Question

in a random sample of five mobile devices, the mean repair cost was $80.00 and the standard deviation was $11.00. assume the population is normally distributed and use a t - distribution to find the margin of error and construct a 90% confidence interval for the population mean. interpret the results.
the 90% confidence interval for the population mean μ is (□,□) (round to two decimal places as needed.)
the margin of error is $□ (round to two decimal places as needed.)
interpret the results. choose the correct answer below.
○ a. if a large sample of mobile devices are taken approximately 90% of them will have repair costs between the bounds of the confidence interval.
○ b. it can be said that 90% of mobile devices have a repair cost between the bounds of the confidence interval.
○ c. with 90% confidence, it can be said that the population mean repair cost is between the bounds of the confidence interval.
○ d. with 90% confidence, it can be said that the repair cost is between the bounds of the confidence interval.

Explanation:

Step1: Find the degree of freedom and critical value

Degree of freedom \(df=n - 1=5-1 = 4\). For a \(90\%\) confidence interval (\(\alpha=1 - 0.90=0.10\), \(\alpha/2=0.05\)), using the t - distribution table or a calculator, the critical value \(t_{\alpha/2}\) with \(df = 4\) is \(t_{0.05,4}=2.132\).

Step2: Calculate the margin of error

The formula for the margin of error \(E=t_{\alpha/2}\frac{s}{\sqrt{n}}\). Given \(s = 11\), \(n = 5\), and \(t_{\alpha/2}=2.132\).

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Step3: Calculate the confidence interval

The confidence interval for the population mean \(\mu\) is \(\bar{x}-E<\mu<\bar{x} + E\). Given \(\bar{x}=80\), \(E\approx10.47\)

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Step4: Interpret the confidence interval

The confidence interval is about the population mean. A confidence interval of \(90\%\) means that with \(90\%\) confidence, the population mean repair cost is between the bounds of the confidence interval.

Answer:

The margin of error \(E\approx\$10.47\). The \(90\%\) confidence interval for the population mean \(\mu\) is \((69.53,90.47)\). The correct interpretation is C. With \(90\%\) confidence, it can be said that the population mean repair cost is between the bounds of the confidence interval.