hypothesis testing calculator power

Power & Sample Size Calculator

Use this advanced sample size calculator to calculate the sample size required for a one-sample statistic, or for differences between two proportions or means (two independent samples). More than two groups supported for binomial data. Calculate power given sample size, alpha, and the minimum detectable effect (MDE, minimum effect of interest).

Experimental design

Data parameters

Related calculators

• Using the power & sample size calculator

Parameters for sample size and power calculations

Calculator output.

• Why is sample size determination important?
• What is statistical power?

Post-hoc power (Observed power)

• Sample size formula
• Types of null and alternative hypotheses in significance tests
• Absolute versus relative difference and why it matters for sample size determination

    Using the power & sample size calculator

This calculator allows the evaluation of different statistical designs when planning an experiment (trial, test) which utilizes a Null-Hypothesis Statistical Test to make inferences. It can be used both as a sample size calculator and as a statistical power calculator . Usually one would determine the sample size required given a particular power requirement, but in cases where there is a predetermined sample size one can instead calculate the power for a given effect size of interest.

1. Number of test groups. The sample size calculator supports experiments in which one is gathering data on a single sample in order to compare it to a general population or known reference value (one-sample), as well as ones where a control group is compared to one or more treatment groups ( two-sample, k-sample ) in order to detect differences between them. For comparing more than one treatment group to a control group the sample size adjustments based on the Dunnett's correction are applied. These are only approximately accurate and subject to the assumption of about equal effect size in all k groups, and can only support equal sample sizes in all groups and the control. Power calculations are not currently supported for more than one treatment group due to their complexity.

2. Type of outcome . The outcome of interest can be the absolute difference of two proportions (binomial data, e.g. conversion rate or event rate), the absolute difference of two means (continuous data, e.g. height, weight, speed, time, revenue, etc.), or the relative difference between two proportions or two means (percent difference, percent change, etc.). See Absolute versus relative difference for additional information. One can also calculate power and sample size for the mean of just a single group. The sample size and power calculator uses the Z-distribution (normal distribution) .

3. Baseline The baseline mean (mean under H 0 ) is the number one would expect to see if all experiment participants were assigned to the control group. It is the mean one expects to observe if the treatment has no effect whatsoever.

4. Minimum Detectable Effect . The minimum effect of interest, which is often called the minimum detectable effect ( MDE , but more accurately: MRDE, minimum reliably detectable effect) should be a difference one would not like to miss , if it existed. It can be entered as a proportion (e.g. 0.10) or as percentage (e.g. 10%). It is always relative to the mean/proportion under H 0 ± the superiority/non-inferiority or equivalence margin. For example, if the baseline mean is 10 and there is a superiority alternative hypothesis with a superiority margin of 1 and the minimum effect of interest relative to the baseline is 3, then enter an MDE of 2 , since the MDE plus the superiority margin will equal exactly 3. In this case the MDE (MRDE) is calculated relative to the baseline plus the superiority margin, as it is usually more intuitive to be interested in that value.

If entering means data, one needs to specify the mean under the null hypothesis (worst-case scenario for a composite null) and the standard deviation of the data (for a known population or estimated from a sample).

5. Type of alternative hypothesis . The calculator supports superiority , non-inferiority and equivalence alternative hypotheses. When the superiority or non-inferiority margin is zero, it becomes a classical left or right sided hypothesis, if it is larger than zero then it becomes a true superiority / non-inferiority design. The equivalence margin cannot be zero. See Types of null and alternative hypothesis below for an in-depth explanation.

6. Acceptable error rates . The type I error rate, α , should always be provided. Power, calculated as 1 - β , where β is the type II error rate, is only required when determining sample size. For an in-depth explanation of power see What is statistical power below. The type I error rate is equivalent to the significance threshold if one is doing p-value calculations and to the confidence level if using confidence intervals.

