Question: The amount of postage was recorded for a random sample of n = 400 parcels handled by the postal system on a particular day. The sample mean and standard deviation were \(\bar{X}=3.47\) and s = £2.50. It is desired to obtain an interval estimate of \(\mu \), the mean amount of postage on all parcels handled by the postal system that day.

(a) Estimate \(\sigma \left( {\bar{X}} \right)\), the standard deviation of the sampling distribution of \(\bar{X}\) .

(b) Calculate the confidence limits \(\bar{X}\) ± 2s{ \(\bar{X}\) }.

(c) Obtain a 90 percent confidence interval for p..

(d) If 100 independent random samples of 400 supermarkets each had been selected and if a 90 percent confidence interval for \(\mu \) had been calculated for each of the 100 samples, what proportion of the 100 confidence intervals would be expected to contain \(\mu \) ?