People with Raynaud's syndrome are prone to experiencing a sudden impairment of blood circulation in their fingers and toes. In an experiment aimed at studying the extent of this impairment, each participant submerged a forefinger in water and the resulting heat output (cal/cm2/min) was measured. For 10 subjects with the syndrome, the average heat output was 0.64, while for 10 non-sufferers, the average output was 2.05. Let μ1 and μ2 represent the true mean heat outputs for the two types of subjects. It is assumed that the two distributions of heat output are normal, with σ1 = 0.2 and σ2 = 0.4.
(a) Consider testing H0: μ1 - μ2 = -1.0 versus Ha: μ1 - μ2 < -1.0. Calculate the p-value of the test. (Answer: 0.0019)
(b) What is the probability of a type II error when the actual difference between μ1 and μ2 is -1.2? (with a significance level of 0.1) (Answer: 0.8212)
(c) Assuming that the two sample sizes are the same, what sample sizes are required to ensure that β = 0.1, α = 0.01, when μ1 - μ2 = -1.2? (Answer: 66 from each population)