Hypothesis Test Calculator (z-test & t-test)

Hypothesis Testing Guide

What do I need to know about Steps in Hypothesis Testing?

1. State H₀ (null hypothesis) and H₁ (alternative hypothesis). 2. Choose significance level α (typically 5%). 3. Calculate test statistic: z = (x̄ − μ₀) / (σ/√n). 4. Find critical value from z-table (or t-table for t-test). 5. Decision rule: reject H₀ if |z| > z_critical (two-tailed) or z > z_critical (one-tailed upper). Or equivalently: reject H₀ if p-value < α. Example: H₀: μ = 50. H₁: μ > 50. x̄ = 52.3, σ = 8, n = 36. z = (52.3-50)/(8/6) = 2.3/1.333 = 1.725. One-tailed at α=0.05: critical z = 1.645. Since our calculated z (1.725) exceeds this critical value, we reject H₀ and conclude there is significant evidence the true mean exceeds 50.

What should I know about Type I and Type II Errors?

Type I error (α): rejecting H₀ when it is actually true. The significance level α is the probability of a Type I error. Choosing α = 0.05: 5% chance of incorrectly rejecting a true null hypothesis. Type II error (β): failing to reject H₀ when it is actually false. Power of test = 1 − β = probability of correctly rejecting a false H₀. There is a trade-off: reducing α (stricter) increases the chance of a Type II error. Context matters: in medical testing for a serious disease, a Type II error — missing a true diagnosis — is usually considered far worse than a Type I error, which is why screening tests are often deliberately set to be more sensitive at the cost of more false positives.

What should I know about p-value Interpretation?

The p-value is the probability of observing a test statistic as extreme as or more extreme than the one calculated, assuming H₀ is true. p-value < α: reject H₀ — statistically significant result. p-value > α: fail to reject H₀ — insufficient evidence. Common misinterpretation: the p-value is NOT the probability that H₀ is true. It is NOT the probability of getting this result by chance. It IS the probability of getting this extreme a result if H₀ were true. A p-value of 0.03 means: if H₀ were true, there's only a 3% chance of observing a result this extreme purely by random variation — evidence against the null hypothesis, but not proof it's false.

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

Use z-test when: population standard deviation σ is known. Sample size is large (n > 30, approximately). Use t-test when: population σ is unknown and estimated from sample. Sample is small (n < 30). The t-distribution has heavier tails than the normal — it is more conservative (harder to reject H₀), reflecting the additional uncertainty from estimating σ. For large n, t approaches z — the test choice matters mainly for small samples. Degrees of freedom for t-test: df = n − 1. The critical t-value is always slightly larger than the equivalent z-value for the same confidence level, since the t-distribution accounts for the extra uncertainty of estimating the standard deviation from a small sample.