Hypothesis Test Calculator

Z-Test & T-Test Proportion Tests Auto Decision Free · No Signup

Free online hypothesis test calculator for means and proportions. Perform Z-tests, T-tests, one-proportion and two-proportion tests with automatic reject/fail-to-reject decision, step-by-step KaTeX formulas, and Python scipy export.

Hypothesis Test
Null hypothesis value

Result

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Enter parameters and click Perform Test

Run hypothesis tests for means, proportions, or group comparisons.

Distribution Visualization

Python Compiler

What Is Hypothesis Testing?

Hypothesis testing is a statistical procedure used to determine whether sample data provides sufficient evidence to reject a claim (the null hypothesis) about a population parameter. It is fundamental to scientific research, quality control, and data-driven decision making.

1. State H₀ & H₁ 2. Choose α level 3. Compute Test stat 4. Find p-value 5. Decide Reject?
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State Hypotheses

Define H₀ (null — no effect) and H₁ (alternative — the claim you want to test). H₀ always contains the equality.

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Calculate Evidence

Compute a test statistic that measures how far the sample result is from the null hypothesis value in standard-error units.

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Make Decision

Compare the p-value to α. If p ≤ α, reject H₀. Otherwise, fail to reject H₀. Never “accept” H₀.

Test Formulas

Z-test for mean (σ known):  z = (x̄ − μ₀) / (σ / √n)
Use when the population standard deviation is known or n is very large.
T-test for mean (σ unknown):  t = (x̄ − μ₀) / (s / √n),  df = n − 1
Use when the population SD is unknown. Uses sample SD s with t-distribution.
One-proportion Z-test:  z = (p̂ − p₀) / √(p₀(1−p₀) / n)
Tests a sample proportion against a claimed population proportion.
Two-proportion Z-test:  z = (p̂₁ − p̂₂) / √(p̂(1−p̂)(1/n₁ + 1/n₂))
Pooled p̂ = (x₁ + x₂) / (n₁ + n₂). Tests if two population proportions are equal.

Decision Making

If p-value ≤ α

Reject H₀. The result is statistically significant. There is sufficient evidence to support H₁.

If p-value > α

Fail to reject H₀. The result is not statistically significant. Insufficient evidence to support H₁.

Common Significance Levels

αConfidencez* (two-tailed)Use Case
0.1090%±1.645Exploratory analysis, social science
0.0595%±1.960Standard for most research
0.0199%±2.576High-stakes, medical trials

Type I and Type II Errors

❌ Type I Error (α)

False positive. Rejecting H₀ when it is actually true. The probability is controlled by α (significance level).

Example: Concluding a drug works when it does not.

⚠️ Type II Error (β)

False negative. Failing to reject H₀ when it is actually false. Related to sample size and effect size.

Example: Missing a real drug effect due to small sample.

✅ Correct: Reject true H₁

H₀ is false and we correctly reject it. This is the desired outcome of a well-powered test.

✅ Correct: Fail to reject true H₀

H₀ is true and we correctly fail to reject it. No false alarm.

Statistical Power = 1 − β — the probability of correctly rejecting a false H₀. Increase power by increasing sample size, using a larger α, or when the true effect size is large.

When to Use Each Test

TestData TypeWhen to Use
Z-test (mean)Continuousσ is known, or n > 30 with known population SD
T-test (mean)Continuousσ is unknown (use sample s). Works for any n with normal population
One-proportion ZCategoricalCompare sample proportion to a claimed value. Need np₀ ≥ 5 and n(1−p₀) ≥ 5
Two-proportion ZCategoricalCompare proportions from two independent groups. Uses pooled proportion under H₀

Tip: In practice, the population σ is almost never known, so the T-test is the most commonly used test for means. The Z-test is primarily used for proportions and in textbook problems.

Frequently Asked Questions

Use the Z-test when the population standard deviation (σ) is known or the sample size is very large. Use the T-test when σ is unknown and you rely on the sample standard deviation (s). The T-distribution has heavier tails, especially for small samples, giving more conservative results.
Choose a one-tailed test if you predict a specific direction (e.g., the new treatment is better). Choose a two-tailed test if you are testing for any difference in either direction. This decision must be made before collecting data to avoid bias.
Rejecting H₀ means the data provides sufficient evidence that the null hypothesis is unlikely. It does not prove H₁ is true. It means that if H₀ were true, the probability of seeing results this extreme is less than α.
Type I error (α) is rejecting H₀ when it is actually true (false positive). Type II error (β) is failing to reject H₀ when it is actually false (false negative). Alpha is controlled by the significance level you choose.
Use a proportion test when your data is categorical (yes/no, pass/fail). A one-proportion test compares a sample proportion to a claimed value. A two-proportion test compares proportions from two independent groups, such as treatment vs. control conversion rates.
Z-test assumes known σ and a normal population (or large n). T-test assumes a normal population (or large n). Proportion tests need np₀ and n(1−p₀) both at least 5. All tests assume random, independent samples.

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