Test the Hypothesis Calculator

Hypothesis Test Inputs

Test type

Alternative direction

Significance level α

Sample mean x̄

Hypothesized mean μ₀

Population standard deviation σ

Sample standard deviation s

Sample size n

Sample 1 mean x̄₁

Sample 2 mean x̄₂

Sample 1 standard deviation s₁

Sample 2 standard deviation s₂

Sample 1 size n₁

Sample 2 size n₂

Hypothesized mean difference

One proportion successes x

One proportion sample size n

Hypothesized proportion p₀

Group 1 successes x₁

Group 1 size n₁

Group 2 successes x₂

Group 2 size n₂

Hypothesized proportion difference

Variance test sample size n

Sample variance s²

Hypothesized standard deviation σ₀

Observed counts

Expected counts

Formula Used

Mean z test: z = (x̄ − μ₀) / (σ / √n).

Mean t test: t = (x̄ − μ₀) / (s / √n), with df = n − 1.

Welch t test: t = [(x̄₁ − x̄₂) − Δ₀] / √(s₁²/n₁ + s₂²/n₂).

Proportion z test: z = (p̂ − p₀) / √[p₀(1 − p₀) / n].

Chi square test: χ² = Σ[(O − E)² / E], or χ² = (n − 1)s² / σ₀² for variance.

Decision rule: reject H₀ when p value ≤ α.

How to Use This Calculator

Select the hypothesis test that matches your data.
Choose a two tailed, greater than, or less than alternative.
Enter alpha, sample statistics, and count data where needed.
Press the calculate button to view the result above the form.
Download the result as CSV or PDF for your records.

Example Data Table

Scenario	Test	Inputs	Question
Average score	One sample t	x̄ = 52, μ₀ = 50, s = 10, n = 25	Is the mean different from 50?
Survey support	One proportion z	x = 56, n = 100, p₀ = 0.50	Is support above the benchmark?
Product categories	Goodness of fit	O = 18, 22, 20, 24, 16	Do counts follow the expected pattern?

Understanding Hypothesis Testing

Hypothesis testing is a structured way to compare a claim with sample evidence. It helps you decide whether a result is unusual, expected, or not strong enough. This calculator supports common tests used in introductory and applied statistics. You can test a population mean, a difference between means, a proportion, a difference between proportions, a variance, or a frequency table.

Why the Test Matters

Every test begins with two statements. The null hypothesis says there is no effect, no change, or a stated benchmark. The alternative hypothesis says the effect is different, greater, or smaller. A test statistic measures how far the sample result sits from the null value. A p value then describes how surprising the sample would be if the null statement were true.

Choosing the Right Method

Use a z test for a mean when the population standard deviation is known. Use a one sample t test when it is unknown. Use a two sample t test when comparing two independent averages. Use a proportion test for pass rates, defect rates, or survey shares. Use a chi square test for variance, goodness of fit, or independence in a table.

Reading the Output

The decision depends on the significance level. A common choice is 0.05, but you can set another value. If the p value is less than or equal to alpha, reject the null hypothesis. If it is larger, do not reject the null hypothesis. This wording avoids saying the null is proven true. Statistics usually measures evidence, not certainty.

Good Data Practices

Results are useful only when the data matches the method. Samples should be random when possible. Observations should be independent. For proportion and chi square tests, expected counts should not be too small. For t tests, severe outliers can distort the conclusion. Always review assumptions before using the final decision.

Practical Use

This tool shows formulas, degrees of freedom, critical values, and confidence intervals when they apply. It also explains each decision in simple language. Export options help you save results for class notes, audits, lab reports, or project records. Keep notes about sources, dates, units, and assumptions so another reader can repeat the same calculation with the same inputs.

FAQs

What is a hypothesis test?

A hypothesis test compares sample evidence with a stated claim. It estimates whether the sample result is unusual under the null hypothesis. The output gives a statistic, p value, and decision rule.

What does the p value mean?

The p value is the probability of seeing evidence this extreme, assuming the null hypothesis is true. A small p value shows stronger evidence against the null claim.

When should I use a t test?

Use a t test for means when the population standard deviation is unknown. It is common for small and medium samples. The calculator uses degrees of freedom to adjust the result.

When should I use a z test?

Use a z test when the population standard deviation is known, or when testing proportions with suitable sample counts. The normal distribution is used for the p value.

What alpha value should I enter?

Alpha is your chosen risk of rejecting a true null hypothesis. Many classes use 0.05. Some research uses 0.01 or 0.10, depending on the field and risk level.

Does failing to reject prove the null?

No. It means the sample did not provide enough evidence against the null hypothesis. More data, better design, or a different test may change the conclusion.

Can I test two proportions?

Yes. Enter successes and sample sizes for both groups. The calculator finds both proportions, the observed difference, the standard error, and the z statistic.

Why are assumptions important?

Assumptions protect the accuracy of the p value and confidence interval. Independence, random sampling, and adequate expected counts help make the conclusion more reliable.