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 = 0.61, and for n = 9 nonsufferers, the average output was 2.05. Let 41 and uz denote the true average heat outputs for the sufferers and nonsufferers, respectively. Assume that the two distributions of heat output are normal with 01 = 0.3 and 2 = 0.4. (a) Consider testing Ho: 41 - M2 = -1.0 versus H: U1 - H2 < -1.0 at level 0.01. Describe in words what He says, and then carry out the test. OH, says that the average heat output for sufferers is more than 1 cal/cm2/min below that of non-sufferers. OH, says that the average heat output for sufferers is the same as that of non-sufferers. He 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 = State the conclusion in the problem context. Fail to reject Ho. The data suggests that the average heat output for sufferers is less than 1 cal/cm2/min below that of non-sufferers. Reject Ho. The data suggests that the average heat output for sufferers is more than 1 cal/cm2/min below that of non-sufferers. Reject Ho. The data suggests that the average heat output for sufferers is the same as that of non-sufferers. Fail to reject Ho. 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 41 and uz is H1 - H2 = -1.1? (Round your answer to four decimal places.) (c) Assuming that m = n, what sample sizes are required to ensure that B = 0.1 when M1 - 42 = -1.1? (Round your answer up to the nearest whole number.) subjects You may need to use the appropriate table in the Appendix of Tables to answer this question.

Answer 1

Ha says that the average heat output for sufferers is more than 1 below that of non sufferers

std error of difference σ1-2=				√(σ21/n1+σ22/n2)    =	0.1667
test stat z                                       =				(x1-x2-Δo)/σ1-2     =	-2.64
p value : =					0.0041

reject Ho, the data suggest that ...is more than,,,,,,,,,,,

b)

for 0.01 level and left tailed test critival value Za rejection region: x <= A +Za*ox or type II error =probability of not rejecting B = -2.33 -1.3883 P(Z >4-1.3883- A)/ox)) = P(Z >-1.73) 0.9582

c)

required sample size             =n

=(Zα/2+Zβ)2(σ12+σ22)/(Δo-Δ)2

=

326