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 45 45 home theater systems has a mean price of ​$ 137.00 137.00. Assume the population standard deviation is ​$ 16.60 16.60. Construct a​ 90% confidence interval for the population mean. The​ 90% confidence interval is ​( nothing ​, nothing ​). ​(Round to two decimal places as​ needed.) Construct a​ 95% confidence interval for the population mean. The​ 95% confidence interval is ​( nothing ​, nothing ​). ​(Round to two decimal places as​ needed.) Interpret the results. Choose the correct answer below. A. With​ 90% confidence, it can be said that the population mean price lies in the first interval. With​ 95% confidence, it can be said that the population mean price lies in the second interval. The​ 95% confidence interval is wider than the​ 90%. B. With​ 90% confidence, it can be said that the population mean price lies in the first interval. With​ 95% confidence, it can be said that the population mean price lies in the second interval. The​ 95% confidence interval is narrower than the​ 90%. C. With​ 90% confidence, it can be said that the sample mean price lies in the first interval. With​ 95% confidence, it can be said that the sample mean price lies in the second interval. The​ 95% confidence interval is wider than the​ 90%.

Answer 1

Solution :

Given that,

Point estimate = sample mean = = 137.00

Population standard deviation =    = 16.60

Sample size = n = 45

1) At 90% confidence level

= 1 - 90%  

= 1 - 0.90 =0.10

/2 = 0.05

Z/2 = Z_{0.05 = 1.645}

Margin of error = E = Z/2 * ( /n)

= 1.645 * ( 16.60 /   45)

= 4.07

At 90% confidence interval estimate of the population mean is,

  ± E

137.00 ± 4.07

( 132.93, 141.07 )

2) At 95% confidence level

= 1 - 95%  

= 1 - 0.95 =0.05

/2 = 0.025

Z/2 = Z_{0.025 = 1.96}

Margin of error = E = Z/2 * ( /n)

= 1.96 * ( 16.60 /   45)

= 4.85

At 95% confidence interval estimate of the population mean is,

  ± E

137.00 ± 4.85   

( 132.15, 141.85)

correct option is =A

With​ 90% confidence, it can be said that the population mean price lies in the first interval. With​ 95% confidence, it can be said that the population mean price lies in the second interval. The​ 95% confidence interval is wider than the​ 90%