Persons having Raynaud'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 = 9 subjects with the syndrome, the average heat output was x = 0.62,and for n = 9 nonsufferers, the average output was 2.08. Let μ1 and μ2 denote the true average heat outputs for the sufferers and nonsufferers, respectively. Assume that the two distributions of heat output are normal with σ1 = 0.3 and σ2 = 0.4. a. Consider testing H0: μ1 − μ2 = −1.0 versus Ha: μ1 − μ2 < −1.0 at level 0.01. Describe in words what Ha says, and then carry out the test. Ha says that the average heat output for sufferers is more than 1 cal/cm2/min below that of non-sufferers.
b. Calculate the test statistic and P-value.
c. What is the probability of a type II error when the actual difference between μ1 and μ2 is μ1 − μ2 = −1.2?
d. Assuming that m = n, what sample sizes are required to ensure that β = 0.1 when μ1 − μ2 = −1.2?