In the advertisement for a large university, the dean of the school of business

Question

In the advertisement for a large university, the dean of the school of business claims that the weekly salary of the school's graduates 1 year after graduation is normally distributed with a mean $1200 and a standard deviation of $400. A second-year student in the business school who has just completed his statistics course would like to check whether the claim about the mean is correct. He does a survey of 100 randomly chosen people who graduated 1 year earlier and determines their weekly salary. (Calculations are to be done to the 4th decimal place unless it is unnecessary)

a. What is the sampling distribution of the mean of the 100 graduates' weekly salary?

b. What is the standard error of the sampling distribution of the mean of the 100 graduates' weekly salary?

c. Now suppose that the sample mean is $1100. If the dean's claim is justified, waht is the probability of observing a sample mean as low as this result?

d. Suppose again that the sample mean is $1100. Estimate the population mean of the weekly salary with 95% confidence.

e. What is the approximate sample size required to estimate the population mean to within $100? Assume the confidence level is to 99%

Explanation / Answer

a) sampling distribution of the mean of the 100 graduates' weekly salary would be aprroximately normal with

mean =1200

b)and std error =std deviation/(n)1/2 =40

c) hence 1-P(X<1100)=P(Z<(1100-1200)/40)=P(Z<-2.5)=0.0062

d)for 95% Ci, z=1.96

hence Ci=mean +/- z*std error =1021.6 ; 1178.4

e)for 99% CI, z=2.5758

margin of error E=100

and std deviation =400

hence sample size n=(S*Z/E)2 ~107

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