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Null Hypothesis

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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 Null Hypothesis?

Null hypothesis symbol, formula of null hypothesis, types of null hypothesis, null hypothesis examples, principle of null hypothesis, how do you find null hypothesis, null hypothesis in statistics, null hypothesis and alternative hypothesis, null hypothesis and alternative hypothesis examples, null hypothesis – practice problems.

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.

Null Hypothesis Meaning

Null Hypothesis represents 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.

The 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.

• 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.

The null hypothesis is usually formulated to be tested against an alternative hypothesis (H 1 or H [Tex]\alpha [/Tex] ) which suggests that there is an effect, difference or relationship present in the population.

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.

In statistical hypothesis testing, researchers begin by stating the null hypothesis, often based on theoretical considerations or previous research. The null hypothesis is then tested against an alternative hypothesis (Ha), which represents the researcher’s claim or the hypothesis they seek to support.

The process of hypothesis testing involves collecting sample data and using statistical methods to assess the likelihood of observing the data if the null hypothesis were true. This assessment is typically done by calculating a test statistic, which measures the difference between the observed data and what would be expected under the null 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.

Difference Between Null Hypothesis and 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.

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.

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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.

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?

Summary – Null Hypothesis and Alternative Hypothesis

The null hypothesis (H 0 ) and alternative hypothesis (H a ) are fundamental concepts in statistical hypothesis testing. The null hypothesis represents the default assumption, stating that there is no significant effect, difference, or relationship between variables. It serves as the baseline against which the alternative hypothesis is tested. In contrast, the alternative hypothesis represents the researcher’s hypothesis or the claim to be tested, suggesting that there is a significant effect, difference, or relationship between variables. The relationship between the null and alternative hypotheses is such that they are complementary, and statistical tests are conducted to determine whether the evidence from the data is strong enough to reject the null hypothesis in favor of the alternative hypothesis. This decision is based on the strength of the evidence and the chosen level of significance. Ultimately, the choice between the null and alternative hypotheses depends on the specific research question and the direction of the effect being investigated.

FAQs on Null Hypothesis

What does null hypothesis stands for.

The null hypothesis, denoted as H 0 ​, is a fundamental concept in statistics used for hypothesis testing. It represents the statement that there is no effect or no difference, and it is the hypothesis that the researcher typically aims to provide evidence against.

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.

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.

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9.1 Null and Alternative Hypotheses

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

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

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

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

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

Mathematical Symbols Used in H 0 and H a :

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

Example 9.1

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

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

Example 9.2

We want to test whether the mean GPA of students in American colleges is different from 2.0 (out of 4.0). The null and alternative hypotheses are the following: 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 fewer than five years to graduate from college, on the average. The null and alternative hypotheses are the following: 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

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

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

• H 0 : p __ 0.40
• H a : p __ 0.40

Collaborative Exercise

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

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• Authors: Barbara Illowsky, Susan Dean
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• Publication date: Mar 27, 2020
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• Book URL: https://openstax.org/books/statistics/pages/1-introduction
• Section URL: https://openstax.org/books/statistics/pages/9-1-null-and-alternative-hypotheses

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9.1: 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.

\(H_0\): 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.

\(H_{0}\) 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.

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 .

Module 9: Hypothesis Testing With One Sample

Null and alternative hypotheses, learning outcomes.

• Describe hypothesis testing in general and in practice

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 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.

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

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

After you have determined which hypothesis the sample supports, you make adecision. 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

Concept 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 (=, ≤ or ≥) Always write the alternative hypothesis , typically denoted with H a or H 1 , using less than, greater than, or not equals symbols, i.e., (≠, >, 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.

