A-Level Hypothesis testing: worked solution

Question

A supporter claims NewPalace average 40 minutes to their first goal, this time being normal with variance 280 minutes². A random sample of 50 matches undergoes a 10% two-tailed test of the claim (H0: μ=40 versus H1: μ≠40). Find the set of sample mean times that would lead to rejecting the claim. [4]

Under H0 the time to the first goal is Normal with mean 40 and variance 280. Step 1. Write down the distribution of the sample mean, not of a single match. For a sample of size n, the sample mean has the same mean but variance divided by n. Here n = 50, so Var(Ȳ) = 280/50 = 5.6. Under H0, Ȳ ~ N(40, 5.6), and the standard error is √5.6 = 2.3664 (4 d.p.). Step 2. Find the critical z-values. The test is two-tailed at the 10% significance level, so the 10% is split, 5% in each tail. The z-value with 5% above it is 1.6449, so the critical values are z = −1.6449 and z = +1.6449. Step 3. Convert those z-values back into values of the sample mean. H0 is rejected when (ȳ − 40)/√5.6 < −1.6449 or (ȳ − 40)/√5.6 > 1.6449. Rearranging, ȳ = 40 ± 1.6449 × √5.6 1.6449 × 2.3664 = 3.8924 lower critical value = 40 − 3.8924 = 36.1076 upper critical value = 40 + 3.8924 = 43.8924 Step 4. State the set, to 3 significant figures. A time cannot be negative, so the lower tail is bounded below by 0. The claim is rejected for 0 < ȳ < 36.1 or ȳ > 43.9 (3 s.f.).