Question

You are given the sample mean and the population standard deviation. Use this information to construct the​ 90% and​ 95% confidence intervals for the population mean. Interpret the results and compare the widths of the confidence intervals. From a random sample of 40 business​ days, the mean closing price of a certain stock was ​$122.05. Assume the population standard deviation is ​$10.53.

Answer 1

Here sample size n=40 is more than 30 and hence we assume that the underlying distribution is approximately normal.

Moreover, population SD is given so we use Z statistic based confidence intervals. Then the 100(1-a)% CI is

  

For a=.10 (i.e. 90%) interval is (119.3114,  124.7886)

Thus in repeated sampling the true mean is contained in the interval (119.3114,  124.7886) 90% times.

The width is  5.4772

For a=.05 (i.e. 95%) interval is (118.7868,   125.3132)
Thus in repeated sampling the true mean is contained in the interval (118.7868,   125.3132) 95% times.

The width is 6.5264, which is more than that of 90% confidence interval.