The sample size calculator will output the sample size of the single group or of all groups, as well as the total sample size required. If used to solve for power it will output the power as a proportion and as a percentage.

    Why is sample size determination important?

While this online software provides the means to determine the sample size of a test, it is of great importance to understand the context of the question, the "why" of it all.

Estimating the required sample size before running an experiment that will be judged by a statistical test (a test of significance, confidence interval, etc.) allows one to:

• determine the sample size needed to detect an effect of a given size with a given probability
• be aware of the magnitude of the effect that can be detected with a certain sample size and power
• calculate the power for a given sample size and effect size of interest

This is crucial information with regards to making the test cost-efficient. Having a proper sample size can even mean the difference between conducting the experiment or postponing it for when one can afford a sample of size that is large enough to ensure a high probability to detect an effect of practical significance.

For example, if a medical trial has low power, say less than 80% (β = 0.2) for a given minimum effect of interest, then it might be unethical to conduct it due to its low probability of rejecting the null hypothesis and establishing the effectiveness of the treatment. Similarly, for experiments in physics, psychology, economics, marketing, conversion rate optimization, etc. Balancing the risks and rewards and assuring the cost-effectiveness of an experiment is a task that requires juggling with the interests of many stakeholders which is well beyond the scope of this text.

    What is statistical power?

Statistical power is the probability of rejecting a false null hypothesis with a given level of statistical significance , against a particular alternative hypothesis. Alternatively, it can be said to be the probability to detect with a given level of significance a true effect of a certain magnitude. This is what one gets when using the tool in "power calculator" mode. Power is closely related with the type II error rate: β, and it is always equal to (1 - β). In a probability notation the type two error for a given point alternative can be expressed as [1] :

β(T α ; μ 1 ) = P(d(X) ≤ c α ; μ = μ 1 )

It should be understood that the type II error rate is calculated at a given point, signified by the presence of a parameter for the function of beta. Similarly, such a parameter is present in the expression for power since POW = 1 - β [1] :

POW(T α ; μ 1 ) = P(d(X) > c α ; μ = μ 1 )

In the equations above c α represents the critical value for rejecting the null (significance threshold), d(X) is a statistical function of the parameter of interest - usually a transformation to a standardized score, and μ 1 is a specific value from the space of the alternative hypothesis.

One can also calculate and plot the whole power function, getting an estimate of the power for many different alternative hypotheses. Due to the S-shape of the function, power quickly rises to nearly 100% for larger effect sizes, while it decreases more gradually to zero for smaller effect sizes. Such a power function plot is not yet supported by our statistical software, but one can calculate the power at a few key points (e.g. 10%, 20% ... 90%, 100%) and connect them for a rough approximation.

Statistical power is directly and inversely related to the significance threshold. At the zero effect point for a simple superiority alternative hypothesis power is exactly 1 - α as can be easily demonstrated with our power calculator. At the same time power is positively related to the number of observations, so increasing the sample size will increase the power for a given effect size, assuming all other parameters remain the same.

Power calculations can be useful even after a test has been completed since failing to reject the null can be used as an argument for the null and against particular alternative hypotheses to the extent to which the test had power to reject them. This is more explicitly defined in the severe testing concept proposed by Mayo & Spanos (2006).

Computing observed power is only useful if there was no rejection of the null hypothesis and one is interested in estimating how probative the test was towards the null . It is absolutely useless to compute post-hoc power for a test which resulted in a statistically significant effect being found [5] . If the effect is significant, then the test had enough power to detect it. In fact, there is a 1 to 1 inverse relationship between observed power and statistical significance, so one gains nothing from calculating post-hoc power, e.g. a test planned for α = 0.05 that passed with a p-value of just 0.0499 will have exactly 50% observed power (observed β = 0.5).

I strongly encourage using this power and sample size calculator to compute observed power in the former case, and strongly discourage it in the latter.

