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. Form=9subjects with the syndrome, the average heat output wasx=0.61,and forn=9nonsufferers, the average output was2.09. Let₁and₂denote the true average heat outputs for the sufferers and nonsufferers, respectively. Assume that the two distributions of heat output are normal with₁=0.3and₂=0.5.(a) Consider testingH₀:₁₂=1.0versusH_a:₁₂<1.0at level 0.01. Describe in words whatH_asays, and then carry out the test.

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

H_asays that the average heat output for sufferers is less than 1cal/cm²/minbelow that of non-sufferers.

H_asays that the average heat output for sufferers is more than 1cal/cm²/minbelow that of non-sufferers.

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

z=P-value =

State the conclusion in the problem context.

Fail to rejectH₀. The data suggests that the average heat output for sufferers is the same as that of non-sufferers.

RejectH₀. The data suggests that the average heat output for sufferers is the same as that of non-sufferers.

RejectH₀. The data suggests that the average heat output for sufferers is more than 1 cal/cm²/min below that of non-sufferers.

Fail to rejectH₀. The data suggests that the average heat output for sufferers is less than 1 cal/cm²/min below that of non-sufferers.

(b) What is the probability of a type II error when the actual difference between₁and₂is₁₂=1.2?

(Round your answer to four decimal places.)

(c) Assuming thatm=n, what sample sizes are required to ensure that= 0.1when₁₂=1.2?

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

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