Further Maths · Further Probability & Statistics. The idea in plain words, the rules to remember, the mistake to avoid and a worked example, written and checked by Mega Lecture.
When a sample is small and the population variance is unknown, the variance must be estimated from the sample. The test statistic then follows a t-distribution, not a normal one, and the population must be normal. The shape of t depends on the degrees of freedom, ν.
Choosing the right test is half the work. One sample against a stated mean: one-sample t-test. Two measurements on the same subjects: paired t-test on the differences. Two independent small samples with a common variance: 2-sample t-test with a pooled variance estimate. Known variances or large samples: a normal (z) test.
A confidence interval uses the same ingredients: estimate ± (table value) × (standard error).
Students use a 2-sample test when the data are paired (the same people measured twice), or use n in place of n − 1 for the degrees of freedom. Ask first whether each value in one sample is matched with one value in the other; if so, work with the differences and ν = n − 1.
Answer: Reject H0: there is evidence at the 5% level that the population mean is greater than 50.
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