    Sample size formula

The formula for calculating the sample size of a test group in a one-sided test of absolute difference is:

where Z 1-α is the Z-score corresponding to the selected statistical significance threshold α , Z 1-β is the Z-score corresponding to the selected statistical power 1-β , σ is the known or estimated standard deviation, and δ is the minimum effect size of interest. The standard deviation is estimated analytically in calculations for proportions, and empirically from the raw data for other types of means.

The formula applies to single sample tests as well as to tests of absolute difference between two samples. A proprietary modification is employed when calculating the required sample size in a test of relative difference . This modification has been extensively tested under a variety of scenarios through simulations.

    Types of null and alternative hypotheses in significance tests

When doing sample size calculations, it is important that the null hypothesis (H 0 , the hypothesis being tested) and the alternative hypothesis is (H 1 ) are well thought out. The test can reject the null or it can fail to reject it. Strictly logically speaking it cannot lead to acceptance of the null or to acceptance of the alternative hypothesis. A null hypothesis can be a point one - hypothesizing that the true value is an exact point from the possible values, or a composite one: covering many possible values, usually from -∞ to some value or from some value to +∞. The alternative hypothesis can also be a point one or a composite one.

In a Neyman-Pearson framework of NHST (Null-Hypothesis Statistical Test) the alternative should exhaust all values that do not belong to the null, so it is usually composite. Below is an illustration of some possible combinations of null and alternative statistical hypotheses: superiority, non-inferiority, strong superiority (margin > 0), equivalence.

All of these are supported in our power and sample size calculator.

Careful consideration has to be made when deciding on a non-inferiority margin, superiority margin or an equivalence margin . Equivalence trials are sometimes used in clinical trials where a drug can be performing equally (within some bounds) to an existing drug but can still be preferred due to less or less severe side effects, cheaper manufacturing, or other benefits, however, non-inferiority designs are more common. Similar cases exist in disciplines such as conversion rate optimization [2] and other business applications where benefits not measured by the primary outcome of interest can influence the adoption of a given solution. For equivalence tests it is assumed that they will be evaluated using a two one-sided t-tests (TOST) or z-tests, or confidence intervals.

Note that our calculator does not support the schoolbook case of a point null and a point alternative, nor a point null and an alternative that covers all the remaining values. This is since such cases are non-existent in experimental practice [3][4] . The only two-sided calculation is for the equivalence alternative hypothesis, all other calculations are one-sided (one-tailed) .

    Absolute versus relative difference and why it matters for sample size determination

When using a sample size calculator it is important to know what kind of inference one is looking to make: about the absolute or about the relative difference, often called percent effect, percentage effect, relative change, percent lift, etc. Where the fist is μ 1 - μ the second is μ 1 -μ / μ or μ 1 -μ / μ x 100 (%). The division by μ is what adds more variance to such an estimate, since μ is just another variable with random error, therefore a test for relative difference will require larger sample size than a test for absolute difference. Consequently, if sample size is fixed, there will be less power for the relative change equivalent to any given absolute change.

For the above reason it is important to know and state beforehand if one is going to be interested in percentage change or if absolute change is of primary interest. Then it is just a matter of fliping a radio button.

    References

1 Mayo D.G., Spanos A. (2010) – "Error Statistics", in P. S. Bandyopadhyay & M. R. Forster (Eds.), Philosophy of Statistics, (7, 152–198). Handbook of the Philosophy of Science . The Netherlands: Elsevier.

