Hypothesis Testing Calculator

Related: confidence interval calculator, type ii error.

The first step in hypothesis testing is to calculate the test statistic. The formula for the test statistic depends on whether the population standard deviation (σ) is known or unknown. If σ is known, our hypothesis test is known as a z test and we use the z distribution. If σ is unknown, our hypothesis test is known as a t test and we use the t distribution. Use of the t distribution relies on the degrees of freedom, which is equal to the sample size minus one. Furthermore, if the population standard deviation σ is unknown, the sample standard deviation s is used instead. To switch from σ known to σ unknown, click on $\boxed{\sigma}$ and select $\boxed{s}$ in the Hypothesis Testing Calculator.

Next, the test statistic is used to conduct the test using either the p-value approach or critical value approach. The particular steps taken in each approach largely depend on the form of the hypothesis test: lower tail, upper tail or two-tailed. The form can easily be identified by looking at the alternative hypothesis (H a ). If there is a less than sign in the alternative hypothesis then it is a lower tail test, greater than sign is an upper tail test and inequality is a two-tailed test. To switch from a lower tail test to an upper tail or two-tailed test, click on $\boxed{\geq}$ and select $\boxed{\leq}$ or $\boxed{=}$, respectively.

In the p-value approach, the test statistic is used to calculate a p-value. If the test is a lower tail test, the p-value is the probability of getting a value for the test statistic at least as small as the value from the sample. If the test is an upper tail test, the p-value is the probability of getting a value for the test statistic at least as large as the value from the sample. In a two-tailed test, the p-value is the probability of getting a value for the test statistic at least as unlikely as the value from the sample.

To test the hypothesis in the p-value approach, compare the p-value to the level of significance. If the p-value is less than or equal to the level of signifance, reject the null hypothesis. If the p-value is greater than the level of significance, do not reject the null hypothesis. This method remains unchanged regardless of whether it's a lower tail, upper tail or two-tailed test. To change the level of significance, click on $\boxed{.05}$. Note that if the test statistic is given, you can calculate the p-value from the test statistic by clicking on the switch symbol twice.

In the critical value approach, the level of significance ($\alpha$) is used to calculate the critical value. In a lower tail test, the critical value is the value of the test statistic providing an area of $\alpha$ in the lower tail of the sampling distribution of the test statistic. In an upper tail test, the critical value is the value of the test statistic providing an area of $\alpha$ in the upper tail of the sampling distribution of the test statistic. In a two-tailed test, the critical values are the values of the test statistic providing areas of $\alpha / 2$ in the lower and upper tail of the sampling distribution of the test statistic.

To test the hypothesis in the critical value approach, compare the critical value to the test statistic. Unlike the p-value approach, the method we use to decide whether to reject the null hypothesis depends on the form of the hypothesis test. In a lower tail test, if the test statistic is less than or equal to the critical value, reject the null hypothesis. In an upper tail test, if the test statistic is greater than or equal to the critical value, reject the null hypothesis. In a two-tailed test, if the test statistic is less than or equal the lower critical value or greater than or equal to the upper critical value, reject the null hypothesis.

When conducting a hypothesis test, there is always a chance that you come to the wrong conclusion. There are two types of errors you can make: Type I Error and Type II Error. A Type I Error is committed if you reject the null hypothesis when the null hypothesis is true. Ideally, we'd like to accept the null hypothesis when the null hypothesis is true. A Type II Error is committed if you accept the null hypothesis when the alternative hypothesis is true. Ideally, we'd like to reject the null hypothesis when the alternative hypothesis is true.

Hypothesis testing is closely related to the statistical area of confidence intervals. If the hypothesized value of the population mean is outside of the confidence interval, we can reject the null hypothesis. Confidence intervals can be found using the Confidence Interval Calculator . The calculator on this page does hypothesis tests for one population mean. Sometimes we're interest in hypothesis tests about two population means. These can be solved using the Two Population Calculator . The probability of a Type II Error can be calculated by clicking on the link at the bottom of the page.

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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 .

