Question 11.6: Mental Illness A researcher wishes to see if there is a diff......
Mental Illness
A researcher wishes to see if there is a difference between men and women in the number of cases of mental disorders. She selects a sample of 30 males and 24 females and classifies them according to their mental disorder. The results are shown.
\begin{array}{|l|c|c|c|c|} \hline \textbf{Gender }& \textbf{Anxiety }& \textbf{Depression }& \textbf{Schizophrenia }& \textbf{Total }\\ \hline \text{Male }& 8 & 12 & 10 & 30 \\ \text{Female }& \underline{12} & \underline{9} & \underline{3} & \underline{24} \\ \text{Total }& 20 & 21&13 & 54 \\ \hline \end{array}
At α = 0.10, can the researcher conclude that there is a difference in the types of disorders?
Step 1 State the hypotheses and identify the claim.
H_{0} : The type of mental disorder is independent of the gender of the person.
H_{1} : The type of mental disorder is related to the gender of the person (claim).
Step 2 Find the critical value. The critical value is 4.605 since the degrees of freedom are (2-1)(3-1)=2.
Step 3 Compute the test value. First compute the expected values.
\begin{array}{lll}E_{1,1}=\frac{(30)(20)}{54}=11.11 & E_{1,2}=\frac{(30)(21)}{54}=11.67 & E_{1,3}=\frac{(30)(13)}{54}=7.22 \\E_{2,1}=\frac{(24)(20)}{54}=8.89 & E_{2,2}=\frac{(24)(21)}{54}=9.33 & E_{2,3}=\frac{(24)(13)}{54}=5.78\end{array}
The completed table is shown.
\begin{array}{|l|l|l|l|c|} \hline \text{Gender }& \text{Anxiety }& \text{Depression }& \text{Schizophrenia }& \text{Total }\\ \hline \text{Male }& 8(11.11) & 12(11.67) & 10(7.22) & 30 \\ \text{Female }& \underline{12}(8.89) &\underline{9}(9.33) & \underline{3}(5.78) & \underline{24} \\ \text{Total }&20&21 &13 & 54\\ \hline \end{array}
The test value is
\begin{aligned}\chi^{2}= & \sum \frac{(O-E)^{2}}{E} \\= & \frac{(8-11.11)^{2}}{11.11}+\frac{(12-11.67)^{2}}{11.67}+\frac{(10-7.22)^{2}}{7.22} \\& +\frac{(12-8.89)^{2}}{8.89}+\frac{(9-9.33)^{2}}{9.33}+\frac{(3-5.78)^{2}}{5.78} \\= & 0.871+0.009+1.070+1.088+0.012+1.337=4.387\end{aligned}
Step 4 Make the decision. The decision is to not reject the null hypothesis since 4.387<4.605. See Figure 11-8.
Step 5 Summarize the results. There is not enough evidence to support the claim that the mental disorder is related to the gender of the individual.
In this case, the P-value is between 0.10 and 0.90. The TI-84 gives a P-value of 0.112. Again, this supports the decision and summary as stated previously.
