Do men and women participate in sports for the same reasons? One goal for sports participants is social comparison—the desire to win or to do better than other people. Another is mastery—the desire to improve one’s skills or to try one’s best. A study on why students participate in sports collected data from independent random samples of $67$male and $67$female undergraduates at a large university. Each student was classified into one of four categories based on his or her responses to a questionnaire about sports goals. The four categories were high social comparison– high mastery (HSC-HM), high social comparison– low mastery (HSC-LM), low social comparison–high mastery (LSC-HM), and low social comparison–low mastery (LSC-LM). One purpose of the study was to compare the goals of male and female students. Here are the data displayed in a two-way table:

$\begin{array}{|lcc|}\hline \text{Goal}& \text{Female}& \text{Male}\\ \text{HSC-HM}& 14& 31\\ \text{HSC-LM}& 21& 18\\ \text{LSC-HM}& 21& 5\\ \text{LSC-LM}& 25& 13\\ \hline\end{array}$

(a) Calculate the conditional distribution (in proportions) of the reported sports goals for each gender.

(b) Make an appropriate graph for comparing the conditional distributions in part (a).

(c) Write a few sentences comparing the distributions of sports goals for male and female undergraduates.

Step by step solution

01

Part(a) Step 1: Given Information

Given

$n_1=67$

$n_2=67$

02

Part(a) Step 2: Explanation

The conditional probability for each gender is every value in the given table divided by the sample size of $67$(since the sample size is the same for males and females).

$\begin{array}{|lcc|}\hline & \text{Female}& \text{Male}\\ \mathrm{HSC}-\mathrm{HM}& \frac{14}{67}\approx 0.2089& \frac{31}{67}\approx 0.4627\\ \mathrm{HSC}-\mathrm{LM}& \frac{7}{67}\approx 0.1045& \frac{18}{67}\approx 0.2687\\ \mathrm{LSC}-\mathrm{HM}& \frac{21}{67}\approx 0.3134& \frac{5}{67}\approx 0.0746\\ \mathrm{LSC}-\mathrm{LM}& \frac{26}{67}\approx 0.3881& \frac{13}{67}\approx 0.1940\\ \hline\end{array}$

03

Part(b) Step 1: Given Information

Given

$n_1=67$

$n_2=67$

04

Part(b) Step 2: Explanation

Create a comparative histogram.

The width of all bars has to be equal and the height has to be equal to the conditional probability found in exercise $27a$

.

05

Part(c) Step 1: Given Information 

Given

$n_1=67$

$n_2=67$

06

Part(c) Step 2: Explanation

Most goals were in LSC-HM and LSC-LM for Females and in HSC-HM and HSC-LM for Males.

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