Answered: ersons having Raynaud's syndrome are…

Algebra & Trigonometry with Analytic Geometry

13th Edition

ISBN:9781133382119

Author:Swokowski

Publisher:Swokowski

Chapter5: Inverse, Exponential, And Logarithmic Functions

Section5.6: Exponential And Logarithmic Equations

Problem 64E

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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 = 0.61, and for n = 9 nonsufferers, the average output was 2.09. Let ?₁ and ?₂ denote the true average heat outputs for the sufferers and nonsufferers, respectively. Assume that the two distributions of heat output are normal with ?₁ = 0.3 and ?₂ = 0.5.

(a) Consider testing H₀: ?₁ − ?₂ = −1.0 versus H_a: ?₁ − ?₂ < −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 the same as that of non-sufferers.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.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.

(b) What is the probability of a type II error when the actual difference between ?₁ and ?₂ is

?₁ − ?₂ = −1.2?

(Round your answer to four decimal places.)

(c) Assuming that m = n, what sample sizes are required to ensure that ? = 0.1 when

?₁ − ?₂ = −1.2?

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