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).
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Obtain a $90\mathrm{\%}$ confidence interval for the difference, ${\mu }_1-{\mu }_2$, between the mean ages at arrest for prisoners with chronic PTSD and remitted PTSD.

The $90\mathrm{\%}$ confidence interval is from 5.934 to 12.066 (Round to three decimal places as needed.)

Step 1 ：Given values are as follows: mean of chronic PTSD group (${\overline{x}}_1$) 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 (${\overline{x}}_2$) 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 $c{i}_{\text{lower}}=({\overline{x}}_1-{\overline{x}}_2)-t\times se$ and $c{i}_{\text{upper}}=({\overline{x}}_1-{\overline{x}}_2)+t\times se$. Substituting the given values, we get $c{i}_{\text{lower}}\approx 6.019$ and $c{i}_{\text{upper}}\approx 11.981$.

Step 6 ：Final Answer: The $90\mathrm{\%}$ 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 $\boxed{{\displaystyle 6.019}}$ to $\boxed{{\displaystyle 11.981}}$.