny the hypothesis. Let us learn more here with definition, symbol, principle, types and example, in this article.

Table of contents:

• Comparison with Alternative Hypothesis

Null Hypothesis Definition

The null hypothesis is a kind of hypothesis which explains the population parameter whose purpose is to test the validity of the given experimental data. This hypothesis is either rejected or not rejected based on the viability of the given population or sample . In other words, the null hypothesis is a hypothesis in which the sample observations results from the chance. It is said to be a statement in which the surveyors wants to examine the data. It is denoted by H 0 .

Null Hypothesis Symbol

In statistics, the null hypothesis is usually denoted by letter H with subscript ‘0’ (zero), such that H 0 . It is pronounced as H-null or H-zero or H-nought. At the same time, the alternative hypothesis expresses the observations determined by the non-random cause. It is represented by H 1 or H a .

Null Hypothesis Principle

The principle followed for null hypothesis testing is, collecting the data and determining the chances of a given set of data during the study on some random sample, assuming that the null hypothesis is true. In case if the given data does not face the expected null hypothesis, then the outcome will be quite weaker, and they conclude by saying that the given set of data does not provide strong evidence against the null hypothesis because of insufficient evidence. Finally, the researchers tend to reject that.

Null Hypothesis Formula

Here, the hypothesis test formulas are given below for reference.

The formula for the null hypothesis is:

H 0 :  p = p 0

The formula for the alternative hypothesis is:

H a = p >p 0 , < p 0 ≠ p 0

The formula for the test static is:

Remember that,  p 0  is the null hypothesis and p – hat is the sample proportion.

Also, read:

Types of Null Hypothesis

There are different types of hypothesis. They are:

Simple Hypothesis

It completely specifies the population distribution. In this method, the sampling distribution is the function of the sample size.

Composite Hypothesis

The composite hypothesis is one that does not completely specify the population distribution.

Exact Hypothesis

Exact hypothesis defines the exact value of the parameter. For example μ= 50

Inexact Hypothesis

This type of hypothesis does not define the exact value of the parameter. But it denotes a specific range or interval. For example 45< μ <60

Null Hypothesis Rejection

Sometimes the null hypothesis is rejected too. If this hypothesis is rejected means, that research could be invalid. Many researchers will neglect this hypothesis as it is merely opposite to the alternate hypothesis. It is a better practice to create a hypothesis and test it. The goal of researchers is not to reject the hypothesis. But it is evident that a perfect statistical model is always associated with the failure to reject the null hypothesis.

How do you Find the Null Hypothesis?

The null hypothesis says there is no correlation between the measured event (the dependent variable) and the independent variable. We don’t have to believe that the null hypothesis is true to test it. On the contrast, you will possibly assume that there is a connection between a set of variables ( dependent and independent).

When is Null Hypothesis Rejected?

The null hypothesis is rejected using the P-value approach. If the P-value is less than or equal to the α, there should be a rejection of the null hypothesis in favour of the alternate hypothesis. In case, if P-value is greater than α, the null hypothesis is not rejected.

Null Hypothesis and Alternative Hypothesis

Now, let us discuss the difference between the null hypothesis and the alternative hypothesis.

Null Hypothesis Examples

Here, some of the examples of the null hypothesis are given below. Go through the below ones to understand the concept of the null hypothesis in a better way.

If a medicine reduces the risk of cardiac stroke, then the null hypothesis should be “the medicine does not reduce the chance of cardiac stroke”. This testing can be performed by the administration of a drug to a certain group of people in a controlled way. If the survey shows that there is a significant change in the people, then the hypothesis is rejected.

Few more examples are:

1). Are there is 100% chance of getting affected by dengue?

Ans: There could be chances of getting affected by dengue but not 100%.

2). Do teenagers are using mobile phones more than grown-ups to access the internet?

Ans: Age has no limit on using mobile phones to access the internet.

3). Does having apple daily will not cause fever?

Ans: Having apple daily does not assure of not having fever, but increases the immunity to fight against such diseases.

4). Do the children more good in doing mathematical calculations than grown-ups?

Ans: Age has no effect on Mathematical skills.

In many common applications, the choice of the null hypothesis is not automated, but the testing and calculations may be automated. Also, the choice of the null hypothesis is completely based on previous experiences and inconsistent advice. The choice can be more complicated and based on the variety of applications and the diversity of the objectives. 

The main limitation for the choice of the null hypothesis is that the hypothesis suggested by the data is based on the reasoning which proves nothing. It means that if some hypothesis provides a summary of the data set, then there would be no value in the testing of the hypothesis on the particular set of data. 

Frequently Asked Questions on Null Hypothesis

What is meant by the null hypothesis.

In Statistics, a null hypothesis is a type of hypothesis which explains the population parameter whose purpose is to test the validity of the given experimental data.

What are the benefits of hypothesis testing?

Hypothesis testing is defined as a form of inferential statistics, which allows making conclusions from the entire population based on the sample representative.

When a null hypothesis is accepted and rejected?

The null hypothesis is either accepted or rejected in terms of the given data. If P-value is less than α, then the null hypothesis is rejected in favor of the alternative hypothesis, and if the P-value is greater than α, then the null hypothesis is accepted in favor of the alternative hypothesis.

Why is the null hypothesis important?

The importance of the null hypothesis is that it provides an approximate description of the phenomena of the given data. It allows the investigators to directly test the relational statement in a research study.

How to accept or reject the null hypothesis in the chi-square test?

If the result of the chi-square test is bigger than the critical value in the table, then the data does not fit the model, which represents the rejection of the null hypothesis.

Put your understanding of this concept to test by answering a few MCQs. Click ‘Start Quiz’ to begin!

Select the correct answer and click on the “Finish” button Check your score and answers at the end of the quiz

Visit BYJU’S for all Maths related queries and study materials

Your result is as below

Request OTP on Voice Call

• Share Share

Register with BYJU'S & Download Free PDFs

Register with byju's & watch live videos.

Module 9: Hypothesis Testing With One Sample

Creating hypotheses, learning outcomes.

