A random sample of 160 car accidents are selected and categorized by the age of the driver determined to be at fault. The results are listed below. The age distribution of drivers for the given categories is $18 \%$ for the under 26 group, $39 \%$ for the $26 - 45$ group, $31 \%$ for the $45 - 65$ group, and $12 \%$ for the group over 65 . Find the rejection region used to test the claim that all ages have crash rates proportional to their number of drivers. Use $\alpha = 0.05$.
$\begin{array}{l|cccc}
\text { Age } & \text { Under 26 } & 26-45 & 46-65 & \text { Over } 65 \
\hline \text { Drivers } & 66 & 39 & 25 & 30
\end{array}$

A) $\chi ^ { 2 } 6.251$
B) $\chi ^ { 2 } 7.815$
C) $\chi ^ { 2 } 11.143$
D) $\chi ^ { 2 } 9.348$

Question

A random sample of 160 car accidents are selected and categorized by the age of the driver determined to be at fault. The results are listed below. The age distribution of drivers for the given categories is $18 \%$ for the under 26 group, $39 \%$ for the $26 - 45$ group, $31 \%$ for the $45 - 65$ group, and $12 \%$ for the group over 65 . Find the rejection region used to test the claim that all ages have crash rates proportional to their number of drivers. Use $\alpha = 0.05$.
$\begin{array}{l|cccc}
\text { Age } & \text { Under 26 } & 26-45 & 46-65 & \text { Over } 65 \
\hline \text { Drivers } & 66 & 39 & 25 & 30
\end{array}$

A) $\chi ^ { 2 } > 6.251$
B) $\chi ^ { 2 } > 7.815$
C) $\chi ^ { 2 } > 11.143$
D) $\chi ^ { 2 } > 9.348$

Quizplus · Accepted Answer

The degrees of freedom for a chi-square test in this context is calculated as the number of categories minus one. Here, there are 4 age categories, so the degrees of freedom is $4 - 1 = 3$. For a significance level of $\alpha = 0.05$ and 3 degrees of freedom, the critical value from the chi-square distribution table is $7.815$. This means the rejection region for the hypothesis test is $\chi^2 > 7.815$.

A Random Sample of 160 Car Accidents Are Selected and Categorized