Question 11.7: Happiness and Income A psychologist randomly selected 100 pe......

It is necessary to make a table showing the number of people in each group who responded yes and the number of people in each group who responded no.

For group 1, 24% of the people responded yes, so $24 \%$ of $100=0.24(100)=24$ responded yes and $100-24=76$ responded no.

For group 2, 33% of the people responded yes, so $33 \%$ of $100=0.33(100)=33$ responded yes and $100-33=67$ responded no.

For group 3, 38% of the people responded yes, so $38 \%$ of $100=0.38(100)=38$ people responded yes and $100-38=62$ people responded no.

For group $4,49 \%$ of the people responded yes, so $49 \%$ of $100=0.49(100)=49$ responded yes, and $100-49=51$ people responded no.

Tabulate the data in a table, and find the sums of the rows and columns as shown.

$\begin{array}{|l|c|c|c|c|c|} \hline \begin{aligned} & \textbf { Household } \\ & \textbf { income }\end{aligned} & \begin{gathered}\textbf { Less than } \\ \mathbf{\$ 3 0 , 0 0 0}\end{gathered} & \begin{gathered}\mathbf{\$ 3 0 , 0 0 0 -} \\ \mathbf{\$ 7 4 , 9 9 9}\end{gathered} & \begin{gathered}\mathbf{\$ 7 5 , 0 0 0 -} \\ \mathbf{\$ 9 9 , 9 9 9}\end{gathered} & \begin{gathered}\mathbf{\$ 1 0 0 , 0 0 0} \\ \textbf{ or more }\end{gathered} & \textbf{Total} \\ \hline Yes & 24 & 33 & 38 & 49 & 144 \\ No & 76 & 67 & 62 & 51 & 256\\ & \overline{100} & \overline{100} &\overline{100}&\overline{100} &\overline{400} \\ \hline \end{array}$

Source: Based on information from Princeton Survey Research Associates International.

Step 1 State the hypotheses and identify the claim.

$H_{0}: p_{1}=p_{2}=p_{3}=p_{4}$ (claim)

$H_{1}$ : At least one proportion differs from the others.

Step 2 Find the critical value. The formula for the degrees of freedom is the same as before: $(R-1)(C-1)=(2-1)(4-1)=1(3)=3$. The critical value is $7.815$.

Step 3 Compute the test value. Since we want to test the claim that the proportions are equal, we use the expected value as $\frac{1}{4} \cdot 400=100$. First compute the expected values as shown previously.

The completed table is shown.

Next calculate the test value.

$\begin{aligned}\chi^{2}= & \sum \frac{(O-E)^{2}}{E} \\= & \frac{(24-36)^{2}}{36}+\frac{(33-36)^{2}}{36}+\frac{(38-36)^{2}}{36}+\frac{(49-36)^{2}}{36} \\& +\frac{(76-64)^{2}}{64}+\frac{(67-64)^{2}}{64}+\frac{(62-64)^{2}}{64}+\frac{(51-64)^{2}}{64} \\= & 4.000+0.250+0.111+4.694+2.250+0.141+0.063+2.641 \\= & 14.150\end{aligned}$

Step 4 Make the decision. Reject the null hypothesis since $14.150>7.815$. See Figure 11-9.

Step 5 Summarize the results. There is enough evidence to reject the claim that there is no difference in the proportions. Hence, the incomes seem to make a difference in the proportions.