Persons with Raynaud's syndrome are prone to experiencing a sudden impairment of blood circulation in their fingers and toes. In an experiment conducted to investigate the extent of this impairment, each participant submerged one of their forefingers in water, and the resulting heat output (cal/cm2/min) was measured. For a group of m = 9 subjects with the syndrome, the average heat output was x = 0.65, while for a group of n = 9 individuals without the syndrome, the average output was 2.07. Let μ1 and μ2 represent the true average heat outputs for the individuals with the syndrome and those without, respectively. It is assumed that the two distributions of heat output follow a normal distribution with σ1 = 0.1 and σ2 = 0.5.
a) Calculate the test statistic and P-value. (Round your test statistic to two decimal places and your P-value to four decimal places.)
b) What is the probability of a type II error when the actual difference between μ1 and μ2 is μ1 − μ2 = −1.4?
c) Assuming that m = n, what sample sizes are required to ensure that β = 0.1 when μ1 − μ2 = −1.4?