Question 8, 10.3.19
HW Score: $66.49 \%, 20.61$ of 31 points
Part 1 of 3
Points: 1.67 of 5
Save
One year, the mean age of an inmate on death row was 39.1 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 9.4 . 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.
\begin{tabular}{l|l|}
$H_{0}:$ & $\mu$ \\
$H_{1}:$ & $\mu$
\end{tabular}
(Type integers or decimals. Do not round.)

Step 1 ：Given values are: mean = 38.3, standard deviation = 9.4, sample size = 32.

Step 2 ：We need to find the t-score for a 95% confidence level with 31 degrees of freedom. This can be done using a t-distribution table or a statistical software or calculator.

Step 3 ：The calculated t-score is approximately 2.04.

Step 4 ：We calculate the margin of error using the formula: margin of error = t-score * (standard deviation / sqrt(sample size)). The calculated margin of error is approximately 3.39.

Step 5 ：We calculate the confidence interval using the formula: lower bound = mean - margin of error, upper bound = mean + margin of error. The calculated confidence interval is approximately (34.91, 41.69).

Step 6 ：This means that we are 95% confident that the true mean age of death-row inmates is between 34.91 and 41.69 years.

Step 7 ：Since the previous mean age of 39.1 years falls within this interval, we do not have enough evidence to reject the null hypothesis that the mean age has not changed.

Step 8 ：The hypotheses for this test are: \(H_{0}: \mu = 39.1\) (The mean age of death-row inmates has not changed.) and \(H_{1}: \mu \neq 39.1\) (The mean age of death-row inmates has changed.)

Step 9 ：The 95% confidence interval for the mean age of death-row inmates is approximately \(\boxed{(34.91, 41.69)}\).