Question

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 = 0.65, and for n = 9 nonsufferers, the average output was 2.06. 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 = 0.1 and σ₂ = 0.5.

(a) Consider testing H₀: μ₁ − μ₂ = −1.0 versus H_a: μ₁ − μ₂ < −1.0 at level 0.01. Describe in words what H_a says, and then carry out the test.

H_a says that the average heat output for sufferers is the same as that of non-sufferers.

H_a says that the average heat output for sufferers is more than 1 cal/cm²/min below that of non-sufferers.    

H_a says that the average heat output for sufferers is less than 1 cal/cm²/min below that of non-sufferers.

Calculate the test statistic and P-value. (Round your test statistic to two decimal places and your P-value to four decimal places.)

z =
P-value

(b) What is the probability of a type II error when the actual difference between μ₁ and μ₂ is μ₁ − μ₂ = −1.5?

(Round your answer to four decimal places.)


(c) Assuming that m = n, what sample sizes are required to ensure that β = 0.1 when μ₁ − μ₂ = −1.5?

(Round your answer up to the nearest whole number.)

subjects

Answer 1

a)

H_a says that the average heat output for sufferers is more than 1 cal/cm²/min below that of non-sufferers.    

b)

c)

n=14