Question

You may need to use the appropriate appendix table or technology to answer this question.

*Barron's* reported that the average number of weeks an
individual is unemployed is 18.5 weeks. Assume that for the
population of all unemployed individuals the population mean length
of unemployment is 18.5 weeks and that the population standard
deviation is 4 weeks. Suppose you would like to select a sample of
60 unemployed individuals for a follow-up study.

(a) Show the sampling distribution of *x*, the sample
mean average for a sample of 60 unemployed individuals.

(b) What is the probability that a simple random sample of 60 unemployed individuals will provide a sample mean within 1 week of the population mean? (Round your answer to four decimal places.)

(c) What is the probability that a simple random sample of 60 unemployed individuals will provide a sample mean within 1/2 week of the population mean? (round your answer to four decimal places.)

Answer #1

μ = 18.5, σ = 4

(a) Sampling distribution of x-bar is approximately normal with μ = 18.5 and s = σ/√n = 4/√60 = 0.5164

(b) We want x-bar to lie between 17.5 weeks and 19.5 weeks

z1 = (17.5 - 18.5)/0.5164 = -1.9365 and z2 = (19.5 - 18.5)/0.5164 = 1.9365

P(17.5 < x-bar < 19.5) = P(-1.9365 < z < 1.9365) = 0.9472

(c) We want x-bar to lie between 18 weeks and 19 weeks

z1 = (18 - 18.5)/0.5164 = -0.9682 and z2 = (19 - 18.5)/0.5164 = 0.9682

P(18 < x-bar < 19) = P(-0.9682 < z < 0.9682 = 0.6671

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