  • 9.1 Null and Alternative Hypotheses
  • Introduction
  • 1.1 Definitions of Statistics, Probability, and Key Terms
  • 1.2 Data, Sampling, and Variation in Data and Sampling
  • 1.3 Frequency, Frequency Tables, and Levels of Measurement
  • 1.4 Experimental Design and Ethics
  • 1.5 Data Collection Experiment
  • 1.6 Sampling Experiment
  • Chapter Review
  • Bringing It Together: Homework
  • 2.1 Stem-and-Leaf Graphs (Stemplots), Line Graphs, and Bar Graphs
  • 2.2 Histograms, Frequency Polygons, and Time Series Graphs
  • 2.3 Measures of the Location of the Data
  • 2.4 Box Plots
  • 2.5 Measures of the Center of the Data
  • 2.6 Skewness and the Mean, Median, and Mode
  • 2.7 Measures of the Spread of the Data
  • 2.8 Descriptive Statistics
  • Formula Review
  • 3.1 Terminology
  • 3.2 Independent and Mutually Exclusive Events
  • 3.3 Two Basic Rules of Probability
  • 3.4 Contingency Tables
  • 3.5 Tree and Venn Diagrams
  • 3.6 Probability Topics
  • Bringing It Together: Practice
  • 4.1 Probability Distribution Function (PDF) for a Discrete Random Variable
  • 4.2 Mean or Expected Value and Standard Deviation
  • 4.3 Binomial Distribution (Optional)
  • 4.4 Geometric Distribution (Optional)
  • 4.5 Hypergeometric Distribution (Optional)
  • 4.6 Poisson Distribution (Optional)
  • 4.7 Discrete Distribution (Playing Card Experiment)
  • 4.8 Discrete Distribution (Lucky Dice Experiment)
  • 5.1 Continuous Probability Functions
  • 5.2 The Uniform Distribution
  • 5.3 The Exponential Distribution (Optional)
  • 5.4 Continuous Distribution
  • 6.1 The Standard Normal Distribution
  • 6.2 Using the Normal Distribution
  • 6.3 Normal Distribution—Lap Times
  • 6.4 Normal Distribution—Pinkie Length
  • 7.1 The Central Limit Theorem for Sample Means (Averages)
  • 7.2 The Central Limit Theorem for Sums (Optional)
  • 7.3 Using the Central Limit Theorem
  • 7.4 Central Limit Theorem (Pocket Change)
  • 7.5 Central Limit Theorem (Cookie Recipes)
  • 8.1 A Single Population Mean Using the Normal Distribution
  • 8.2 A Single Population Mean Using the Student's t-Distribution
  • 8.3 A Population Proportion
  • 8.4 Confidence Interval (Home Costs)
  • 8.5 Confidence Interval (Place of Birth)
  • 8.6 Confidence Interval (Women's Heights)
  • 9.2 Outcomes and the Type I and Type II Errors
  • 9.3 Distribution Needed for Hypothesis Testing
  • 9.4 Rare Events, the Sample, and the Decision and Conclusion
  • 9.5 Additional Information and Full Hypothesis Test Examples
  • 9.6 Hypothesis Testing of a Single Mean and Single Proportion
  • 10.1 Two Population Means with Unknown Standard Deviations
  • 10.2 Two Population Means with Known Standard Deviations
  • 10.3 Comparing Two Independent Population Proportions
  • 10.4 Matched or Paired Samples (Optional)
  • 10.5 Hypothesis Testing for Two Means and Two Proportions
  • 11.1 Facts About the Chi-Square Distribution
  • 11.2 Goodness-of-Fit Test
  • 11.3 Test of Independence
  • 11.4 Test for Homogeneity
  • 11.5 Comparison of the Chi-Square Tests
  • 11.6 Test of a Single Variance
  • 11.7 Lab 1: Chi-Square Goodness-of-Fit
  • 11.8 Lab 2: Chi-Square Test of Independence
  • 12.1 Linear Equations
  • 12.2 The Regression Equation
  • 12.3 Testing the Significance of the Correlation Coefficient (Optional)
  • 12.4 Prediction (Optional)
  • 12.5 Outliers
  • 12.6 Regression (Distance from School) (Optional)
  • 12.7 Regression (Textbook Cost) (Optional)
  • 12.8 Regression (Fuel Efficiency) (Optional)
  • 13.1 One-Way ANOVA
  • 13.2 The F Distribution and the F Ratio
  • 13.3 Facts About the F Distribution
  • 13.4 Test of Two Variances
  • 13.5 Lab: One-Way ANOVA
  • A | Appendix A Review Exercises (Ch 3–13)
  • B | Appendix B Practice Tests (1–4) and Final Exams
  • C | Data Sets
  • D | Group and Partner Projects
  • E | Solution Sheets
  • F | Mathematical Phrases, Symbols, and Formulas
  • G | Notes for the TI-83, 83+, 84, 84+ Calculators