2 Georgiev G.Z. (2017) "The Case for Non-Inferiority A/B Tests", [online] https://blog.analytics-toolkit.com/2017/case-non-inferiority-designs-ab-testing/ (accessed May 7, 2018)

3 Georgiev G.Z. (2017) "One-tailed vs Two-tailed Tests of Significance in A/B Testing", [online] https://blog.analytics-toolkit.com/2017/one-tailed-two-tailed-tests-significance-ab-testing/ (accessed May 7, 2018)

4 Hyun-Chul Cho Shuzo Abe (2013) "Is two-tailed testing for directional research hypotheses tests legitimate?", Journal of Business Research 66:1261-1266

5 Lakens D. (2014) "Observed power, and what to do if your editor asks for post-hoc power analyses" [online] http://daniellakens.blogspot.bg/2014/12/observed-power-and-what-to-do-if-your.html (accessed May 7, 2018)

Cite this calculator & page

If you'd like to cite this online calculator resource and information as provided on the page, you can use the following citation: Georgiev G.Z., "Sample Size Calculator" , [online] Available at: https://www.gigacalculator.com/calculators/power-sample-size-calculator.php URL [Accessed Date: 25 Nov, 2024].

Our statistical calculators have been featured in scientific papers and articles published in high-profile science journals by:

The author of this tool

     Statistical calculators

Hypothesis Testing 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.

Normal, T - Statistical power calculator

Information.

Calculates the test power for the specific sample size and draw a power analysis chart. For the two-tailed test, it calculates the strict interpretation, includes the probability to reject the null assumption in the opposite tail of the true effect

Distribution

Power calculator

Having enough power in your A/B test requires a large enough sample size.

Power is the probability that a test correctly rejects a false null hypothesis - i.e., ensuring an A/B tests is sensitive enough to detect a true effect when there is one. To calculate the sample we need for a certain power, we need several inputs - including baseline conversion rate, minimum detectable effect, A/B split ratio, significance and power.

How to use this calculator:

Determine your baseline conversion rate.

This is the current conversion rate of your control group. In an A/B test, the baseline conversion is the expected rate of conversion (or other desireable outcome) in the control group, or those not being exposed to a new experience.

Choose a minimum detectable effect

This is the smallest difference that can be  consistently detected  in an experiment. In an A/B test, this is the minimum change in desireable outcome you’d want to be able to detect.

Input your values

The output sample size of the calculator will be the minimum viable amount to consistently achieve statistically significant results, based on the power level that you choose. Choosing a higher power means a lower frequency of false negatives, but will also require a commensurate number more samples.

The calculator is automatically set to optimal defaults, but you can adjust the advanced settings to see how they impact your results.

If you are looking to determine if a single test variation is better than a control, use a one-sided test (recommended) . If you want to determine if its different from the control, then use a two-sided test.

A/B split ratio

Most A/B tests are conducted with a 50%/50% split across test and control users (represented as an input of 0.5 in this calculator), but this can be tuned to your own experimental design.

Significance (α)

Alpha is the probability that a statistically significant difference is detected when one does not exist. 0.05 is a common default for Alpha , but you can choose a higher or lower value to adjust the probability that an observed difference isn’t due to chance, but requires a larger sample size.

Statistical Power (1 - β)

As shared above,  statistical power  is the probability that the minimum detectable effect will be detected, assuming it exists. If you’d like to calculate a minimum detectable effect or A/B test duration automatically based on your data each time you run a test, sign up for Statsig!

Join the #1 experimentation community

Try statsig today.

Teach yourself statistics

How to Find the Power of a Statistical Test

When a researcher designs a study to test a hypothesis, he/she should compute the power of the test (i.e., the likelihood of avoiding a Type II error).

How to Compute the Power of a Hypothesis Test

To compute the power of a hypothesis test, use the following three-step procedure.

• Define the region of acceptance . Previously, we showed how to compute the region of acceptance for a hypothesis test.
• Specify the critical parameter value. The critical parameter value is an alternative to the value specified in the null hypothesis. The difference between the critical parameter value and the value from the null hypothesis is called the effect size . That is, the effect size is equal to the critical parameter value minus the value from the null hypothesis.
• Compute power. Assume that the true population parameter is equal to the critical parameter value, rather than the value specified in the null hypothesis. Based on that assumption, compute the probability that the sample estimate of the population parameter will fall outside the region of acceptance. That probability is the power of the test.

