The contingency table below shows the results of a random sample of 200 state representatives that was conducted to see whether their opinions on a bill are related to their party affiliation. $$\begin{array} { l | c c c }
& { \text { Opinion } } \
\hline \text { Party } & \text { Approve } & \text { Disapprove } & \text { No Opinion } \
\text { Republican } & 42 & 20 & 14 \
\text { Democrat } & 50 & 24 & 18 \
\text { Independent } & 10 & 16 & 6
\end{array}$$ Find the chi-square test statistic, χ $$\chi ^ { 2 }$$ to test the claim of independence.
A)8.030
B)11.765
C)7.662
D)9.483

Question

The contingency table below shows the results of a random sample of 200 state representatives that was conducted to see whether their opinions on a bill are related to their party affiliation. $$\begin{array} { l | c c c } 
&  { \text { Opinion } } \
\hline \text { Party } & \text { Approve } & \text { Disapprove } & \text { No Opinion } \
\text { Republican } & 42 & 20 & 14 \
\text { Democrat } & 50 & 24 & 18 \
\text { Independent } & 10 & 16 & 6
\end{array}$$ Find the chi-square test statistic, χ $$\chi ^ { 2 }$$  to test the claim of independence.
A)8.030
B)11.765
C)7.662
D)9.483

Quizplus · Accepted Answer

The chi-square test statistic is calculated as:$$\chi ^ { 2 } = \sum \frac{(O - E) ^ { 2 }}{E}$$where O is the observed frequency and E is the expected frequency.The expected frequency for each cell is calculated as:$$E = \frac{(	ext { row total })(	ext { column total })}{	ext { grand total }}$$The grand total is 200.The row totals are:Republican: 42 + 20 + 14 = 76Democrat: 50 + 24 + 18 = 92Independent: 10 + 16 + 6 = 32The column totals are:Approve: 42 + 50 + 10 = 102Disapprove: 20 + 24 + 16 = 60No Opinion: 14 + 18 + 6 = 38The expected frequencies are:Republican Approve: (76)(102)/200 = 38.76Republican Disapprove: (76)(60)/200 = 23.24Republican No Opinion: (76)(38)/200 = 14.00Democrat Approve: (92)(102)/200 = 46.92Democrat Disapprove: (92)(60)/200 = 27.60Democrat No Opinion: (92)(38)/200 = 17.48Independent Approve: (32)(102)/200 = 16.32Independent Disapprove: (32)(60)/200 = 9.60Independent No Opinion: (32)(38)/200 = 6.08The chi-square test statistic is:$$\chi ^ { 2 } = \frac{(42 - 38.76) ^ { 2 }}{38.76} + \frac{(20 - 23.24) ^ { 2 }}{23.24} + \frac{(14 - 14.00) ^ { 2 }}{14.00} + \frac{(50 - 46.92) ^ { 2 }}{46.92} + \frac{(24 - 27.60) ^ { 2 }}{27.60} + \frac{(18 - 17.48) ^ { 2 }}{17.48} + \frac{(10 - 16.32) ^ { 2 }}{16.32} + \frac{(16 - 9.60) ^ { 2 }}{9.60} + \frac{(6 - 6.08) ^ { 2 }}{6.08} = 8.030$$Therefore, the correct answer is A.

The Contingency Table Below Shows the Results of a Random