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. Ifconvenient, use technology to construct the confidence intervals.A random sample of 40 home theater systems has a mean price of $126.00. Assume the population standard deviation is$18.60Construct a 90% confidence interval for the population mean.

Answer 1

Given the following below

`\begin{gathered} \bar{x}=\text{ \$126}.00 \\ n=40 \\ \sigma=\text{ \$18.60} \\ At\text{ 90\% the critical z is }\pm1.645 \end{gathered}`

To find the 90% confidence interval, the formula is

`\bar{x}\pm z\frac{\sigma}{\sqrt[]{n}}`

Substituting values into the formula above

`126\pm1.645\frac{18.6}{\sqrt[]{40}}`

Solve the above

`\begin{gathered} 126+_{}1.645\frac{18.6}{\sqrt[]{40}}=130.84\text{ (two decimal places)} \\ 126-1.645\frac{18.6}{\sqrt[]{40}}=121.16\text{ (two decimal place)} \end{gathered}`

At 90% confidence interval, the population mean will be between 121.16 and 130.84 (121.16, 130.84)