• Given a claim about a mean, determine null and alternative hypotheses
• Given a claim about a proportion, determine 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.

Recall: Subscript

Mathematicians use subscripts to distinguish between random variables. A subscript is a small number written to the right of, and a little lower than, a variable. There is always a Null and an Alternative hypothesis. Statisticians use the subscript 0 for the null hypothesis because null is the same as nought or zero. Statisticians use the subscript a or alpha for the alternative hypothesis because the statement is alternative to the null or opposite of the null.

Since the null and alternative hypotheses are contradictory, you must examine evidence to decide whether or not you have enough evidence to reject the null hypothesis. 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.

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.

H 0 : The drug reduces cholesterol by 25%. p = 0.25

H a : The drug does not reduce cholesterol by 25%. p ≠ 0.25

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

H 0 : μ = 66

H a : μ ≠ 66

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

H 0 : μ ≥ 45

H a : μ < 45

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

H 0 : p = 0.40

H a : p > 0.40

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.

• Statistics, Null and Alternative Hypotheses. Provided by : OpenStax. Located at : https://openstax.org/books/statistics/pages/9-1-null-and-alternative-hypotheses . License : CC BY: Attribution . License Terms : Access for free at https://openstax.org/books/statistics/pages/1-introduction
• Introductory Statistics. Authored by : Barbara Illowsky, Susan Dean. Provided by : OpenStax. Located at : https://openstax.org/books/introductory-statistics/pages/1-introduction . License : CC BY: Attribution . License Terms : Access for free at https://openstax.org/books/introductory-statistics/pages/1-introduction
• Prealgebra. Provided by : OpenStax. Located at : https://openstax.org/books/prealgebra/pages/1-introduction . License : CC BY: Attribution . License Terms : Access for free at https://openstax.org/books/prealgebra/pages/1-introduction
• Simple hypothesis testing | Probability and Statistics | Khan Academy. Authored by : Khan Academy. Located at : https://youtu.be/5D1gV37bKXY . License : All Rights Reserved . License Terms : Standard YouTube License

Privacy Policy

If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

To log in and use all the features of Khan Academy, please enable JavaScript in your browser.

AP®︎/College Statistics

Course: ap®︎/college statistics   >   unit 10.

• Idea behind hypothesis testing

Examples of null and alternative hypotheses

• Writing null and alternative hypotheses
• P-values and significance tests
• Comparing P-values to different significance levels
• Estimating a P-value from a simulation
• Estimating P-values from simulations
• Using P-values to make conclusions

Want to join the conversation?

• Upvote Button navigates to signup page
• Downvote Button navigates to signup page
• Flag Button navigates to signup page

Video transcript

• School Guide
• Mathematics
• Number System and Arithmetic
• Trigonometry
• Probability
• Mensuration
• Maths Formulas
• Class 8 Maths Notes
• Class 9 Maths Notes
• Class 10 Maths Notes
• Class 11 Maths Notes
• Class 12 Maths Notes
• Greater Than and Less Than Symbols | Meaning of Signs and Examples
• Numerator in Mathematics
• Rational and Irrational Numbers
• Expanded Form
• Perfect Cubes - Definition, List, Chart and Examples
• Linear Algebra Symbols
• Divisibility Rule of 11
• Perfect Numbers
• Denominator in Maths
• Additive Inverse and Multiplicative Inverse
• Measurement
• Predecessor and Successor
• Binary Multiplication
• Area and Perimeter of Shapes | Formula and Examples
• Probability and Statistics
• Number Symbols
• Square Root of 4
• Factors of 30
• Binary Division

Null Hypothesis

Null Hypothesis , often denoted as H 0, is a foundational concept in statistical hypothesis testing. It represents an assumption that no significant difference, effect, or relationship exists between variables within a population. It serves as a baseline assumption, positing no observed change or effect occurring. The null is t he truth or falsity of an idea in analysis.

In this article, we will discuss the null hypothesis in detail, along with some solved examples and questions on the null hypothesis.

Table of Content

• What Is a Null Hypothesis?

Symbol of Null Hypothesis

Formula of null hypothesis, types of null hypothesis, principle of null hypothesis, how do you find null hypothesis, what is a null hypothesis.

Null Hypothesis in statistical analysis suggests the absence of statistical significance within a specific set of observed data. Hypothesis testing, using sample data, evaluates the validity of this hypothesis. Commonly denoted as H 0 or simply “null,” it plays an important role in quantitative analysis, examining theories related to markets, investment strategies, or economies to determine their validity.

Definition of Null Hypothesis

Null Hypothesis represent a default position, often suggesting no effect or difference, against which researchers compare their experimental results. The Null Hypothesis, often denoted as H 0 , asserts a default assumption in statistical analysis. It posits no significant difference or effect, serving as a baseline for comparison in hypothesis testing.

Null Hypothesis is represented as H 0 , the Null Hypothesis symbolizes the absence of a measurable effect or difference in the variables under examination.

Certainly, a simple example would be asserting that the mean score of a group is equal to a specified value like stating that the average IQ of a population is 100.

The Null Hypothesis is typically formulated as a statement of equality or absence of a specific parameter in the population being studied. It provides a clear and testable prediction for comparison with the alternative hypothesis. The formulation of the Null Hypothesis typically follows a concise structure, stating the equality or absence of a specific parameter in the population.

Mean Comparison (Two-sample t-test)

H 0 : μ 1 = μ 2

This asserts that there is no significant difference between the means of two populations or groups.

Proportion Comparison

H 0 : p 1 − p 2 = 0

This suggests no significant difference in proportions between two populations or conditions.

Equality in Variance (F-test in ANOVA)

H 0 : σ 1 = σ 2

This states that there’s no significant difference in variances between groups or populations.

Independence (Chi-square Test of Independence):

H 0 : Variables are independent

This asserts that there’s no association or relationship between categorical variables.

Null Hypotheses vary including simple and composite forms, each tailored to the complexity of the research question. Understanding these types is pivotal for effective hypothesis testing.

