A teacher figures that final grades in the chemistry department are distributed as: $\mathrm { A } , 25 \% ; \mathrm { B } , 25 \% ; \mathrm { C } , 40 \%$; D, $5 \% ; F , 5 \%$. At the end of a randomly selected semester, the following number of grades were recorded. Calculate the chi-square test statistic $\chi ^ { 2 }$ to determine if the grade distribution for the department is different than expected. Use $\alpha = 0.01$.
$\begin{array}{l|ccccc}
\text { Grade } & \text { A } & \text { B } & \text { C } & \text { D } & \text { F } \
\hline \text { Number } & 36 & 42 & 60 & 8 & 14
\end{array}$

A) $5.25$
B) $6.87$
C) $0.6375$
D) $4.82$

Question

A teacher figures that final grades in the chemistry department are distributed as: $\mathrm { A } , 25 \% ; \mathrm { B } , 25 \% ; \mathrm { C } , 40 \%$; D, $5 \% ; F , 5 \%$. At the end of a randomly selected semester, the following number of grades were recorded. Calculate the chi-square test statistic $\chi ^ { 2 }$ to determine if the grade distribution for the department is different than expected. Use $\alpha = 0.01$.
$\begin{array}{l|ccccc}
\text { Grade } & \text { A } & \text { B } & \text { C } & \text { D } & \text { F } \
\hline \text { Number } & 36 & 42 & 60 & 8 & 14
\end{array}$

A) $5.25$
B) $6.87$
C) $0.6375$
D) $4.82$

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 frequencies are:$\mathrm { A } : 0.25 \times 160 = 40$$\mathrm { B } : 0.25 \times 160 = 40$$\mathrm { C } : 0.40 \times 160 = 64$$\mathrm { D } : 0.05 \times 160 = 8$$\mathrm { F } : 0.05 \times 160 = 8$The chi-square test statistic is then:$\chi ^ { 2 } = \frac{(36 - 40) ^ { 2 }}{40} + \frac{(42 - 40) ^ { 2 }}{40} + \frac{(60 - 64) ^ { 2 }}{64} + \frac{(8 - 8) ^ { 2 }}{8} + \frac{(14 - 8) ^ { 2 }}{8} = 5.25$The critical value of $\chi ^ { 2 }$ with 4 degrees of freedom and $\alpha = 0.01$ is 13.28. Since $\chi ^ { 2 } = 5.25 < 13.28$, we fail to reject the null hypothesis and conclude that the grade distribution for the department is not different than expected.

A Teacher Figures That Final Grades in the Chemistry Department A,25%;B,25%;C,40%\mathrm { A } , 25 \% ; \mathrm { B } , 25 \% ; \mathrm { C } , 40 \%A,25%;B,25%;C,40%

A Teacher Figures That Final Grades in the Chemistry Department $\mathrm { A } , 25 \% ; \mathrm { B } , 25 \% ; \mathrm { C } , 40 \%$