Which hypothesis test should I use?
Short answer: ask three questions. What kind of data do you have (continuous or counts/proportions)? What are you comparing (means, variances or proportions)? How many groups, and are they paired?
| Question | Test |
|---|---|
| One mean vs a target, σ known | 1-sample z |
| One mean vs a target, σ unknown | 1-sample t |
| Two independent means | 2-sample t (Welch if the variances differ) |
| Before/after on the same units | Paired t (on the differences) |
| Three or more means | One-way ANOVA (F test) |
| One variance vs a target | Chi-square test for variance |
| Two variances | F test (or Levene if the data aren't normal) |
| One or two proportions | 1- or 2-proportion z test |
| Counts in categories / independence | Chi-square goodness of fit / contingency table |
| Non-normal, small samples, medians | Nonparametric (sign test, Wilcoxon, Mann-Whitney, Kruskal-Wallis) |
Worked example
Two suppliers' parts are tested for tensile strength, 10 parts from each. Supplier 1: mean 12.4, s = 0.6. Supplier 2: mean 11.8, s = 0.5. The parts are different, so the samples are independent, the data are continuous, and you're comparing means: use a 2-sample t test.
What it means: the two-sided critical t is about 2.1 for roughly 17–18 degrees of freedom at α = 0.05. Since 2.43 is greater than that, the difference is statistically significant. Then ask whether 0.6 units actually matters for the product.
Common exam traps
- Using a 2-sample test on paired data. Measuring the same parts twice calls for a paired t test, which is far more sensitive.
- Running many t tests instead of ANOVA. Each extra test inflates the overall α.
- Thinking "fail to reject" means "proved equal." It only means the data didn't show a difference, possibly because the sample was too small.
- One-tailed vs two-tailed. Choose the tail from the question you're asking, before you see the data.
Try 5 free CQE practice questions, including a calculation problem with fresh numbers. Every wrong answer comes with an explanation. No account needed.
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