• OpenStax, Statistics, Null and Alternative Hypotheses. Provided by : OpenStax. Located at : http://cnx.org/contents/[email protected]:58/Introductory_Statistics . License : CC BY: Attribution
• Introductory Statistics . Authored by : Barbara Illowski, Susan Dean. Provided by : Open Stax. Located at : http://cnx.org/contents/[email protected] . License : CC BY: Attribution . License Terms : Download for free at http://cnx.org/contents/[email protected]
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Exploring the Null Hypothesis: Definition and Purpose

Updated: July 5, 2023 by Ken Feldman

Hypothesis testing is a branch of statistics in which, using data from a sample, an inference is made about a population parameter or a population probability distribution .

First, a hypothesis statement and assumption is made about the population parameter or probability distribution. This initial statement is called the Null Hypothesis and is denoted by H o. An alternative or alternate hypothesis (denoted Ha ), is then stated which will be the opposite of the Null Hypothesis.

The hypothesis testing process and analysis involves using sample data to determine whether or not you can be statistically confident that you can reject or fail to reject the H o. If the H o is rejected, the statistical conclusion is that the alternative or alternate hypothesis Ha is true.

Overview: What is the Null Hypothesis (Ho)? 

Hypothesis testing applies to all forms of statistical inquiry. For example, it can be used to determine whether there are differences between population parameters or an understanding about slopes of regression lines or equality of probability distributions.

In all cases, the first thing you do is state the Null and Alternate Hypotheses. The word Null in the context of hypothesis testing means “nothing” or “zero.”  

As an example, if we wanted to test whether there was a difference in two population means based on the calculations from two samples, we would state the Null Hypothesis in the form of: 

Ho: mu1 = mu2 or mu1- mu2 = 0  

In other words, there is no difference, or the difference is zero. Note that the notation is in the form of a population parameter, not a sample statistic. 

Since you are using sample data to make your inferences about the population, it’s possible you’ll make an error. In the case of the Null Hypothesis, we can make one of two errors.

•   Type 1 , or alpha error: An alpha error is when you mistakenly reject the Null and believe that something significant happened. In other words, you believe that the means of the two populations are different when they aren’t.
• Type 2, or beta error: A beta error is when you fail to reject the null when you should have.  In this case, you missed something significant and failed to take action. 

A classic example is when you get the results back from your doctor after taking a blood test. If the doctor says you have an infection when you really don’t, that is an alpha error. That is thinking that there is something significant going on when there isn’t. We also call that a false positive. The doctor rejected the null that “there was zero infection” and missed the call.

On the other hand, if the doctor told you that everything was OK when you really did have an infection, then he made a beta, or type 2, error. He failed to reject the Null Hypothesis when he should have. That is called a false negative.

The decision to reject or not to reject the Null Hypothesis is based on three numbers. 

• Alpha, which you get to choose. Alpha is the risk you are willing to assume of falsely rejecting the Null. The typical values for alpha are 1%, 5%, or 10%. Depending on the importance of the conclusion, you only want to falsely claim a difference when there is none, 1%, 5%, or 10% of the time.
• Beta, which is typically 20%. This means you’re willing to be wrong 20% of the time in failing to reject the null when you should have. 
• P-value, which is calculated from the data. The p-value is the actual risk you have in being wrong if you reject the null. You would like that to be low.  

Your decision as to what to do about the null is made by comparing the alpha value (your assumed risk) with the p-value (actual risk). If the actual risk is lower than your assumed risk, you can feel comfortable in rejecting the null and claiming something has happened. But, if the actual risk is higher than your assumed risk you will be taking a bigger risk than you want by rejecting the null.

RELATED: NULL VS. ALTERNATIVE HYPOTHESIS

3 benefits of the null hypothesis .

The stating and testing of the null hypothesis is the foundation of hypothesis testing. By doing so, you set the parameters for your statistical inference.

1. Statistical assurance of determining differences between population parameters

Just looking at the mathematical difference between the means of two samples and making a decision is woefully inadequate. By statistically testing the null hypothesis, you will have more confidence in any inferences you want to make about populations based on your samples.

2. Statistically based estimation of the probability of a population distribution

Many statistical tests require assumptions of specific distributions. Many of these tests assume that the population follows the normal distribution . If it doesn’t, the test may be invalid.  

