Q.R11.5 - The Practice of Statistics for AP

Question

Who were the popular kids at your elementary school? Did they get good grades or have good looks? Were they good at sports? 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. One of the questions they asked was “What would you most like to do at school: make good grades, be good at sports, or be popular?” The two-way table below summarizes the students’ responses.

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

(a) Construct an appropriate graph to compare male and female responses. Write a few sentences describing the relationship between gender and goals. 

(b) Is there convincing evidence of an association between gender and goals for elementary school students? Carry out a test at the $\alpha =0.05$ level and report your conclusion.

 (c) Which cell contributes most to the chi-square statistic in part (b)? Explain 

Step-by-Step Solution

Answer

b. There is a link between student gender and their goal.

c. The disparity between female and male sport preferences is the greatest.

Part (a)Step 1: Given information

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. 

Part(a) Step 2: Explanation

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.

Part(b) Step 1: Given information

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.

Part (b) Step 2: Explanation

Chi square statistic

$\sum \frac{(O-E{)}^2}{E}~{\chi }_{(r-1)\times (c-1)}$

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

The expected values are as follows: (E)

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

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

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

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

$P\left({\chi }_2>21.456\right)=0.000022$

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.

Part(c) Step 1: Given information

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)?

Part(c) Step 2: Explanation

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 $(=17.26).$