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 $132.00. assume the population standard deviation is $16.60. construct a 90% confidence interval for the population mean. the 90% confidence interval is (□,□). (round to two decimal places as needed.) construct a 95% confidence interval for the population mean. the 95% confidence interval is (□,□). (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 narrower 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 wider 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%.

Explanation:

Step1: Recall the formula for confidence interval

The formula for a confidence interval when the population standard deviation \(\sigma\) is known is \(\bar{x}\pm z_{\alpha/2}\frac{\sigma}{\sqrt{n}}\), where \(\bar{x}\) is the sample mean, \(z_{\alpha/2}\) is the z - score, \(\sigma\) is the population standard deviation, and \(n\) is the sample size.

Step2: Find \(z\) - scores

For a 90% confidence interval, \(\alpha = 1 - 0.90=0.10\), and \(\alpha/2 = 0.05\). From the standard normal table, \(z_{0.05}\approx1.645\).
For a 95% confidence interval, \(\alpha=1 - 0.95 = 0.05\), and \(\alpha/2=0.025\). From the standard normal table, \(z_{0.025}\approx1.96\).

Step3: Calculate the margin of error for 90% confidence interval

Given \(\bar{x}=132\), \(\sigma = 16.6\), \(n = 50\).
The margin of error \(E_1=z_{\alpha/2}\frac{\sigma}{\sqrt{n}}=1.645\times\frac{16.6}{\sqrt{50}}\)

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The 90% confidence interval is \(\bar{x}-E_1<\mu<\bar{x} + E_1\), \(132-3.86<\mu<132 + 3.86\), \((128.14,135.86)\)

Step4: Calculate the margin of error for 95% confidence interval

The margin of error \(E_2=z_{\alpha/2}\frac{\sigma}{\sqrt{n}}=1.96\times\frac{16.6}{\sqrt{50}}\)

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The 95% confidence interval is \(\bar{x}-E_2<\mu<\bar{x}+E_2\), \(132 - 4.60<\mu<132+4.60\), \((127.40,136.60)\)

Step5: Interpret the results

Confidence intervals are about estimating the population mean. A higher confidence level (95% compared to 90%) requires a larger \(z\) - score, which leads to a larger margin of error. So the 95% confidence interval is wider than the 90% confidence interval.

Answer:

• The 90% confidence interval is \((128.14,135.86)\)
• The 95% confidence interval is \((127.40,136.60)\)
• 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 wider than the 90%