Equality Null Hypothesis (Simple Null Hypothesis)

The Equality Null Hypothesis, also known as the Simple Null Hypothesis, is a fundamental concept in statistical hypothesis testing that assumes no difference, effect or relationship between groups, conditions or populations being compared.

Non-Inferiority Null Hypothesis

In some studies, the focus might be on demonstrating that a new treatment or method is not significantly worse than the standard or existing one.

Superiority Null Hypothesis

The concept of a superiority null hypothesis comes into play when a study aims to demonstrate that a new treatment, method, or intervention is significantly better than an existing or standard one.

Independence Null Hypothesis

In certain statistical tests, such as chi-square tests for independence, the null hypothesis assumes no association or independence between categorical variables.

Homogeneity Null Hypothesis

In tests like ANOVA (Analysis of Variance), the null hypothesis suggests that there’s no difference in population means across different groups.

Examples of Null Hypothesis

• Medicine: Null Hypothesis: “No significant difference exists in blood pressure levels between patients given the experimental drug versus those given a placebo.”
• Education: Null Hypothesis: “There’s no significant variation in test scores between students using a new teaching method and those using traditional teaching.”
• Economics: Null Hypothesis: “There’s no significant change in consumer spending pre- and post-implementation of a new taxation policy.”
• Environmental Science: Null Hypothesis: “There’s no substantial difference in pollution levels before and after a water treatment plant’s establishment.”

The principle of the null hypothesis is a fundamental concept in statistical hypothesis testing. It involves making an assumption about the population parameter or the absence of an effect or relationship between variables.

In essence, the null hypothesis (H 0 ) proposes that there is no significant difference, effect, or relationship between variables. It serves as a starting point or a default assumption that there is no real change, no effect or no difference between groups or conditions.

Null Hypothesis Rejection

Rejecting the Null Hypothesis occurs when statistical evidence suggests a significant departure from the assumed baseline. It implies that there is enough evidence to support the alternative hypothesis, indicating a meaningful effect or difference. Null Hypothesis rejection occurs when statistical evidence suggests a deviation from the assumed baseline, prompting a reconsideration of the initial hypothesis.

Identifying the Null Hypothesis involves defining the status quotient, asserting no effect and formulating a statement suitable for statistical analysis.

When is Null Hypothesis Rejected?

The Null Hypothesis is rejected when statistical tests indicate a significant departure from the expected outcome, leading to the consideration of alternative hypotheses. It occurs when statistical evidence suggests a deviation from the assumed baseline, prompting a reconsideration of the initial hypothesis.

Null Hypothesis and Alternative Hypothesis

In the realm of hypothesis testing, the null hypothesis (H 0 ) and alternative hypothesis (H₁ or Ha) play critical roles. The null hypothesis generally assumes no difference, effect, or relationship between variables, suggesting that any observed change or effect is due to random chance. Its counterpart, the alternative hypothesis, asserts the presence of a significant difference, effect, or relationship between variables, challenging the null hypothesis. These hypotheses are formulated based on the research question and guide statistical analyses.

Null Hypothesis vs Alternative Hypothesis

The null hypothesis (H 0 ) serves as the baseline assumption in statistical testing, suggesting no significant effect, relationship, or difference within the data. It often proposes that any observed change or correlation is merely due to chance or random variation. Conversely, the alternative hypothesis (H 1 or Ha) contradicts the null hypothesis, positing the existence of a genuine effect, relationship or difference in the data. It represents the researcher’s intended focus, seeking to provide evidence against the null hypothesis and support for a specific outcome or theory. These hypotheses form the crux of hypothesis testing, guiding the assessment of data to draw conclusions about the population being studied.

Example of Alternative and Null Hypothesis

Let’s envision a scenario where a researcher aims to examine the impact of a new medication on reducing blood pressure among patients. In this context:

Null Hypothesis (H 0 ): “The new medication does not produce a significant effect in reducing blood pressure levels among patients.”

Alternative Hypothesis (H 1 or Ha): “The new medication yields a significant effect in reducing blood pressure levels among patients.”

The null hypothesis implies that any observed alterations in blood pressure subsequent to the medication’s administration are a result of random fluctuations rather than a consequence of the medication itself. Conversely, the alternative hypothesis contends that the medication does indeed generate a meaningful alteration in blood pressure levels, distinct from what might naturally occur or by random chance.

Also, Check

Solved Examples on Null Hypothesis

Example 1: A researcher claims that the average time students spend on homework is 2 hours per night.

Null Hypothesis (H 0 ): The average time students spend on homework is equal to 2 hours per night. Data: A random sample of 30 students has an average homework time of 1.8 hours with a standard deviation of 0.5 hours. Test Statistic and Decision: Using a t-test, if the calculated t-statistic falls within the acceptance region, we fail to reject the null hypothesis. If it falls in the rejection region, we reject the null hypothesis. Conclusion: Based on the statistical analysis, we fail to reject the null hypothesis, suggesting that there is not enough evidence to dispute the claim of the average homework time being 2 hours per night.

Example 2: A company asserts that the error rate in its production process is less than 1%.

Null Hypothesis (H 0 ): The error rate in the production process is 1% or higher. Data: A sample of 500 products shows an error rate of 0.8%. Test Statistic and Decision: Using a z-test, if the calculated z-statistic falls within the acceptance region, we fail to reject the null hypothesis. If it falls in the rejection region, we reject the null hypothesis. Conclusion: The statistical analysis supports rejecting the null hypothesis, indicating that there is enough evidence to dispute the company’s claim of an error rate of 1% or higher.

Null Hypothesis – Practice Problems

Q1. A researcher claims that the average time spent by students on homework is less than 2 hours per day. Formulate the null hypothesis for this claim?

Q2. A manufacturing company states that their new machine produces widgets with a defect rate of less than 5%. Write the null hypothesis to test this claim?

Q3. An educational institute believes that their online course completion rate is at least 60%. Develop the null hypothesis to validate this assertion?

Q4. A restaurant claims that the waiting time for customers during peak hours is not more than 15 minutes. Formulate the null hypothesis for this claim?

Q5. A study suggests that the mean weight loss after following a specific diet plan for a month is more than 8 pounds. Construct the null hypothesis to evaluate this statement?

