Chapter 11, Problem 5CRE

(a)

To determine

To graph: The male and female responses.

Explanation of Solution

Given information:

Total number of students in a sample $=478$

The two way table of the responses is given below

	Goal
Gender	Grades	Popular	Sports
Female	$130$	$91$	$30$
Male	$117$	$50$	$60$

Graph:

The graph of comparison is illustrated below

  

Interpretation:

From the above, it can be clearly seen that males are more likely to choose Grades and Sports rather than females while females are more likely to choose Popular than males.

For both males and females, most of them are likely to choose Grades.

(b)

To determine

To check: Whether there is any association between gender and goals.

Answer to Problem 5CRE

There is an association between goal and gender of students.

Explanation of Solution

Formula used:

Chi square statistic

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

Calculation:

The observed figures (O) are given in sub part a

The expected values will be (E)

	Goal
Gender	Grades	Popular	Sports	Total
Females	$\frac{251\times 247}{478}=129.70$	$\frac{251\times 141}{478}=74.04$	$\frac{251\times 90}{478}=47.26$	$251$
Male	$\frac{227\times 247}{478}=117.30$	$\frac{227\times 141}{478}=66.96$	$\frac{227\times 90}{478}=42.74$	$227$
Total	$247$	$141$	$90$	$478$

The null and alternate hypotheses are

  $H_0:$ There is no association between goal and gender.

  $H_1:$ There is an association between goal and gender.

Test statistic

Under null hypothesis

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

The test value is $21.456$

P value

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

Conclusion:

The p value is less than level of significance, hence, there is sufficient evidence to reject null hypothesis at $5\%$ level of significance and thus conclude that there is an association between goal and gender of students.

(c)

To determine

To find: The cell contributing most to the chi square statistic.

Answer to Problem 5CRE

Sports − Female

Sports - Male

Explanation of Solution

The cell contributing most will be the one which has highest difference between observed and expected counts.

In this case, the difference $\left(=17.26\right)$ is highest for female sport choice and male sports choice