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. If​ convenient, use technology to construct the confidence intervals. A random sample of 50 home theater systems has a mean price of ​$149.00. Assume the population standard deviation is ​$18.70.

Construct a​ 90% confidence interval for the population mean. The​ 90% confidence interval is ​( _____​, ______​). ​(Round to two decimal places as​ needed.)

Answer 1

At 90% confidence interval the critical value is z_0.05 = 1.645

The 90% confidence interval for population mean is

+/- z_0.05 *

= 149 +/- 1.645 * 18.7/

= 149 +/- 4.35

= 144.65, 153.35

We are 90% confident that the true population mean lies within the confidence bound 144.65 and 153.35.

At 95% confidence interval the critical value is z_0.025 = 1.96

The 95% confidence interval for population mean is

+/- z_0.025 *

= 149 +/- 1.96 * 18.7/

= 149 +/- 5.18

= 143.82, 154.18

We are 95% confident that the true population mean lies within the confidence bound 143.82 and 154.18.

The width of the 95% confidence interval is wider than the 90% confidence interval. Because if the confidence level increases, the margin of error also increases, so that the width of the interval becomes wider.