What is the difference between Z-test and T-test?

Use Z-test when the population standard deviation sigma is known or the sample size is large n greater than 30. Use T-test when sigma is unknown and you use the sample standard deviation s. The T-distribution has heavier tails especially for small samples.

How do I choose between one-tailed and two-tailed test?

Choose one-tailed if you predict a specific direction like the new treatment is better. Choose two-tailed if you test for any difference in either direction. This decision must be made before collecting data.

What does reject H0 mean?

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 alpha.

What is the difference between Type I and Type II error?

Type I error alpha is rejecting H0 when it is actually true which is a false positive. Type II error beta is failing to reject H0 when it is actually false which is a false negative. Alpha is controlled by the significance level.

When should I use a proportion test?

Use a proportion test when your data is categorical like yes or no pass or fail. One-proportion tests compare a sample proportion to a claimed value. Two-proportion tests compare proportions from two independent groups.

Hypothesis Test Calculator

Z & t tests · proportions · p-value · distribution plot

Test Type

Sample Mean (x̄)

Population Mean (μ₀)

Null hypothesis value

Population SD (σ)

Sample Size (n)

Significance Level (α)

Alternative Hypothesis

Sample 1

Successes (x₁)

Size (n₁)

Sample 2

Successes (x₂)

Size (n₂)

Significance Level (α)

Alternative Hypothesis

Quick Examples

Result

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Enter parameters and click Perform Test

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

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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.

📝

State Hypotheses

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

📊

Calculate Evidence

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

⚖️

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

α	Confidence	z* (two-tailed)	Use Case
0.10	90%	±1.645	Exploratory analysis, social science
0.05	95%	±1.960	Standard for most research
0.01	99%	±2.576	High-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

Test	Data Type	When to Use
Z-test (mean)	Continuous	σ is known, or n > 30 with known population SD
T-test (mean)	Continuous	σ is unknown (use sample s). Works for any n with normal population
One-proportion Z	Categorical	Compare sample proportion to a claimed value. Need np₀ ≥ 5 and n(1−p₀) ≥ 5
Two-proportion Z	Categorical	Compare 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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Hypothesis Test Calculator

Sample 1

Sample 2

Result

Enter parameters and click Perform Test

Distribution Visualization

Python Compiler

What Is Hypothesis Testing?

State Hypotheses

Calculate Evidence

Make Decision

Test Formulas

Decision Making

Common Significance Levels

Type I and Type II Errors

❌ Type I Error (α)

⚠️ Type II Error (β)

✅ Correct: Reject true H₁

✅ Correct: Fail to reject true H₀

When to Use Each Test

Frequently Asked Questions

Support This Free Tool

Sample 1

Sample 2

Result

Enter parameters and click Perform Test

Distribution Visualization

Python Compiler

Related Tools

What Is Hypothesis Testing?

State Hypotheses

Calculate Evidence

Make Decision

Test Formulas

Decision Making

Common Significance Levels

Type I and Type II Errors

❌ Type I Error (α)

⚠️ Type II Error (β)

✅ Correct: Reject true H₁

✅ Correct: Fail to reject true H₀

When to Use Each Test

Frequently Asked Questions

Support This Free Tool