The following examples illustrate how this works. The first example involves a mean score; and the second example, a proportion.

Sample Size Calculator

The steps required to compute the power of a hypothesis test can be time-consuming and complex. Stat Trek's Sample Size Calculator does this work for you - quickly and accurately. The calculator is easy to use, and it is free. You can find the Sample Size Calculator in Stat Trek's main menu under the Stat Tools tab. Or you can tap the button below.

Example 1: Power of the Hypothesis Test of a Mean Score

Two inventors have developed a new, energy-efficient lawn mower engine. One inventor says that the engine will run continuously for 5 hours (300 minutes) on a single ounce of regular gasoline. Suppose a random sample of 50 engines is tested. The engines run for an average of 295 minutes, with a standard deviation of 20 minutes. The inventor tests the null hypothesis that the mean run time is 300 minutes against the alternative hypothesis that the mean run time is not 300 minutes, using a 0.05 level of significance.

The other inventor says that the new engine will run continuously for only 290 minutes on a ounce of gasoline. Find the power of the test to reject the null hypothesis, if the second inventor is correct.

Solution: The steps required to compute power are presented below.

• Define the region of acceptance . In a previous lesson, we showed that the region of acceptance for this problem consists of the values between 294.46 and 305.54 (see previous lesson ).
• Specify the critical parameter value . The null hypothesis tests the hypothesis that the run time of the engine is 300 minutes. We are interested in determining the probability that the hypothesis test will reject the null hypothesis, if the true run time is actually 290 minutes. Therefore, the critical parameter value is 290. (Another way to express the critical parameter value is through effect size. The effect size is equal to the critical parameter value minus the hypothesized value. Thus, effect size is equal to 290 - 300 or -10.)

Therefore, we need to compute the probability that the sampled run time will be less than 294.46 or greater than 305.54. To do this, we make the following assumptions:

• The sampling distribution of the mean is normally distributed. (Because the sample size is relatively large, this assumption can be justified by the central limit theorem .)
• The mean of the sampling distribution is the critical parameter value, 290.
• The standard error of the sampling distribution is 2.83. The standard error of the sampling distribution was computed in a previous lesson (see previous lesson ).

Given these assumptions, we first assess the probability that the sample run time will be less than 294.46. This is easy to do, using the Normal Calculator . We enter the following values into the calculator: normal random variable = 294.46; mean = 290; and standard deviation = 2.83. Given these inputs, we find that the cumulative probability is 0.942. This means the probability that the sample mean will be less than 294.46 is 0.942.

Next, we assess the probability that the sample mean is greater than 305.54. Again, we use the Normal Calculator . We enter the following values into the calculator: normal random variable = 305.54; mean = 290; and standard deviation = 2.83. Given these inputs, we find that the probability that the sample mean is less than 305.54 (i.e., the cumulative probability) is 1.0. Thus, the probability that the sample mean is greater than 305.54 is 1 - 1.0 or 0.0.

Example 2: Power of the Hypothesis Test of a Proportion

A major corporation offers a large bonus to all of its employees if at least 80 percent of the corporation's 1,000,000 customers are very satisfied. The company conducts a survey of 100 randomly sampled customers to determine whether or not to pay the bonus. The null hypothesis states that the proportion of very satisfied customers is at least 0.80. If the null hypothesis cannot be rejected, given a significance level of 0.05, the company pays the bonus.

Suppose the true proportion of satisfied customers is 0.75. Find the power of the test to reject the null hypothesis.

• Define the region of acceptance . In a previous lesson, we showed that the region of acceptance for this problem consists of the values between 0.734 and 1.00. (see previous lesson ).
• Specify the critical parameter value . The null hypothesis tests the hypothesis that the proportion of very satisfied customers is 0.80. We are interested in determining the probability that the hypothesis test will reject the null hypothesis, if the true satisfaction level is 0.75. Therefore, the critical parameter value is 0.75. (Another way to express the critical parameter value is through effect size. The effect size is equal to the critical parameter value minus the hypothesized value. Thus, effect size is equal to [0.75 - 0.80] or - 0.05.)