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

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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 19 February 2024, from https://www.scribbr.co.uk/stats/null-and-alternative-hypothesis/

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Hypothesis Testing Calculator

Understanding Hypothesis Testing: A Guide to the Hypothesis Testing Calculator

Hypothesis testing is a crucial statistical method used to make informed decisions about data and draw conclusions. Whether you’re a student, researcher, or professional, a Hypothesis Testing Calculator can be an invaluable tool in your statistical toolkit. Let’s explore what hypothesis testing is and how this calculator can assist you:

Hypothesis Testing Basics:

  • Null Hypothesis (H0): This is the default assumption or claim that there is no significant difference or effect. It’s often denoted as H0.
  • Alternative Hypothesis (Ha): This is the statement that contradicts the null hypothesis. It suggests that there is a significant difference or effect. It’s denoted as Ha.
  • Significance Level (α): This is the predetermined threshold (e.g., 0.05 or 5%) used to determine statistical significance. If the calculated p-value is less than α, you reject the null hypothesis.
  • p-value: This is the probability of observing the results (or more extreme results) if the null hypothesis is true. A small p-value suggests that the results are unlikely under the null hypothesis.

Key Features of the Hypothesis Testing Calculator:

  • Input Parameters: The calculator typically requires you to input sample data, choose the type of test (e.g., t-test, chi-square test), specify the null and alternative hypotheses, and set the significance level.
  • Calculations: Once you input the data and parameters, the calculator performs the necessary statistical tests and calculations. It generates results such as the test statistic, degrees of freedom, and the p-value.
  • Interpretation: Based on the results, the calculator helps you determine whether to reject or fail to reject the null hypothesis. It provides an interpretation of the findings, which is crucial for drawing conclusions.
  • Visual Representation: Some calculators may offer visual aids like graphs or charts to help you better understand the data distribution and test results.

Significance of the Hypothesis Testing Calculator:

  • Scientific Research: Researchers across various fields use hypothesis testing to validate their hypotheses and draw meaningful conclusions from data.
  • Quality Control: Industries use hypothesis testing to ensure the quality and consistency of products and processes.
  • Medical Studies: In medical research, hypothesis testing helps assess the effectiveness of treatments or interventions.
  • Academics: Students and educators use hypothesis testing to teach and learn statistical concepts and conduct experiments.
  • Data-Driven Decisions: Businesses use hypothesis testing to make data-driven decisions, such as whether to launch a new product based on market research.

Conclusion:

The Hypothesis Testing Calculator is a powerful tool that simplifies complex statistical analysis and enables data-driven decision-making. Whether you’re conducting experiments, analyzing survey data, or performing quality control, understanding hypothesis testing and using this calculator can help you make informed choices and contribute to evidence-based research and decision-making.

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Examples of null and alternative hypotheses

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Video transcript

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.

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Test statistics | Definition, Interpretation, and Examples

Published on July 17, 2020 by Rebecca Bevans . Revised on June 22, 2023.

The test statistic is a number calculated from a statistical test of a hypothesis. It shows how closely your observed data match the distribution expected under the null hypothesis of that statistical test.

The test statistic is used to calculate the p value of your results, helping to decide whether to reject your null hypothesis.

Table of contents

What exactly is a test statistic, types of test statistics, interpreting test statistics, reporting test statistics, other interesting articles, frequently asked questions about test statistics.

A test statistic describes how closely the distribution of your data matches the distribution predicted under the null hypothesis of the statistical test you are using.

The distribution of data is how often each observation occurs, and can be described by its central tendency and variation around that central tendency. Different statistical tests predict different types of distributions, so it’s important to choose the right statistical test for your hypothesis.

The test statistic summarizes your observed data into a single number using the central tendency, variation, sample size, and number of predictor variables in your statistical model.

