Chapter 9.1, Problem 5E

Persons having Reynaud’s syndrome are apt to suffer a sudden impairment of blood circulation in fingers and toes. In an experiment to study the extent of this impairment, each subject immersed a forefinger in water and the resulting heat output (cal/cm²/min) was measured. For m = 10 subjects with the syndrome, the average heat output was $\bar{X}$ = .64, and for n = 10 nonsufferers, the average output was 2.05. Let µ₁ and µ₂ denote the true average heat outputs for the two types of subjects. Assume that the two distributions of heat output are normal with σ₁ = .2 and σ₂ = .4.

a. Consider testing H₀: µ₁ − µ₂ = −1.0 versus Ha: µ₁ = µ₂, < −1.0 at level .01. Describe in words what Ha says, and then carry out the test.

b. What is the probability of a type II error when the actual difference between µ₁ and µ₂ is µ₁ − µ₂ = 21.2?

c. Assuming that m 5 n, what sample sizes are required to ensure that β = .1 when µ₁ – μ₂ = −1.2?