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Given in the question that, a study was performed to examine the factors that determine social status for children in grades 4, 5, and 6. Researchers administered a questionnaire to a random sample of 478 students in these grades.

We need to construct an appropriate graph to compare male and female responses.

A sample's total number of students $=478$

Below is a two-way table of the responses.

$\begin{array}{|llll|}\hline & \text{Goal}& & \\ \text{Gender}& \text{Grades}& \text{Popular}& \text{Sports}\\ \text{Female}& 130& 91& 30\\ \text{Male}& 117& 50& 60\\ \hline\end{array}$

The following graph depicts the comparison.

Interpretation: As shown above, males are more likely than females and choose Grades and Sports, but females are more likely choose the Popular.

The majority of boys and females are likely to pick Grades.

Given in the question that, a two way table

We need to find that whether there is a convincing evidence of an association between gender and goals for elementary school students.

Chi square statistic

$鈭�\frac{(O-E{)}^2}{E}~{蠂}_{(r-1)脳(c-1)}$

Figures discovered Subpart a contains$\left(O\right)$.

The expected values are as follows: (E)

$\begin{array}{|lllll|}\hline & \text{Goal}& & & \\ \text{Gender}& \text{Grades}& \text{Popular}& \text{Sports}& \text{Total}\\ \text{Females}& \frac{251脳247}{478}=129.70& \frac{251脳141}{478}=74.04& \frac{251脳90}{478}=47.26& 251\\ \text{Male}& \frac{227脳247}{478}=117.30& \frac{227脳141}{478}=66.96& \frac{227脳90}{478}=42.74& 227\\ \text{Total}& 247& 141& 90& 478\\ \hline\end{array}$

The alternative and null hypotheses are

$H_0$: There is no link between gender and aim.

$H_1$: There is a link between gender and objective.

Test statistic

Under null hypothesis

$\frac{(129.70-130{)}^2}{129.70}+\frac{(74.04-91{)}^2}{74.04}+\frac{(47.26-30{)}^2}{47.26}+\frac{(117.30-117{)}^2}{117.30}+\frac{(66.96-50{)}^2}{66.96}+\frac{(42.74-60{)}^2}{42.74}~{蠂}_2$

The $P$value for the test is $21.456$.

$P\left({蠂}_2>21.456\right)=0.000022$

Because the p value is smaller than the threshold of significance, there is adequate evidence to reject the null hypothesis at the $5$percent level of significance, implying that there is a link between goal and student gender.

Given in the question that, A study was performed to examine the factors that determine social status for children in grades 4, 5, and 6. Researchers administered a questionnaire to a random sample of 478 students in these grades.

We need to find the cell which contributes most to the chi-square statistic in part (b)?

The cell with the greatest disparity between observed and predicted numbers will contribute the most.

In this situation, the disparity between female and male sport preferences is the greatest$(=17.26).$