Question 11.7: Happiness and Income A psychologist randomly selected 100 pe......
Happiness and Income
A psychologist randomly selected 100 people from each of four income groups and asked them if they were “very happy.” For people who made less than $30,000, 24% responded yes. For people who made $30,000 to $74,999, 33% responded yes. For people who made $75,000 to $90,999, 38% responded yes, and for people who made $100,000 or more, 49% responded yes. At α = 0.05, test the claim that there is no difference in the proportion of people in each economic group who were very happy.
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.
\begin{aligned}E_{1,1}=\frac{(144)(100)}{400}=36 & E_{1,2}=\frac{(144)(100)}{400}=36 & E_{1,3}=\frac{(144)(100)}{400}=36 & E_{1,4}=\frac{(144)(100)}{400}=36 \\E_{2,1}=\frac{(256)(100)}{400}=64 & E_{2,2}=\frac{(256)(100)}{400}=64 & E_{2,3}=\frac{(256)(100)}{400}=64 & E_{2,4}=\frac{(256)(100)}{400}=64\end{aligned}The completed table is shown.
\begin{array}{|l|c|c|c|c|c|} \hline \begin{array}{l}\textbf { Household } \\ \textbf { income }\end{array} & \begin{array}{c}\textbf { Less than } \\ \ \mathbf{\$3 0 , 0 0 0}\end{array} & \begin{array}{c}\mathbf{\$ 3 0 , 0 0 0 -} \\ \mathbf{\$ 7 4 , 9 9 9}\end{array} & \begin{array}{c}\mathbf{\$ 7 5 , 0 0 0 -} \\ \mathbf{\$ 9 9 , 9 9 9}\end{array} & \begin{array}{c}\mathbf{\$ 1 0 0 , 0 0 0} \\ \textbf { or more }\end{array} &\textbf{ Total} \\ \hline Yes & 24(36) & 33(36) & 38(36) & 49(36) & 144 \\ No & \underline{76}(64) & \underline{67}(64) & \underline{62}(64) & \underline{51}(64) & \underline{256} \\ & 100 \qquad & 100 \qquad& 100 \qquad& 100 \qquad& 400\\ \hline \end{array}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.