Generally, the test statistic is calculated as the pattern in your data (i.e., the correlation between variables or difference between groups) divided by the variance in the data (i.e., the standard deviation ).

  • Null hypothesis ( H 0 ): There is no correlation between temperature and flowering date.
  • Alternate hypothesis ( H A or H 1 ): There is a correlation between temperature and flowering date.

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null hypothesis symbol calculator

Below is a summary of the most common test statistics, their hypotheses, and the types of statistical tests that use them.

Different statistical tests will have slightly different ways of calculating these test statistics, but the underlying hypotheses and interpretations of the test statistic stay the same.

In practice, you will almost always calculate your test statistic using a statistical program (R, SPSS, Excel, etc.), which will also calculate the p value of the test statistic. However, formulas to calculate these statistics by hand can be found online.

  • a regression coefficient of 0.36
  • a t value comparing that coefficient to the predicted range of regression coefficients under the null hypothesis of no relationship

The t value of the regression test is 2.36 – this is your test statistic.

For any combination of sample sizes and number of predictor variables, a statistical test will produce a predicted distribution for the test statistic. This shows the most likely range of values that will occur if your data follows the null hypothesis of the statistical test.

The more extreme your test statistic – the further to the edge of the range of predicted test values it is – the less likely it is that your data could have been generated under the null hypothesis of that statistical test.

The agreement between your calculated test statistic and the predicted values is described by the p value . The smaller the p value, the less likely your test statistic is to have occurred under the null hypothesis of the statistical test.

Because the test statistic is generated from your observed data, this ultimately means that the smaller the p value, the less likely it is that your data could have occurred if the null hypothesis was true.

Test statistics can be reported in the results section of your research paper along with the sample size, p value of the test, and any characteristics of your data that will help to put these results into context.

Whether or not you need to report the test statistic depends on the type of test you are reporting.

By surveying a random subset of 100 trees over 25 years we found a statistically significant ( p < 0.01) positive correlation between temperature and flowering dates ( R 2 = 0.36, SD = 0.057).

In our comparison of mouse diet A and mouse diet B, we found that the lifespan on diet A  ( M = 2.1 years; SD = 0.12) was significantly shorter than the lifespan on diet B ( M = 2.6 years; SD = 0.1), with an average difference of 6 months ( t (80) = -12.75; p < 0.01).

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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.

  • Confidence interval
  • Descriptive statistics
  • Measures of central tendency
  • Correlation coefficient

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

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  • Nonresponse bias
  • Regression to the mean

A test statistic is a number calculated by a  statistical test . It describes how far your observed data is from the  null hypothesis  of no relationship between  variables or no difference among sample groups.

The test statistic tells you how different two or more groups are from the overall population mean , or how different a linear slope is from the slope predicted by a null hypothesis . Different test statistics are used in different statistical tests.

The formula for the test statistic depends on the statistical test being used.

Generally, the test statistic is calculated as the pattern in your data (i.e. the correlation between variables or difference between groups) divided by the variance in the data (i.e. the standard deviation ).

The test statistic you use will be determined by the statistical test.

You can choose the right statistical test by looking at what type of data you have collected and what type of relationship you want to test.

The test statistic will change based on the number of observations in your data, how variable your observations are, and how strong the underlying patterns in the data are.

For example, if one data set has higher variability while another has lower variability, the first data set will produce a test statistic closer to the null hypothesis , even if the true correlation between two variables is the same in either data set.

Statistical significance is a term used by researchers to state that it is unlikely their observations could have occurred under the null hypothesis of a statistical test . Significance is usually denoted by a p -value , or probability value.

Statistical significance is arbitrary – it depends on the threshold, or alpha value, chosen by the researcher. The most common threshold is p < 0.05, which means that the data is likely to occur less than 5% of the time under the null hypothesis .

When the p -value falls below the chosen alpha value, then we say the result of the test is statistically significant.

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  • 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.

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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.

\alpha

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.

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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.

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Hypothesis testing involves the careful construction of two statements: the null hypothesis and the alternative hypothesis. These hypotheses can look very similar but are actually different.

How do we know which hypothesis is the null and which one is the alternative? We will see that there are a few ways to tell the difference.