Null Hypothesis – Frequently Asked Questions

How to form a null hypothesis.

A null hypothesis is formed based on the assumption that there is no significant difference or effect between the groups being compared or no association between variables being tested. It often involves stating that there is no relationship, no change, or no effect in the population being studied.

When Do we reject the Null Hypothesis?

In statistical hypothesis testing, if the p-value (the probability of obtaining the observed results) is lower than the chosen significance level (commonly 0.05), we reject the null hypothesis. This suggests that the data provides enough evidence to refute the assumption made in the null hypothesis.

What is a Null Hypothesis in Research?

In research, the null hypothesis represents the default assumption or position that there is no significant difference or effect. Researchers often try to test this hypothesis by collecting data and performing statistical analyses to see if the observed results contradict the assumption.

What Are Alternative and Null Hypotheses?

The null hypothesis (H0) is the default assumption that there is no significant difference or effect. The alternative hypothesis (H1 or Ha) is the opposite, suggesting there is a significant difference, effect or relationship.

What Does it Mean to Reject the Null Hypothesis?

Rejecting the null hypothesis implies that there is enough evidence in the data to support the alternative hypothesis. In simpler terms, it suggests that there might be a significant difference, effect or relationship between the groups or variables being studied.

How to Find Null Hypothesis?

Formulating a null hypothesis often involves considering the research question and assuming that no difference or effect exists. It should be a statement that can be tested through data collection and statistical analysis, typically stating no relationship or no change between variables or groups.

How is Null Hypothesis denoted?

The null hypothesis is commonly symbolized as H 0 in statistical notation.

What is the Purpose of the Null hypothesis in Statistical Analysis?

The null hypothesis serves as a starting point for hypothesis testing, enabling researchers to assess if there’s enough evidence to reject it in favor of an alternative hypothesis.

What happens if we Reject the Null hypothesis?

Rejecting the null hypothesis implies that there is sufficient evidence to support an alternative hypothesis, suggesting a significant effect or relationship between variables.

Is it Possible to Prove the Null Hypothesis?

No, statistical testing aims to either reject or fail to reject the null hypothesis based on evidence from sample data. It does not prove the null hypothesis to be true.

What are Test for Null Hypothesis?

Various statistical tests, such as t-tests or chi-square tests, are employed to evaluate the validity of the Null Hypothesis in different scenarios.

Please Login to comment...

Similar reads.

• Geeks Premier League 2023
• Math-Concepts
• Geeks Premier League
• School Learning
• How to Use ChatGPT with Bing for Free?
• 7 Best Movavi Video Editor Alternatives in 2024
• How to Edit Comment on Instagram
• 10 Best AI Grammar Checkers and Rewording Tools
• 30 OOPs Interview Questions and Answers (2024)

Improve your Coding Skills with Practice

What kind of Experience do you want to share?

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.

This book may not be used in the training of large language models or otherwise be ingested into large language models or generative AI offerings without OpenStax's permission.

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

© Dec 6, 2023 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.

• Search Search Please fill out this field.

What Is a Null Hypothesis?

How a null hypothesis works, the alternative hypothesis, examples of a null hypothesis.

• Null Hypothesis and Investments
• Null Hypothesis FAQs
• Corporate Finance
• Financial Ratios

Null Hypothesis: What Is It and How Is It Used in Investing?

Adam Hayes, Ph.D., CFA, is a financial writer with 15+ years Wall Street experience as a derivatives trader. Besides his extensive derivative trading expertise, Adam is an expert in economics and behavioral finance. Adam received his master's in economics from The New School for Social Research and his Ph.D. from the University of Wisconsin-Madison in sociology. He is a CFA charterholder as well as holding FINRA Series 7, 55 & 63 licenses. He currently researches and teaches economic sociology and the social studies of finance at the Hebrew University in Jerusalem.

A null hypothesis is a type of statistical hypothesis that proposes that no statistical significance exists in a set of given observations. Hypothesis testing is used to assess the credibility of a hypothesis by using sample data. Sometimes referred to simply as the "null," it is represented as H 0 .

The null hypothesis, also known as the conjecture, is used in quantitative analysis to test theories about markets, investing strategies, or economies to decide if an idea is true or false.

Key Takeaways

• A null hypothesis is a type of conjecture in statistics that proposes that there is no difference between certain characteristics of a population or data-generating process.
• The alternative hypothesis proposes that there is a difference.
• Hypothesis testing provides a method to reject a null hypothesis within a certain confidence level.
• If you can reject the null hypothesis, it provides support for the alternative hypothesis.
• Null hypothesis testing is the basis of the principle of falsification in science.

Investopedia / Alex Dos Diaz

A null hypothesis is a type of conjecture in statistics that proposes that there is no difference between certain characteristics of a population or data-generating process. For example, a gambler may be interested in whether a game of chance is fair. If it is fair, then the expected earnings per play come to zero for both players. If the game is not fair, then the expected earnings are positive for one player and negative for the other. To test whether the game is fair, the gambler collects earnings data from many repetitions of the game, calculates the average earnings from these data, then tests the null hypothesis that the expected earnings are not different from zero.

If the average earnings from the sample data are sufficiently far from zero, then the gambler will reject the null hypothesis and conclude the alternative hypothesis—namely, that the expected earnings per play are different from zero. If the average earnings from the sample data are near zero, then the gambler will not reject the null hypothesis, concluding instead that the difference between the average from the data and zero is explainable by chance alone.

The null hypothesis assumes that any kind of difference between the chosen characteristics that you see in a set of data is due to chance. For example, if the expected earnings for the gambling game are truly equal to zero, then any difference between the average earnings in the data and zero is due to chance.

Analysts look to reject   the null hypothesis because doing so is a strong conclusion. This requires strong evidence in the form of an observed difference that is too large to be explained solely by chance. Failing to reject the null hypothesis—that the results are explainable by chance alone—is a weak conclusion because it allows that factors other than chance may be at work but may not be strong enough for the statistical test to detect them.

A null hypothesis can only be rejected, not proven.

