Question

one year, the mean age of an inmate on death row was 40.1 years. a sociologist wondered whether the mean age of a death - row inmate has changed since then. she randomly selects 32 death - row inmates and finds that their mean age is 38.3, with a standard deviation of 9.3. construct a 95% confidence interval about the mean age. what does the interval imply? click the icon to view the table of critical t - values. choose the correct hypotheses. ( h_0:mu = 40.1 ) ( h_1:mu
eq40.1 ) (type integers or decimals. do not round.) construct a 95% confidence interval about the mean age. with 95% confidence, the mean age of a death row inmate is between (square) years and (square) years. (round to two decimal places as needed)

Explanation:

Step1: Determine the critical value

Since the sample size \(n = 32\), the degrees of freedom \(df=n - 1=32-1 = 31\). For a 95% confidence interval, \(\alpha=1 - 0.95=0.05\), and \(\frac{\alpha}{2}=0.025\). Looking up the \(t\) - distribution table (or using a calculator), \(t_{\frac{\alpha}{2},df}=t_{0.025,31}\approx 2.04\)

Step2: Calculate the margin of error

The formula for the margin of error \(E\) for a confidence interval for the population mean when the population standard deviation \(\sigma\) is unknown is \(E = t_{\frac{\alpha}{2}}\frac{s}{\sqrt{n}}\), where \(s = 9.3\) (sample standard deviation) and \(n = 32\) (sample size).

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Step3: Calculate the confidence interval

The formula for the confidence interval is \(\bar{x}-E<\mu<\bar{x} + E\), where \(\bar{x}=38.3\) (sample mean)

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Answer:

With 95% confidence, the mean age of a death - row inmate is between \(34.95\) years and \(41.65\) years.