The Null Hypothesis

The null hypothesis reflects that there will be no observed effect in our experiment. In a mathematical formulation of the null hypothesis, there will typically be an equal sign. This hypothesis is denoted by H 0 .

The null hypothesis is what we attempt to find evidence against in our hypothesis test. We hope to obtain a small enough p-value that it is lower than our level of significance alpha and we are justified in rejecting the null hypothesis. If our p-value is greater than alpha, then we fail to reject the null hypothesis.

If the null hypothesis is not rejected, then we must be careful to say what this means. The thinking on this is similar to a legal verdict. Just because a person has been declared "not guilty", it does not mean that he is innocent. In the same way, just because we failed to reject a null hypothesis it does not mean that the statement is true.

For example, we may want to investigate the claim that despite what convention has told us, the mean adult body temperature is not the accepted value of 98.6 degrees Fahrenheit . The null hypothesis for an experiment to investigate this is “The mean adult body temperature for healthy individuals is 98.6 degrees Fahrenheit.” If we fail to reject the null hypothesis, then our working hypothesis remains that the average adult who is healthy has a temperature of 98.6 degrees. We do not prove that this is true.

If we are studying a new treatment, the null hypothesis is that our treatment will not change our subjects in any meaningful way. In other words, the treatment will not produce any effect in our subjects.

The Alternative Hypothesis

The alternative or experimental hypothesis reflects that there will be an observed effect for our experiment. In a mathematical formulation of the alternative hypothesis, there will typically be an inequality, or not equal to symbol. This hypothesis is denoted by either H a or by H 1 .

The alternative hypothesis is what we are attempting to demonstrate in an indirect way by the use of our hypothesis test. If the null hypothesis is rejected, then we accept the alternative hypothesis. If the null hypothesis is not rejected, then we do not accept the alternative hypothesis. Going back to the above example of mean human body temperature, the alternative hypothesis is “The average adult human body temperature is not 98.6 degrees Fahrenheit.”

If we are studying a new treatment, then the alternative hypothesis is that our treatment does, in fact, change our subjects in a meaningful and measurable way.

The following set of negations may help when you are forming your null and alternative hypotheses. Most technical papers rely on just the first formulation, even though you may see some of the others in a statistics textbook.

  • Null hypothesis: “ x is equal to y .” Alternative hypothesis “ x is not equal to y .”
  • Null hypothesis: “ x is at least y .” Alternative hypothesis “ x is less than y .”
  • Null hypothesis: “ x is at most y .” Alternative hypothesis “ x is greater than y .”
  • An Example of a Hypothesis Test
  • Hypothesis Test for the Difference of Two Population Proportions
  • What Is a P-Value?
  • How to Conduct a Hypothesis Test
  • Hypothesis Test Example
  • Chi-Square Goodness of Fit Test
  • How to Do Hypothesis Tests With the Z.TEST Function in Excel
  • The Difference Between Type I and Type II Errors in Hypothesis Testing
  • Type I and Type II Errors in Statistics
  • The Runs Test for Random Sequences
  • What 'Fail to Reject' Means in a Hypothesis Test
  • What Is the Difference Between Alpha and P-Values?
  • An Example of Chi-Square Test for a Multinomial Experiment
  • Null Hypothesis Definition and Examples
  • What Is a Hypothesis? (Science)
  • Null Hypothesis Examples

Sign Test Calculator

Instructions: This calculator conducts a Sign Test. Please select the null and alternative hypotheses, enter the number of positives (+) and the number of negatives (-), along with the significance level, and the results of the sign test will be displayed for you (please disregard the ties):

null hypothesis symbol calculator

How to do a Sign Test?

More about the sign test for you to understand in a better way the results presented above: A sign test is a parametric test used to assess claims about a population median. It is typically used when the assumptions for a z-test for one mean are not met (namely, when the distribution departs significantly from normality). The test has, as every other hypothesis test, two non-overlapping hypotheses, the null and the alternative hypothesis. The null hypothesis is a statement about the population median, under the assumption of no effect, and the alternative hypothesis is the complementary hypothesis to the null hypothesis.

