# Hypothesis Test Setup Calculator

Free hypothesis test calculator. Set up and evaluate one-sample z-tests and t-tests with null/alternative hypotheses, test statistics, critical values.

## What this calculates

Set up and evaluate a complete hypothesis test. Enter your sample data, choose the test type and tail, and get the null/alternative hypotheses, test statistic, critical values, p-value, and conclusion.

## Inputs

- **Test Type** — options: Z-Test (known σ or large n), T-Test (unknown σ, small n) — Choose z-test when population σ is known; t-test when using sample s.
- **Tail Type** — options: Two-Tailed (≠), Left-Tailed (<), Right-Tailed (>) — Direction of the alternative hypothesis.
- **Sample Mean (x̄)** — The mean of your sample.
- **Hypothesized Mean (μ₀)** — The population mean under the null hypothesis.
- **Standard Deviation (σ or s)** — min 0.0001 — Population σ for z-test, or sample s for t-test.
- **Sample Size (n)** — min 2 — Number of observations in the sample.
- **Significance Level (α)** — options: 0.01 (1%), 0.05 (5%), 0.10 (10%) — The probability threshold for rejecting H₀.

## Outputs

- **Null Hypothesis (H₀)** — formatted as text — The null hypothesis statement.
- **Alternative Hypothesis (H₁)** — formatted as text — The alternative hypothesis statement.
- **Test Statistic** — The calculated z or t statistic.
- **Critical Value(s)** — formatted as text — The critical value(s) for the rejection region.
- **P-Value** — The p-value for the test.
- **Decision** — formatted as text — Whether to reject or fail to reject H₀.
- **Conclusion** — formatted as text — A plain language conclusion.

## Details

Hypothesis testing follows a structured framework:

Steps

- State H₀ (null) and H₁ (alternative)

- Choose α (significance level)

- Calculate test statistic

- Find p-value or compare to critical value

- Make a decision (reject or fail to reject H₀)

Test Statistic:
z = (x̄ - μ₀) / (σ/√n)  or  t = (x̄ - μ₀) / (s/√n)

Decision Rule

- If p-value < α → Reject H₀

- If p-value ≥ α → Fail to reject H₀

## Frequently Asked Questions

**Q: What is the difference between 'reject' and 'fail to reject'?**

A: Rejecting H₀ means the data provides enough evidence to support the alternative hypothesis. 'Failing to reject' H₀ does NOT prove H₀ is true -- it means there is insufficient evidence to conclude otherwise. We never 'accept' H₀; we only fail to reject it.

**Q: How do I choose between a one-tailed and two-tailed test?**

A: Use a two-tailed test when you want to detect any difference from the hypothesized value (H₁: μ ≠ μ₀). Use a one-tailed test when you only care about a specific direction (H₁: μ > μ₀ or H₁: μ < μ₀). The choice should be made BEFORE looking at the data.

**Q: What is a Type I and Type II error?**

A: Type I error (α): Rejecting H₀ when it is actually true (false positive). Type II error (β): Failing to reject H₀ when it is actually false (false negative). The significance level α directly controls the Type I error rate. Reducing α increases the risk of Type II error.

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Source: https://vastcalc.com/calculators/statistics/hypothesis-test
Category: Statistics
Last updated: 2026-04-21