A-Level Hypothesis testing: worked solution

Question

To keep the mean height at most 12.8 cm, the manager upgrades the line so population variance becomes 0.04 cm². He then tests at the 2% level with a large sample of n boxes. Find the critical region in terms of n.

Let μ be the population mean height of the boxes after the upgrade. The manager wants the mean to be at most 12.8 cm, so he is testing against the possibility that it has risen above 12.8: H₀: μ = 12.8 H₁: μ > 12.8 This is a one-tailed test at the 2% significance level. The population variance is now 0.04 cm², so the population standard deviation is σ = √0.04 = 0.2 cm. The sample size n is large, so by the Central Limit Theorem the sample mean is approximately normal: under H₀, X̄ ~ N(12.8, 0.04/n), with standard error √(0.04/n) = 0.2/√n. Standardise: Z = (X̄ − 12.8)/(0.2/√n) ~ N(0, 1) approximately. H₀ is rejected for large values of x̄, i.e. in the upper 2% tail of Z. The critical value z satisfies P(Z ≥ z) = 0.02, so z = 2.0537489. Reject H₀ when (x̄ − 12.8)/(0.2/√n) ≥ 2.0537489 x̄ − 12.8 ≥ 2.0537489 × 0.2/√n x̄ ≥ 12.8 + 0.41074.../√n Critical region: x̄ ≥ 12.8 + 0.411/√n (3 s.f.).