What is the sign test in statistics

  • The sign test is a non-parametric test, and as such, it does not require the sample to come from a normally distributed population
  • The sign test is very flexible and can be used in many contexts where it is possible to measure the outcome as "positive" or "negative" (such as being above or below the median, etc.)
  • Depending on our knowledge about the "no effect" situation, the sign-test can be two-tailed, left-tailed or right-tailed
  • The main principle of hypothesis testing is that the null hypothesis is rejected if the test statistic obtained is sufficiently unlikely under the assumption that the null hypothesis is true
  • If the sample size is small enough, then we need to use a comparison with a critical value (which depends on the significance level provided) that is obtained from a sign test table (check the back of your book).
  • If the sample size is large enough, then normal approximation can be used, and an appropriate z-test can be used.

How do you find the test value of a sign?

If \(X^+\) and \(X^-\) are the number of positive and negative signs, respectively, then the test statistic is computed as \(X = \min\{X^+, X^-\}\). The null hypothesis of the sign test is rejected if \(X \le X*\), where \(X*\) is the critical value for the Sign Test, for the significance level provided and the type of tails specified. If the sample size is large enough, a formula for a z-statistic can be used, and it is

If the sample size is large enough, the null hypothesis is rejected when the z-statistic lies on the rejection region, which is determined by the significance level (\(\alpha\)) and the type of tail (two-tailed, left-tailed or right-tailed).

The sign test can be used in case that the assumptions are not met for a one-sample t-test. If instead, the assumptions are met, then you can use our t-test for one mean calculator .

Applications of the Sign Test

The sign test is one of the most versatile tests in non-parametric statistics. It takes many shapes, starting with the basic test for a population median, but with simple adaptations it can be turned into a runs test or into a Wilcoxon signed rank test

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Statistical Power Calculator

The statistical power is a power of a binary hypothesis test. It is the probability that effectively rejects the null hypothesis value (H 0 ) when the alternative hypothesis value (H 1 ) is true. In this calculator, calculate the statistical power of a test (p = 1 - β) from the beta value.

Null Hypothesis Test

null hypothesis symbol calculator

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  1. Null Hypothesis

    null hypothesis symbol calculator

  2. How to Write a Null Hypothesis (with Examples and Templates)

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  3. PPT

    null hypothesis symbol calculator

  4. How to type null hypothesis symbol in word

    null hypothesis symbol calculator

  5. How To Find The Null And Alternative Hypothesis

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  6. PPT

    null hypothesis symbol calculator

VIDEO

  1. calculator 🧮 symbol in MSWORD 🤩🥰#ytshorts #freefire #trending #jayshreeram

  2. What's a Null Hypothesis, an Alternative Hypothesis, and Statistical Power

  3. What is Null Hypothesis Testing Mean? || Academic Research || Ettienne-Murphy

  4. #null hypothesis #p value

  5. Testing of Hypothesis,Null, alternative hypothesis, type-I & -II Error etc @VATAMBEDUSRAVANKUMAR

  6. Null Hypothesis vs Alternate Hypothesis

COMMENTS

  1. Hypothesis Testing Calculator with Steps

    Hypothesis Testing Calculator Solve Example 1 • Example 2 The first step in hypothesis testing is to calculate the test statistic. The formula for the test statistic depends on whether the population standard deviation (σ) is known or unknown. If σ is known, our hypothesis test is known as a z test and we use the z distribution.

  2. How to Write a Null Hypothesis (5 Examples)

    How to Write a Null Hypothesis (5 Examples) A hypothesis test uses sample data to determine whether or not some claim about a population parameter is true. Whenever we perform a hypothesis test, we always write a null hypothesis and an alternative hypothesis, which take the following forms:

  3. Null & Alternative Hypotheses

    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 (H0): There's no effect in the population. Alternative hypothesis (Ha or H1): There's an effect in the population.

  4. Null Hypothesis: Definition, Rejecting & Examples

    By Jim Frost 6 Comments What is a Null Hypothesis? The null hypothesis in statistics states that there is no difference between groups or no relationship between variables. It is one of two mutually exclusive hypotheses about a population in a hypothesis test.

  5. 9.1: Null and Alternative Hypotheses

    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.

  6. 9.1 Null and Alternative Hypotheses

    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

  7. P-value Calculator

    A P-value calculator is used to determine the statistical significance of an observed result in hypothesis testing. It takes as input the observed test statistic, the null hypothesis, and the relevant parameters of the statistical test (such as degrees of freedom), and computes the p-value.

