Q 8.73 - Elementary Statistics

Question

Political Prisoners. A. Ehlers et al. studied various characteristics of political prisoners from the former East Germany and presented their findings in the paper "Posttraumatic Stress Disorder (PTSD) Following Political Imprisonment: The Role of Mental Defeat. Alienation, and Perceived Permanent Change". According to the article. the mean duration of imprisonment for 32 patients with chronic PTSD was $33.4$ months. Assuming that $\sigma =42$months, determine a $95\%$ confidence interval for the mean duration of imprisonment, $\mu$. of all East German political prisoners with chronic PTSD. Interpret your answer in words.

Step-by-Step Solution

Answer

$95\%$confidence interval of $\mu$ is $(18.8477,47.9523)$.

Step 1: Given Information

It is given that

$\sigma =42$

$n=32$

Sample mean is $33.4$ months.

Step 2: Determination of Confidence Level

$\overline{x}=33.4$

For $95\%$ confidence interval of $\mu$,

Confidence Level $=100\times 0.95\%$

$\Rightarrow 1-\alpha =0.95$

$\Rightarrow \alpha =0.05$

$\Rightarrow z_{\frac{a}{2}}=z_{\frac{0.05}{2}}=z_{0.025}=1.96$

Step 3: Determination of 95 % Confidence Interval for μ

$95\%$ confidence level of $\mu$ is given by

$\overline{x}-z_{0.025}\times \frac{\sigma }{\sqrt{n}},\overline{x}+z_{0.025}\times \frac{\sigma }{\sqrt{n}}$

$=33.4-1.96\times \frac{42}{\sqrt{32}},33.4+1.96\times \frac{42}{\sqrt{32}}$

$=33.40-14.5523,33.40+14.5523$

$=18.8477,47.9523$

Hence, $95\%$ confidence level of $\mu$ is $\left(18.8477,47.9523\right)$