Therefore, we need to compute the probability that the sample proportion will be less than 0.734. To do this, we take the following steps:

• Assume that the sampling distribution of the mean is normally distributed. (Because the sample size is relatively large, this assumption can be justified by the central limit theorem .)
• Assume that the mean of the sampling distribution is the critical parameter value, 0.75. (This assumption is justified because, for the purpose of calculating power, we assume that the true population proportion is equal to the critical parameter value. And the mean of all possible sample proportions is equal to the population proportion. Hence, the mean of the sampling distribution is equal to the critical parameter value.)

σ P = sqrt[ P * ( 1 - P ) / n ]

σ P = sqrt[ ( 0.75 * 0.25 ) / 100 ] = 0.0433

Statistical Power Calculator using the t-distribution*

*plot adapted from behavioral research data analysis with r.

Hypothesis Testing Calculator

Welcome to our complete hypothesis testing calculator, the ideal tool for doing accurate and trustworthy statistical studies. Our calculator is meant to fulfill the demands of students, researchers, and professionals while also simplifying the hypothesis testing procedure.

• One sample z test hypothesis calculator
• One sample t test hypothesis calculator
• One proportion z test hypothesis calculator
• Two sample z test hypothesis calculator
• Two sample t test hypothesis calculator (equal and unequal variance)
• Two proportion z test hypothesis calculator

Why Use Our Hypothesis Testing Calculator?

Hypothesis testing is an important part of statistical analysis because it allows you to make population-level inferences based on sample data. Our calculator makes this process easier by providing user-friendly interfaces and step-by-step directions for performing different tests. Here are several significant advantages:

• Accuracy: Our calculators are designed to provide precise calculations, ensuring your results are reliable.
• Variety: With a range of calculators available, you can perform different types of hypothesis tests as needed.
• User-Friendly: Easy-to-use interfaces make it simple for anyone to perform complex statistical analyses.
• Free Access: Our tools are available for free, making high-quality statistical analysis accessible to everyone.

How to Use Our Hypothesis Testing Calculator

Using our hypothesis testing calculator is straightforward. Simply select the type of test you need from the list above, input your data, and follow the prompts. Our calculators will guide you through each step, ensuring you understand the process and obtain accurate results.

What is a Null Hypothesis (H 0 )?

The null hypothesis, denoted as H 0 , is the default assumption in hypothesis testing. It posits that there is no significant effect or difference between groups or conditions. Essentially, it represents the status quo or the idea that any observed differences are due to random chance.

Examples of Null Hypotheses:

• In a clinical trial: H 0 : "There is no difference in the effectiveness of Drug A and Drug B."
• In a manufacturing process: H 0 : "The mean length of the produced parts is equal to the specified length."
• In a survey: H 0 : "The proportion of voters who support Candidate X is 50%."

Related Calculators:

• List of all calculators
• P-value calculator
• Critical value Calculator

What is an Alternative Hypothesis (H a )?

The alternative hypothesis, denoted as H a , is the statement that contradicts the null hypothesis. It suggests that there is a significant effect or difference. The alternative hypothesis represents what the researcher aims to prove or the presence of an effect they are testing for.

Examples of Alternative Hypotheses:

• In a clinical trial: H a : "There is a difference in the effectiveness of Drug A and Drug B."
• In a manufacturing process: H a : "The mean length of the produced parts is not equal to the specified length."
• In a survey: H a : "The proportion of voters who support Candidate X is not 50%."

The Importance of Hypothesis Testing

Hypothesis testing is fundamental in statistical analysis as it allows researchers to make data-driven decisions. By comparing the null and alternative hypotheses, researchers can determine the likelihood that their observations are due to chance or if there is evidence to support a significant effect.