  8. What symbols are used to represent null hypotheses?

    Χ 2 = 8.41 + 8.67 + 11.6 + 5.4 = 34.08. Step 3: Find the critical chi-square value. Since there are four groups (round and yellow, round and green, wrinkled and yellow, wrinkled and green), there are three degrees of freedom.. For a test of significance at α = .05 and df = 3, the Χ 2 critical value is 7.82.. Step 4: Compare the chi-square value to the critical value

  9. Null and Alternative Hypotheses

    The null and alternative hypotheses are two competing claims that researchers weigh evidence for and against using a statistical test: Null hypothesis (H0): There's no effect in the population. Alternative hypothesis (HA): There's an effect in the population. The effect is usually the effect of the independent variable on the dependent ...

  10. Hypothesis Testing Calculator

    Null Hypothesis (H0): This is the default assumption or claim that there is no significant difference or effect. It's often denoted as H0. Alternative Hypothesis (Ha): This is the statement that contradicts the null hypothesis. It suggests that there is a significant difference or effect. It's denoted as Ha.

  11. Hypothesis Testing Calculator

     This Hypothesis Testing Calculator determines whether an alternative hypothesis is true or not. Based on whether it is true or not determines whether we accept or reject the hypothesis. We accept true hypotheses and reject false hypotheses. The null hypothesis is the hypothesis that is claimed and that we will test against.

  12. Examples of null and alternative hypotheses

    Does the null hypothesis need to have the equality symbol only? In the second question about average hours of sleep, Sal writes the null hypothesis H0: u >= 8 hours. However, in every other null hypothesis I've seen on Khan Academy, the null hypothesis has a "=" not a ">=" or "<=".

  13. Null & Alternative Hypothesis

    P-value (the probability value) is the value p of the statistic used to test the null hypothesis. If p < α then we reject the null hypothesis.

  14. Hypothesis Test Calculator

    This section answers some common questions about . Use this Hypothesis Test Calculator for quick results in Python and R. Learn the step-by-step hypothesis test process and why hypothesis testing is important.

  15. Null and Alternative Hypotheses

    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.

  16. Test statistics

    Null hypothesis (H0): There is no correlation between temperature and flowering date. Alternate hypothesis (HA or H1): There is a correlation between temperature and flowering date. To test this hypothesis you perform a regression test, which generates a t value as its test statistic.

  17. Null hypothesis

    It is often symbolized as H0 . The statement that is being tested against the null hypothesis is the alternative hypothesis. [2] Symbols may include H1 and Ha . A statistical significance test starts with a random sample from a population.

  18. Null Hypothesis

    Math Article Null Hypothesis 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.

  19. Null Hypothesis

    Less Than Symbol- Meaning and Examples | Less Than Sign Binary Division Null Hypothesis Read Null Hypothesis, often denoted as H0, 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.

  20. Decision Rule Calculator

    This calculator tells you whether you should reject or fail to reject a null hypothesis based on the value of the test statistic, the format of the test (one-tailed or two-tailed), and the significance level you have chosen to use. One-tailed or two-tailed hypothesis? One-tailed Two-tailed Significance level 0.01 0.05 0.10

  21. Null Hypothesis and Alternative Hypothesis

    Most technical papers rely on just the first formulation, even though you may see some of the others in a statistics textbook. Null hypothesis: " x is equal to y .". Alternative hypothesis " x is not equal to y .". Null hypothesis: " x is at least y .". Alternative hypothesis " x is less than y .". Null hypothesis: " x is at ...

  22. Sign Test Calculator

    Sign Test Calculator. Instructions: This calculator conducts a Sign Test. Please select the null and alternative hypotheses, enter the number of positives (+) and the number of negatives (-), along with the significance level, and the results of the sign test will be displayed for you (please disregard the ties): Significance Level ( \alpha α) =.

  23. Statistical Power Calculator

    Statistical Power Calculator. The statistical power is a power of a binary hypothesis test. It is the probability that effectively rejects the null hypothesis value (H 0) when the alternative hypothesis value (H 1) is true. In this calculator, calculate the statistical power of a test (p = 1 - β) from the beta value. Calculator. Formula.