3. Assess the strength of your conclusions as to what to do with the null hypothesis

Hypothesis testing calculations will provide some relative strength to your decisions as to whether you reject or fail to reject the null hypothesis.

Why is the Null Hypothesis important to understand?

The interpretation of the statistics relative to the null hypothesis is what’s important.

1. Properly write the null hypothesis to properly capture what you are seeking to prove

The null is always written in the same format. That is, the lack of difference or some other condition. The alternative hypothesis can be written in three formats depending on what you want to prove. 

2. Frame your statement and select an appropriate alpha risk

You don’t want to place too big of a hurdle or burden on your decision-making relative to action on the null hypothesis by selecting an alpha value that is too high or too low.

3. There are decision errors when deciding on how to respond to the Null Hypothesis

Since your decision relative to rejecting or not rejecting the null is based on statistical calculations, it is important to understand how that decision works. 

An industry example of using the Null Hypothesis 

The new director of marketing just completed the rollout of a new marketing campaign targeting the Hispanic market. Early indications showed that the campaign was successful in increasing sales in the Hispanic market. 

He came to that conclusion by comparing a sample of sales prior to the campaign and current sales after implementation of the campaign. He was anxious to proudly tell his boss how successful the campaign was. But, he decided to first check with his Lean Six Sigma Black Belt to see whether she agreed with his conclusion.

The Black Belt first asked the director his tolerance for risk of being wrong by telling the boss the campaign was successful when in fact, it wasn’t. That was the alpha value. The Director picked 5% since he was new and didn’t want to make a false claim so early in his career. He also picked 20% as his beta value.  

When the Black Belt was done analyzing the data, she found out that the p-value was 15%.  That meant if the director told the VP the campaign worked, there was a 15% chance he would be wrong and that the campaign probably needed some revising. Since he was only willing to be wrong 5% of the time, the decision was to not reject the null since his 5% assumed risk was less than the 15% actual risk.

3 best practices when thinking about the Null Hypothesis 

Using hypothesis testing to help make better data-driven decisions requires that you properly address the Null Hypothesis. 

1. Always use the proper nomenclature when stating the Null Hypothesis 

The null will always be in the form of decisions regarding the population, not the sample. 

2. The Null Hypothesis will always be written as the absence of some parameter or process characteristic

The writing of the Alternate Hypothesis can vary, so be sure you understand exactly what condition you are testing against. 

3. Pick a reasonable alpha risk so you’re not always failing to reject the Null Hypothesis

Being too cautious will lead you to make beta errors, and you’ll never learn anything about your population data. 

Frequently Asked Questions (FAQ) about the Null Hypothesis

What form should the null hypothesis be written in.

The Null Hypothesis should always be in the form of no difference or zero and always refer to the state of the population, not the sample. 

What is an alpha error? 

An alpha error, or Type 1 error, is rejecting the Null Hypothesis and claiming a significant event has occurred when, in fact, that is not true and the Null should not have been rejected.

How do I use the alpha error and p-value to decide on what decision I should make about the Null Hypothesis? 

The most common way of answering this is, “If the p-value is low (less than the alpha), the Null should be rejected. If the p-value is high (greater than the alpha) then the Null should not be rejected.”

Becoming familiar with the Null Hypothesis (Ho)

The proper writing of the Null Hypothesis is the basis for applying hypothesis testing to help you make better data-driven decisions. The format of the Null will always be in the form of zero, or the non-existence of some condition. It will always refer to a population parameter and not the sample you use to do your hypothesis testing calculations.

Be aware of the two types of errors you can make when deciding on what to do with the Null. Select reasonable risks values for your alpha and beta risks. By comparing your alpha risk with the calculated risk computed from the data, you will have sufficient information to make a wise decision as to whether you should reject the Null Hypothesis or not.

About the Author

Ken Feldman

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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

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Video transcript