AP Stats Hypothesis Testing Calculator

A hypothesis test asks whether sample evidence is strong enough to reject a claim about a population mean. This page runs a one-sample z-test for a mean when the population standard deviation is known.

What Is Hypothesis Testing?

Hypothesis testing is the formal process of deciding whether sample data give convincing evidence against a claim about a population. You start with a null hypothesis, the claim you test, and an alternative hypothesis, the claim you suspect instead. You compute a test statistic and a p-value, then reject the null hypothesis only when the p-value is small enough, usually below 0.05.

How to Use This Calculator

Enter the sample mean, the claimed population mean, the known population standard deviation, and the sample size. Choose α, then click Calculate. The result gives the standard error, z-statistic, two-tailed p-value, and the decision at your α. The curve shades both tails beyond |z|, which is where a two-tailed p-value comes from.

How to Calculate a One-Sample Z-Test

Formulas: standard error SE = σ / √n, test statistic z = (x̄ − μ₀) / SE, and two-tailed p-value p = 2 × P(Z > |z|). Reject H₀ if p < α.

Worked example: x̄ = 52, μ₀ = 50, σ = 10, n = 36, α = 0.05. SE = 10 / √36 = 1.667. z = (52 − 50) / 1.667 = 1.20. Two-tailed p ≈ 0.2301. Since 0.2301 > 0.05, fail to reject H₀: the sample does not give convincing evidence that the true mean differs from 50.

Tips and Common Mistakes

• Use σ, the population standard deviation, for a z-test. If only the sample SD (s) is known, use a t-test instead.
• Do not say “accept H₀” — the correct conclusion is “fail to reject H₀.”
• A two-tailed test is the default here; a one-tailed p-value would be half of this p-value.
• Check conditions: random sample, and either a normal population or a large n (usually n ≥ 30).
• A small p-value is evidence against H₀, not the probability that H₀ is true.

Frequently Asked Questions

What is the null hypothesis in this calculator?

H₀: μ = μ₀, the claimed population mean you entered. The alternative is two-sided: μ ≠ μ₀.

When should I use a z-test instead of a t-test?

Use a z-test when σ is known. In AP Statistics, most mean tests use a t-test because σ is rarely known.

What does the p-value mean?

Assuming H₀ is true, it is the probability of getting a sample mean at least as far from μ₀ as yours, in either direction.

Why is the decision “fail to reject” and not “accept”?

A non-significant result does not prove H₀ is true; it only means the evidence was not strong enough to reject it.

What if my z is negative?

The sign shows direction (sample mean below μ₀). For a two-tailed test, only |z| matters for the p-value.

Does a larger sample always help?

Yes, in general: larger n gives a smaller standard error, so the same difference from μ₀ produces a larger |z| and a smaller p-value.

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