The average student loan debt of a U.S. college student at the end of four years

Question

The average student loan debt of a U.S. college student at the end of four years of college is estimated to be about $21,000. You take a random sample of 100 college students in the state of Wyoming, and find that the mean debt is $19,250. Assume that the population standard deviation is $2,200.
(a) Construct a 95% confidence interval for the mean debt of all Wyoming college graduates.

(b) Construct a 98% confidence interval for the mean debt of all Wyoming college graduates.
(c) Construct a 99% confidence interval for the mean debt of all Wyoming college graduates.
(d) If the population standard deviation is unknown, and if the sample standard deviation was calculated to be $2,350, construct a 95% confidence interval for the mean debt of all Wyoming college graduates.
(e) If the margin of error is to be reduced to $320, what is the required sample size? Assume the population standard deviation is $2,200 and keep the confidence level at 95%.

Explanation / Answer

We know that the confidence interval is given as

Mean +- Z* SD/sqrt(N) , where n is sample size = 100

a) Mean = 19250 and SD = 2200 , Now in order to calculate the 95% CI , we must look at the z table and find the value as 1.96

so 19250 +- 1.96*2200/sqrt(100), so we get the CI as 19681.2 , 18818.8

b) Mean = 19250 and SD = 2200 , Now in order to calculate the 98% CI , we must look at the z table and find the value as 2.326

so 19250 +- 2.326*2200/sqrt(100), so we get the CI as 19761.72 , 18738.28

c) Mean = 19250 and SD = 2200 , Now in order to calculate the 98% CI , we must look at the z table and find the value as 2.576

so 19250 +- 2.576*2200/sqrt(100), so we get the CI as 19816.72 , 18683.28

d) In this case the the CI becomes

mean +- Z*SD

19250 +- 1.96*2350 = (23856 , 14644)

Please note that we can answer only 4 subparts of a question at a time , as per the answering guidelines

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