Preliminary analyses indicate that you can consider the assumptions for using nonpooled t-procedures satisfied. Researchers randomly and independently selected 34 prisoners diagnosed with chronic posttraumatic stress disorder (PTSD) and 22 prisoners that were diagnosed with PTSD but had since recovered (remitted). \begin{tabular}{l|l} \hline Chronic & Remitted \\ \hline $\bar{x}_{1}=29.2$ & $\bar{x}_{2}=20.2$ \\ $s_{1}=3$ & $s_{2}=8$ \\ $n_{1}=34$ & $n_{2}=22$ \\ \hline \end{tabular} Obtain a $90 \%$ confidence interval for the difference, $\mu_{1}-\mu_{2}$, between the mean ages at arrest for prisoners with chronic PTSD and remitted PTSD. The $90 \%$ confidence interval is from 5.934 to 12.066 (Round to three decimal places as needed.)
Solution
Step 1 ：Given values are as follows: mean of chronic PTSD group (\( \bar{x}_{1} \)) is 29.2, standard deviation of chronic PTSD group (\( s_{1} \)) is 3, sample size of chronic PTSD group (\( n_{1} \)) is 34, mean of remitted PTSD group (\( \bar{x}_{2} \)) is 20.2, standard deviation of remitted PTSD group (\( s_{2} \)) is 8, sample size of remitted PTSD group (\( n_{2} \)) is 22, and level of significance (\( \alpha \)) is 0.10.
Step 2 ：Calculate the degrees of freedom (\( df \)) using the formula \( df = n_{1} + n_{2} - 2 \). Substituting the given values, we get \( df = 54 \).
Step 3 ：Calculate the t-score for a two-tailed test using the formula \( t = \text{ppf}(1 - \alpha/2, df) \). Substituting the given values, we get \( t \approx 1.674 \).
Step 4 ：Calculate the standard error (\( se \)) using the formula \( se = \sqrt{(s_{1}^{2}/n_{1}) + (s_{2}^{2}/n_{2})} \). Substituting the given values, we get \( se \approx 1.782 \).
Step 5 ：Calculate the lower and upper limits of the 90% confidence interval for the difference between the mean ages at arrest for prisoners with chronic PTSD and remitted PTSD using the formulas \( ci_{\text{lower}} = (\bar{x}_{1} - \bar{x}_{2}) - t \times se \) and \( ci_{\text{upper}} = (\bar{x}_{1} - \bar{x}_{2}) + t \times se \). Substituting the given values, we get \( ci_{\text{lower}} \approx 6.019 \) and \( ci_{\text{upper}} \approx 11.981 \).
Step 6 ：Final Answer: The $90 \%$ confidence interval for the difference, \( \mu_{1}-\mu_{2} \), between the mean ages at arrest for prisoners with chronic PTSD and remitted PTSD is approximately from \( \boxed{6.019} \) to \( \boxed{11.981} \).
