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 = 8 subjects with the syndrome, the average heat output was x = 0.61, and for n = 8 nonsufferers, the average output was 2.09. Let uy 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 = 0.1 and 2 = 0.5. (a) Consider testing Ho: M1 - M2 = -1.0 versus H: U1 - uz < -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. O 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. O Fail to 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 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. (b) What is the probability of a type II error when the actual difference between 4, and uz is 41 - 42 = -1.5? (Round your answer to four decimal places.) (C) Assuming that m = n, what sample sizes are required to ensure that B = 0.1 when 41 - M2 = -1.5? (Round your answer up to the nearest whole number.) subjects

Answer 1

a) The test statistic defined under the null hypothesis is = IT, - t - Cho tu 2 1615 5/12 2. 0.61-2.09 -(1.0) 0.12 +0.52 -0.48 0.1803 =-266 value 0.0039 (From standard Normal table we Suice P reject the null value 0.05 & 0.01 and hypothesis at 5% & 1% level of significance conclude that the an werage heat output for sufferers more than I cal/em/min below that of non- sufferers Chol