An important point to note is that we are testing the null hypothesis because there is an element of doubt about its validity. Whatever information that is against the stated null hypothesis is captured in the alternative (alternate) hypothesis (H1).

For the above examples, the alternative hypothesis would be:

• Students score an average that is  not  equal to seven.
• The mean annual return of the mutual fund is  not  equal to 8% per year.

In other words, the alternative hypothesis is a direct contradiction of the null hypothesis.

Here is a simple example: A school principal claims that students in her school score an average of seven out of 10 in exams. The null hypothesis is that the population mean is 7.0. To test this null hypothesis, we record marks of, say, 30 students (sample) from the entire student population of the school (say 300) and calculate the mean of that sample.

We can then compare the (calculated) sample mean to the (hypothesized) population mean of 7.0 and attempt to reject the null hypothesis. (The null hypothesis here—that the population mean is 7.0—cannot be proved using the sample data. It can only be rejected.)

Take another example: The annual return of a particular  mutual fund  is claimed to be 8%. Assume that a mutual fund has been in existence for 20 years. The null hypothesis is that the mean return is 8% for the mutual fund. We take a random sample of annual returns of the mutual fund for, say, five years (sample) and calculate the sample mean. We then compare the (calculated) sample mean to the (claimed) population mean (8%) to test the null hypothesis.

For the above examples, null hypotheses are:

• Example A : Students in the school score an average of seven out of 10 in exams.
• Example B: Mean annual return of the mutual fund is 8% per year.

For the purposes of determining whether to reject the null hypothesis, the null hypothesis (abbreviated H 0 ) is assumed, for the sake of argument, to be true. Then the likely range of possible values of the calculated statistic (e.g., the average score on 30 students’ tests) is determined under this presumption (e.g., the range of plausible averages might range from 6.2 to 7.8 if the population mean is 7.0). Then, if the sample average is outside of this range, the null hypothesis is rejected. Otherwise, the difference is said to be “explainable by chance alone,” being within the range that is determined by chance alone.

How Null Hypothesis Testing Is Used in Investments

As an example related to financial markets, assume Alice sees that her investment strategy produces higher average returns than simply buying and holding a stock . The null hypothesis states that there is no difference between the two average returns, and Alice is inclined to believe this until she can conclude contradictory results.

Refuting the null hypothesis would require showing statistical significance, which can be found by a variety of tests. The alternative hypothesis would state that the investment strategy has a higher average return than a traditional buy-and-hold strategy.

One tool that can determine the statistical significance of the results is the p-value. A p-value represents the probability that a difference as large or larger than the observed difference between the two average returns could occur solely by chance.

A p-value that is less than or equal to 0.05 often indicates whether there is evidence against the null hypothesis. If Alice conducts one of these tests, such as a test using the normal model, resulting in a significant difference between her returns and the buy-and-hold returns (the p-value is less than or equal to 0.05), she can then reject the null hypothesis and conclude the alternative hypothesis.

How Is the Null Hypothesis Identified?

The analyst or researcher establishes a null hypothesis based on the research question or problem that they are trying to answer. Depending on the question, the null may be identified differently. For example, if the question is simply whether an effect exists (e.g., does X influence Y?) the null hypothesis could be H 0 : X = 0. If the question is instead, is X the same as Y, the H0 would be X = Y. If it is that the effect of X on Y is positive, H0 would be X > 0. If the resulting analysis shows an effect that is statistically significantly different from zero, the null can be rejected.

How Is Null Hypothesis Used in Finance?

In finance, a null hypothesis is used in quantitative analysis. A null hypothesis tests the premise of an investing strategy, the markets, or an economy to determine if it is true or false. For instance, an analyst may want to see if two stocks, ABC and XYZ, are closely correlated. The null hypothesis would be ABC ≠ XYZ.

How Are Statistical Hypotheses Tested?

Statistical hypotheses are tested by a four-step process . The first step is for the analyst to state the two hypotheses so that only one can be right. The next step is to formulate an analysis plan, which outlines how the data will be evaluated. The third step is to carry out the plan and physically analyze the sample data. The fourth and final step is to analyze the results and either reject the null hypothesis or claim that the observed differences are explainable by chance alone.

What Is an Alternative Hypothesis?

An alternative hypothesis is a direct contradiction of a null hypothesis. This means that if one of the two hypotheses is true, the other is false.

Sage Publishing. " Chapter 8: Introduction to Hypothesis Testing ," Pages 4–7.

Sage Publishing. " Chapter 8: Introduction to Hypothesis Testing ," Page 4.

Sage Publishing. " Chapter 8: Introduction to Hypothesis Testing ," Page 7.

• Terms of Service
• Editorial Policy
• Privacy Policy
• Your Privacy Choices

• school Campus Bookshelves
• menu_book Bookshelves
• perm_media Learning Objects
• login Login
• how_to_reg Request Instructor Account
• hub Instructor Commons
• Download Page (PDF)
• Download Full Book (PDF)
• Periodic Table
• Physics Constants
• Scientific Calculator
• Reference & Cite
• Tools expand_more
• Readability

selected template will load here

This action is not available.

11.1: Testing the Hypothesis that β = 0

• Last updated
• Save as PDF
• Page ID 26113

The correlation coefficient, \(r\), tells us about the strength and direction of the linear relationship between \(x\) and \(y\). However, the reliability of the linear model also depends on how many observed data points are in the sample. We need to look at both the value of the correlation coefficient \(r\) and the sample size \(n\), together. We perform a hypothesis test of the "significance of the correlation coefficient" to decide whether the linear relationship in the sample data is strong enough to use to model the relationship in the population.

The sample data are used to compute \(r\), the correlation coefficient for the sample. If we had data for the entire population, we could find the population correlation coefficient. But because we have only sample data, we cannot calculate the population correlation coefficient. The sample correlation coefficient, \(r\), is our estimate of the unknown population correlation coefficient.

