1. (Sampling distribution & confidence interval) A random variable has a population mean equal to 1,573 and population variance equal to 952,021. Your interest lies in estimating the population mean of this random variable. (Of course, you do not know what the population mean and population variance are.) With that in mind, you take a representative sample of size 85 from the population of the random variable. You then use this sample data to calculate the sample average as an estimate for the population mean.
(a) Using your knowledge about the central limit theorem (CLT), and assuming that the CLT has already “established itself” / “kicked in” when the sample size is 85, what is the probability that the sample average that you calculated will lie between 1,502 and 1,748?
(b) You then use your sample data to calculate a 92% confidence interval for the population mean. Assuming that you can estimate the variance of the underlying random variable very precisely with your sample data, what should be the width of this confidence interval?
(c) Suppose that now you would like to cut the width of your 92% confidence interval in half by increasing your sample size. What should be the size of your sample in order to achieve this goal?