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Null versus Alternative hypothesis

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This symbol denotes the null hypothesis

Null hypothesis

Alternative Hypothesis

This symbol denotes the alternative hypothesis

Alternative hypothesis

Bedazzled Motor Company claims that its new sedan, the Glitteratti , will average better than 26 miles per gallon in the city. Use μ, the true average mileage of the Glitteratti .

H0: μ = 26 Ha: μ > 26 claim

H0: μ < 26 Ha: μ ≥ 26 claim

H0: μ > 26 claim Ha: μ ≤ 26

H0: μ = 26 Ha: μ < 26 claim

The owner of a soccer team claims that the average attendance at games is over 62,900, and he is therefore justified in moving the team to a city with a larger stadium.

H0: μ < 62,900 Ha: μ ≥ 62,900 claim

H0: μ = 62,900 claim Ha: μ < 62,900

H0: μ > 62,900 Ha: μ ≤ 62,900 claim

H0: μ = 62,900 Ha: μ > 62,900 claim

• 5. Multiple Choice Edit 3 minutes 1 pt Which of the following represents the appropriate null & alternative hypotheses? The mean height of women is greater than 64" H₀: μ > 64" Hₐ: μ ≠ 64" H₀: μ = 64" Hₐ: μ > 64" H₀: p > 64" Hₐ: p ≠ 64" H₀: p = 64" Hₐ: p > 64"

True or False:

The null hypothesis is always µ = #.

The alternate hypothesis is always µ < #

µ > # or

Your null and alternate hypotheses should always have the same number value.

A new diet program would like to claim that their methods result in a mean weight loss of more than ten pounds in two weeks. In order to determine if this is a valid claim, they hire an independent testing agency that then selects 25 people to be placed on this diet. The agency should be testing the null hypothesis H 0 : µ = 10 and the alternative hypothesis:

Ha: μ < 10

Ha: μ > 10

• 11. Multiple Choice Edit 1 minute 1 pt What is the mean? # happening the most the number in the middle biggest-smallest the average

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• Null and Alternative Hypotheses | Definitions & Examples

Null and Alternative Hypotheses | Definitions & Examples

Published on 5 October 2022 by Shaun Turney . Revised on 6 December 2022.

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 ): There’s an effect in the population.

The effect is usually the effect of the independent variable on the dependent variable .

Table of contents

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

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.” On the other hand, 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.

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.

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 ≤).

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.

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.

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

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.

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

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

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

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

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Turney, S. (2022, December 06). Null and Alternative Hypotheses | Definitions & Examples. Scribbr. Retrieved 6 May 2024, from https://www.scribbr.co.uk/stats/null-and-alternative-hypothesis/

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• Q 1 / 8 Score 0 What is a type I error? 29 failing to reject a null hypothesis when it is actually false accepting a null hypothesis when it is actually false accepting an alternative hypothesis when it is actually false rejecting a null hypothesis when it is actually true

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• Q 1 What is a type I error? failing to reject a null hypothesis when it is actually false accepting a null hypothesis when it is actually false accepting an alternative hypothesis when it is actually false rejecting a null hypothesis when it is actually true 30 s
• Q 2 What is the p-value? the probability of a type I error the expected value of the test statistic, assuming the null hypothesis is true the probability of obtaining a test statistic at least as extreme as the one observed, assuming the null hypothesis is true the probability that the null hypothesis is true 30 s
• Q 3 What is the critical region? the set of all possible outcomes the set of all test statistics that would lead to acceptance of the null hypothesis at a given significance level the set of all test statistics that would lead to either rejection or acceptance of the null hypothesis the set of all test statistics that would lead to rejection of the null hypothesis at a given significance level 30 s
• Q 4 What is the test statistic? a calculated value that represents the number of standard deviations a sample mean is from the null hypothesis mean a calculated value that represents the size of the sample a calculated value that represents the probability of the null hypothesis being true a calculated value that represents the number of standard deviations the null hypothesis is from the sample mean 30 s
• Q 5 What is a two-tailed test? a test in which the alternative hypothesis specifies either a greater than or less than relationship between the populations or samples a test in which the null hypothesis is rejected a test in which there is no alternative hypothesis a test in which the alternative hypothesis specifies a difference between the populations or samples but does not specify a greater than or less than relationship 30 s
• Q 6 What is a one-tailed test? a test in which the null hypothesis is rejected a test in which the alternative hypothesis specifies an equal relationship between the populations or samples a test in which the alternative hypothesis specifies either a greater than or less than relationship between the populations or samples a test in which there is no alternative hypothesis 30 s
• Q 7 What is the significance level? the p-value the probability of making a type I error the probability of making a type II error the probability that the null hypothesis is true 30 s
• Q 8 What is a null hypothesis? a statement that there is no difference between any populations or samples a statement that there is a significant difference between specified populations or samples a statement that there is no correlation between variables a statement that there is no significant difference between specified populations or samples 30 s

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9.2: Null and Alternative Hypotheses

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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] .

Null and Alternative Hypotheses

The null hypothesis is a statement that assumes there is no significant difference or relationship between variables in a population. The alternative hypothesis, on the other hand, suggests that there is a significant difference or relationship.

Imagine you are accused of cheating on a test. The null hypothesis would be that you did not cheat, while the alternative hypothesis would be that you did cheat.

Related terms

One-sample z-test : A statistical test used to compare a sample mean to a known population mean.

Significance level : Also known as alpha (α), it is the probability of rejecting the null hypothesis when it is actually true.

Type I error : Rejecting the null hypothesis when it is actually true.

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