• The symbol for the population correlation coefficient is \(\rho\), the Greek letter "rho."
• \(\rho =\) population correlation coefficient (unknown)
• \(r =\) sample correlation coefficient (known; calculated from sample data)

The hypothesis test lets us decide whether the value of the population correlation coefficient \(\rho\) is "close to zero" or "significantly different from zero". We decide this based on the sample correlation coefficient \(r\) and the sample size \(n\).

If the test concludes that the correlation coefficient is significantly different from zero, we say that the correlation coefficient is "significant."

• Conclusion: There is sufficient evidence to conclude that there is a significant linear relationship between \(x\) and \(y\) because the correlation coefficient is significantly different from zero.
• What the conclusion means: There is a significant linear relationship between \(x\) and \(y\). We can use the regression line to model the linear relationship between \(x\) and \(y\) in the population.

If the test concludes that the correlation coefficient is not significantly different from zero (it is close to zero), we say that correlation coefficient is "not significant".

• Conclusion: "There is insufficient evidence to conclude that there is a significant linear relationship between \(x\) and \(y\) because the correlation coefficient is not significantly different from zero."
• What the conclusion means: There is not a significant linear relationship between \(x\) and \(y\). Therefore, we CANNOT use the regression line to model a linear relationship between \(x\) and \(y\) in the population.
• If \(r\) is significant and the scatter plot shows a linear trend, the line can be used to predict the value of \(y\) for values of \(x\) that are within the domain of observed \(x\) values.
• If \(r\) is not significant OR if the scatter plot does not show a linear trend, the line should not be used for prediction.
• If \(r\) is significant and if the scatter plot shows a linear trend, the line may NOT be appropriate or reliable for prediction OUTSIDE the domain of observed \(x\) values in the data.

PERFORMING THE HYPOTHESIS TEST

• Null Hypothesis: \(H_{0}: \rho = 0\)
• Alternate Hypothesis: \(H_{a}: \rho \neq 0\)

WHAT THE HYPOTHESES MEAN IN WORDS:

• Null Hypothesis \(H_{0}\) : The population correlation coefficient IS NOT significantly different from zero. There IS NOT a significant linear relationship(correlation) between \(x\) and \(y\) in the population.
• Alternate Hypothesis \(H_{a}\) : The population correlation coefficient IS significantly DIFFERENT FROM zero. There IS A SIGNIFICANT LINEAR RELATIONSHIP (correlation) between \(x\) and \(y\) in the population.

DRAWING A CONCLUSION:There are two methods of making the decision. The two methods are equivalent and give the same result.

• Method 1: Using the \(p\text{-value}\)
• Method 2: Using a table of critical values

In this chapter of this textbook, we will always use a significance level of 5%, \(\alpha = 0.05\)

Using the \(p\text{-value}\) method, you could choose any appropriate significance level you want; you are not limited to using \(\alpha = 0.05\). But the table of critical values provided in this textbook assumes that we are using a significance level of 5%, \(\alpha = 0.05\). (If we wanted to use a different significance level than 5% with the critical value method, we would need different tables of critical values that are not provided in this textbook.)

METHOD 1: Using a \(p\text{-value}\) to make a decision

To calculate the \(p\text{-value}\) using LinRegTTEST:

On the LinRegTTEST input screen, on the line prompt for \(\beta\) or \(\rho\), highlight "\(\neq 0\)"

The output screen shows the \(p\text{-value}\) on the line that reads "\(p =\)".

(Most computer statistical software can calculate the \(p\text{-value}\).)

If the \(p\text{-value}\) is less than the significance level ( \(\alpha = 0.05\) ):

• Decision: Reject the null hypothesis.
• Conclusion: "There is sufficient evidence to conclude that there is a significant linear relationship between \(x\) and \(y\) because the correlation coefficient is significantly different from zero."

If the \(p\text{-value}\) is NOT less than the significance level ( \(\alpha = 0.05\) )

• Decision: DO NOT REJECT the null hypothesis.
• Conclusion: "There is insufficient evidence to conclude that there is a significant linear relationship between \(x\) and \(y\) because the correlation coefficient is NOT significantly different from zero."

Calculation Notes:

• You will use technology to calculate the \(p\text{-value}\). The following describes the calculations to compute the test statistics and the \(p\text{-value}\):
• The \(p\text{-value}\) is calculated using a \(t\)-distribution with \(n - 2\) degrees of freedom.
• The formula for the test statistic is \(t = \frac{r\sqrt{n-2}}{\sqrt{1-r^{2}}}\). The value of the test statistic, \(t\), is shown in the computer or calculator output along with the \(p\text{-value}\). The test statistic \(t\) has the same sign as the correlation coefficient \(r\).
• The \(p\text{-value}\) is the combined area in both tails.

An alternative way to calculate the \(p\text{-value}\) ( \(p\) ) given by LinRegTTest is the command 2*tcdf(abs(t),10^99, n-2) in 2nd DISTR.

THIRD-EXAM vs FINAL-EXAM EXAMPLE: \(p\text{-value}\) method

• Consider the third exam/final exam example.
• The line of best fit is: \(\hat{y} = -173.51 + 4.83x\) with \(r = 0.6631\) and there are \(n = 11\) data points.
• Can the regression line be used for prediction? Given a third exam score ( \(x\) value), can we use the line to predict the final exam score (predicted \(y\) value)?
• \(H_{0}: \rho = 0\)
• \(H_{a}: \rho \neq 0\)
• \(\alpha = 0.05\)
• The \(p\text{-value}\) is 0.026 (from LinRegTTest on your calculator or from computer software).
• The \(p\text{-value}\), 0.026, is less than the significance level of \(\alpha = 0.05\).
• Decision: Reject the Null Hypothesis \(H_{0}\)
• Conclusion: There is sufficient evidence to conclude that there is a significant linear relationship between the third exam score (\(x\)) and the final exam score (\(y\)) because the correlation coefficient is significantly different from zero.

Because \(r\) is significant and the scatter plot shows a linear trend, the regression line can be used to predict final exam scores.

METHOD 2: Using a table of Critical Values to make a decision

The 95% Critical Values of the Sample Correlation Coefficient Table can be used to give you a good idea of whether the computed value of \(r\) is significant or not . Compare \(r\) to the appropriate critical value in the table. If \(r\) is not between the positive and negative critical values, then the correlation coefficient is significant. If \(r\) is significant, then you may want to use the line for prediction.

