Hypothesis Tests

The hypothesis-test calculators run the four standard z-tests used when the population standard deviation is known (or n is large enough to treat the sample SD as known): one-sample z-test, two-sample z-test for means, and one- and two-proportion z-tests. Each shows the observed test statistic, the rejection-region boundaries, the p-value, and the formal decision at the chosen α.

The Critical Value Calculator is the lookup companion: enter α and tail direction, get z*. Useful when you're working from a textbook table or want to bypass the full test calculation.

When to use these calculators

Use the one-sample z-test when you have a single sample mean and want to test it against a hypothesized population value (μ₀), assuming σ is known. Use the two-sample z-test for comparing the means of two independent groups with known σ in each.

The proportion z-tests apply when the data are proportions (success/failure counts) rather than continuous measurements. The one-proportion test compares a sample proportion to a hypothesized population proportion (p₀); the two-proportion test compares two independent sample proportions to each other.

Every test shows the rejection-region boundaries (cyan shading) and the observed test statistic (amber marker) on the bell curve, so the user can see whether the observed value lands inside or outside the rejection zone — the conventional textbook visualization that makes the reject/fail-to-reject decision obvious. The Critical Value Calculator surfaces z* on its own for tabular lookups.

Frequently Asked Questions

What's the difference between a z-test and a t-test?

A z-test assumes the population standard deviation σ is known (or n is large enough that the sample SD is essentially σ). A t-test estimates σ from the sample, with degrees of freedom that account for that uncertainty. For n ≥ 30 the t-test and z-test agree to within ~1%, so this site's z-tests are commonly used as the default in introductory stats.

How do I choose α?

Convention sets α = 0.05 for most published research, 0.01 for stricter contexts (medical trials, regulatory submissions), and 0.10 for exploratory analyses. Lower α reduces the false-positive rate but requires stronger evidence to reject H₀. The calculators accept any α between 0 and 1.

What's the difference between one-tailed and two-tailed?

One-tailed tests place all of α in a single direction (e.g., 'the new method is better, not just different'). Two-tailed tests split α between both directions ('the new method differs in either direction'). Two-tailed is the conservative default; use one-tailed only when the alternative hypothesis genuinely specifies direction before data collection.

Does failing to reject H₀ mean H₀ is true?

No. Failing to reject H₀ means the sample didn't provide enough evidence to reject it at the chosen α; the test is silent on whether H₀ is actually true. The two-sample test's CI counterpart on this site (linked from each test calc to the matching CI mode) helps surface effect sizes that may be clinically meaningful even when the test is non-significant.

Why do critical values like 1.96 and 2.5758 come up so often?

These are the two-tailed critical z* values at α = 0.05 (z* = 1.96) and α = 0.01 (z* = 2.5758) on the standard normal. They're memorized in textbooks because they correspond to the conventional 95% and 99% confidence levels. The Critical Value Calculator returns these exactly, plus the one-tailed companions (1.6449 at α = 0.05; 2.3263 at α = 0.01).