Testing of Hypotheses
Testing of Hypotheses tests the logic of a statistical test: setting hypotheses, choosing the correct statistic and critical region, and interpreting the decision.
Framework
H₀ states no difference or a specified value; H₁ states the alternative and fixes whether the test is one-tailed or two-tailed. α is the level of significance, the probability of a type I error, that is rejecting a true H₀. A type II error is failing to reject a false H₀, and power = 1 − β.
Test statistics
Large-sample mean: z = (x̄ − μ₀)/(σ/√n), with critical values 1.96 (two-tailed 5%), 2.58 (two-tailed 1%) and 1.645 (one-tailed 5%). Small sample: t = (x̄ − μ₀)/(s/√n) with n − 1 degrees of freedom. Goodness of fit: χ² = Σ(O − E)²/E, with one degree of freedom lost for each parameter estimated. Equality of variances: F = s₁²/s₂² with (n₁ − 1, n₂ − 1) degrees of freedom.
Worked example
n = 100, x̄ = 51.5, σ = 10; test μ = 50 against μ ≠ 50 at 5%. Then z = (51.5 − 50)/(10/10) = 1.5. Since 1.5 < 1.96, H₀ is not rejected: the sample gives no significant evidence of a change.
Traps
Using the two-tailed critical value for a one-tailed test; putting s in place of σ/√n in the denominator; forgetting that estimating a parameter costs a degree of freedom in χ²; reading 'not rejected' as 'proved true'.