Example \(\PageIndex{1}\)

Suppose you computed \(r = 0.801\) using \(n = 10\) data points. \(df = n - 2 = 10 - 2 = 8\). The critical values associated with \(df = 8\) are \(-0.632\) and \(+0.632\). If \(r <\) negative critical value or \(r >\) positive critical value, then \(r\) is significant. Since \(r = 0.801\) and \(0.801 > 0.632\), \(r\) is significant and the line may be used for prediction. If you view this example on a number line, it will help you.

Exercise \(\PageIndex{1}\)

For a given line of best fit, you computed that \(r = 0.6501\) using \(n = 12\) data points and the critical value is 0.576. Can the line be used for prediction? Why or why not?

If the scatter plot looks linear then, yes, the line can be used for prediction, because \(r >\) the positive critical value.

Example \(\PageIndex{2}\)

Suppose you computed \(r = –0.624\) with 14 data points. \(df = 14 – 2 = 12\). The critical values are \(-0.532\) and \(0.532\). Since \(-0.624 < -0.532\), \(r\) is significant and the line can be used for prediction

Exercise \(\PageIndex{2}\)

For a given line of best fit, you compute that \(r = 0.5204\) using \(n = 9\) data points, and the critical value is \(0.666\). Can the line be used for prediction? Why or why not?

No, the line cannot be used for prediction, because \(r <\) the positive critical value.

Example \(\PageIndex{3}\)

Suppose you computed \(r = 0.776\) and \(n = 6\). \(df = 6 - 2 = 4\). The critical values are \(-0.811\) and \(0.811\). Since \(-0.811 < 0.776 < 0.811\), \(r\) is not significant, and the line should not be used for prediction.

Exercise \(\PageIndex{3}\)

For a given line of best fit, you compute that \(r = -0.7204\) using \(n = 8\) data points, and the critical value is \(= 0.707\). Can the line be used for prediction? Why or why not?

Yes, the line can be used for prediction, because \(r <\) the negative critical value.

THIRD-EXAM vs FINAL-EXAM EXAMPLE: critical value method

Consider the third exam/final exam example. The line of best fit is: \(\hat{y} = -173.51 + 4.83x\) with \(r = 0.6631\) and there are \(n = 11\) data points. Can the regression line be used for prediction? Given a third-exam score ( \(x\) value), can we use the line to predict the final exam score (predicted \(y\) value)?

• Use the "95% Critical Value" table for \(r\) with \(df = n - 2 = 11 - 2 = 9\).
• The critical values are \(-0.602\) and \(+0.602\)
• Since \(0.6631 > 0.602\), \(r\) is significant.
• Conclusion:There is sufficient evidence to conclude that there is a significant linear relationship between the third exam score (\(x\)) and the final exam score (\(y\)) because the correlation coefficient is significantly different from zero.

Example \(\PageIndex{4}\)

Suppose you computed the following correlation coefficients. Using the table at the end of the chapter, determine if \(r\) is significant and the line of best fit associated with each r can be used to predict a \(y\) value. If it helps, draw a number line.

• \(r = –0.567\) and the sample size, \(n\), is \(19\). The \(df = n - 2 = 17\). The critical value is \(-0.456\). \(-0.567 < -0.456\) so \(r\) is significant.
• \(r = 0.708\) and the sample size, \(n\), is \(9\). The \(df = n - 2 = 7\). The critical value is \(0.666\). \(0.708 > 0.666\) so \(r\) is significant.
• \(r = 0.134\) and the sample size, \(n\), is \(14\). The \(df = 14 - 2 = 12\). The critical value is \(0.532\). \(0.134\) is between \(-0.532\) and \(0.532\) so \(r\) is not significant.
• \(r = 0\) and the sample size, \(n\), is five. No matter what the \(dfs\) are, \(r = 0\) is between the two critical values so \(r\) is not significant.

Exercise \(\PageIndex{4}\)

For a given line of best fit, you compute that \(r = 0\) using \(n = 100\) data points. Can the line be used for prediction? Why or why not?

No, the line cannot be used for prediction no matter what the sample size is.

Assumptions in Testing the Significance of the Correlation Coefficient

Testing the significance of the correlation coefficient requires that certain assumptions about the data are satisfied. The premise of this test is that the data are a sample of observed points taken from a larger population. We have not examined the entire population because it is not possible or feasible to do so. We are examining the sample to draw a conclusion about whether the linear relationship that we see between \(x\) and \(y\) in the sample data provides strong enough evidence so that we can conclude that there is a linear relationship between \(x\) and \(y\) in the population.

The regression line equation that we calculate from the sample data gives the best-fit line for our particular sample. We want to use this best-fit line for the sample as an estimate of the best-fit line for the population. Examining the scatter plot and testing the significance of the correlation coefficient helps us determine if it is appropriate to do this.

The assumptions underlying the test of significance are:

• There is a linear relationship in the population that models the average value of \(y\) for varying values of \(x\). In other words, the expected value of \(y\) for each particular value lies on a straight line in the population. (We do not know the equation for the line for the population. Our regression line from the sample is our best estimate of this line in the population.)
• The \(y\) values for any particular \(x\) value are normally distributed about the line. This implies that there are more \(y\) values scattered closer to the line than are scattered farther away. Assumption (1) implies that these normal distributions are centered on the line: the means of these normal distributions of \(y\) values lie on the line.
• The standard deviations of the population \(y\) values about the line are equal for each value of \(x\). In other words, each of these normal distributions of \(y\) values has the same shape and spread about the line.
• The residual errors are mutually independent (no pattern).
• The data are produced from a well-designed, random sample or randomized experiment.

Linear regression is a procedure for fitting a straight line of the form \(\hat{y} = a + bx\) to data. The conditions for regression are:

• Linear In the population, there is a linear relationship that models the average value of \(y\) for different values of \(x\).
• Independent The residuals are assumed to be independent.
• Normal The \(y\) values are distributed normally for any value of \(x\).
• Equal variance The standard deviation of the \(y\) values is equal for each \(x\) value.
• Random The data are produced from a well-designed random sample or randomized experiment.

