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 = 10 subjects with the syndrome, the average heat output was x = 0.61, and for n = 10 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 = 0.5.

(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 less than 1 cal/cm²/min below 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.

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 =

State the conclusion in the problem context.

Fail to reject H₀. The data suggests that the average heat output for sufferers is less than 1 cal/cm²/min below that of non-sufferers.Fail to reject H₀. The data suggests that the average heat output for sufferers is the same as that of non-sufferers.    Reject H₀. The data suggests that the average heat output for sufferers is more than 1 cal/cm²/min below that of non-sufferers.Reject H₀. The data suggests that the average heat output for sufferers is the same as that of non-sufferers.

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

z = -2.55

p value =0.0054

Reject H₀. The data suggests that the average heat output for sufferers is more than 1 cal/cm²/min below that of non-sufferers

b)

c)