The slope \(b\) and intercept \(a\) of the least-squares line estimate the slope \(\beta\) and intercept \(\alpha\) of the population (true) regression line. To estimate the population standard deviation of \(y\), \(\sigma\), use the standard deviation of the residuals, \(s\). \(s = \sqrt{\frac{SEE}{n-2}}\). The variable \(\rho\) (rho) is the population correlation coefficient. To test the null hypothesis \(H_{0}: \rho =\) hypothesized value , use a linear regression t-test. The most common null hypothesis is \(H_{0}: \rho = 0\) which indicates there is no linear relationship between \(x\) and \(y\) in the population. The TI-83, 83+, 84, 84+ calculator function LinRegTTest can perform this test (STATS TESTS LinRegTTest).

Formula Review

Least Squares Line or Line of Best Fit:

\[\hat{y} = a + bx\]

\[a = y\text{-intercept}\]

\[b = \text{slope}\]

Standard deviation of the residuals:

\[s = \sqrt{\frac{SEE}{n-2}}\]

\[SSE = \text{sum of squared errors}\]

\[n = \text{the number of data points}\]

Statistics Made Easy

When Do You Reject the Null Hypothesis? (3 Examples)

A hypothesis test is a formal statistical test we use to reject or fail to reject a statistical hypothesis.

We always use the following steps to perform a hypothesis test:

Step 1: State the null and alternative hypotheses.

The null hypothesis , denoted as H 0 , is the hypothesis that the sample data occurs purely from chance.

The alternative hypothesis , denoted as H A , is the hypothesis that the sample data is influenced by some non-random cause.

2. Determine a significance level to use.

Decide on a significance level. Common choices are .01, .05, and .1. 

3. Calculate the test statistic and p-value.

Use the sample data to calculate a test statistic and a corresponding p-value .

4. Reject or fail to reject the null hypothesis.

If the p-value is less than the significance level, then you reject the null hypothesis.

If the p-value is not less than the significance level, then you fail to reject the null hypothesis.

You can use the following clever line to remember this rule:

“If the p is low, the null must go.”

In other words, if the p-value is low enough then we must reject the null hypothesis.

The following examples show when to reject (or fail to reject) the null hypothesis for the most common types of hypothesis tests.

Example 1: One Sample t-test

A  one sample t-test  is used to test whether or not the mean of a population is equal to some value.

For example, suppose we want to know whether or not the mean weight of a certain species of turtle is equal to 310 pounds.

We go out and collect a simple random sample of 40 turtles with the following information:

• Sample size n = 40
• Sample mean weight  x  = 300
• Sample standard deviation s = 18.5

We can use the following steps to perform a one sample t-test:

Step 1: State the Null and Alternative Hypotheses

We will perform the one sample t-test with the following hypotheses:

• H 0 :  μ = 310 (population mean is equal to 310 pounds)
• H A :  μ ≠ 310 (population mean is not equal to 310 pounds)

We will choose to use a significance level of 0.05 .

We can plug in the numbers for the sample size, sample mean, and sample standard deviation into this One Sample t-test Calculator to calculate the test statistic and p-value:

• t test statistic: -3.4187
• two-tailed p-value: 0.0015

Since the p-value (0.0015) is less than the significance level (0.05) we reject the null hypothesis .

We conclude that there is sufficient evidence to say that the mean weight of turtles in this population is not equal to 310 pounds.

Example 2: Two Sample t-test

A  two sample t-test is used to test whether or not two population means are equal.

For example, suppose we want to know whether or not the mean weight between two different species of turtles is equal.

We go out and collect a simple random sample from each population with the following information:

• Sample size n 1 = 40
• Sample mean weight  x 1  = 300
• Sample standard deviation s 1 = 18.5
• Sample size n 2 = 38
• Sample mean weight  x 2  = 305
• Sample standard deviation s 2 = 16.7

We can use the following steps to perform a two sample t-test:

We will perform the two sample t-test with the following hypotheses:

• H 0 :  μ 1  = μ 2 (the two population means are equal)
• H 1 :  μ 1  ≠ μ 2 (the two population means are not equal)

We will choose to use a significance level of 0.10 .

We can plug in the numbers for the sample sizes, sample means, and sample standard deviations into this Two Sample t-test Calculator to calculate the test statistic and p-value:

• t test statistic: -1.2508
• two-tailed p-value: 0.2149

Since the p-value (0.2149) is not less than the significance level (0.10) we fail to reject the null hypothesis .

We do not have sufficient evidence to say that the mean weight of turtles between these two populations is different.

Example 3: Paired Samples t-test

A paired samples t-test is used to compare the means of two samples when each observation in one sample can be paired with an observation in the other sample.

For example, suppose we want to know whether or not a certain training program is able to increase the max vertical jump of college basketball players.

To test this, we may recruit a simple random sample of 20 college basketball players and measure each of their max vertical jumps. Then, we may have each player use the training program for one month and then measure their max vertical jump again at the end of the month:

We can use the following steps to perform a paired samples t-test:

We will perform the paired samples t-test with the following hypotheses:

• H 0 :  μ before = μ after (the two population means are equal)
• H 1 :  μ before ≠ μ after (the two population means are not equal)

We will choose to use a significance level of 0.01 .

We can plug in the raw data for each sample into this Paired Samples t-test Calculator to calculate the test statistic and p-value:

• t test statistic: -3.226
• two-tailed p-value: 0.0045

Since the p-value (0.0045) is less than the significance level (0.01) we reject the null hypothesis .

We have sufficient evidence to say that the mean vertical jump before and after participating in the training program is not equal.

Bonus: Decision Rule Calculator 

You can use this decision rule calculator to automatically determine whether you should reject or fail to reject a null hypothesis for a hypothesis test based on the value of the test statistic.

Published by Zach

Leave a reply cancel reply.

Your email address will not be published. Required fields are marked *