Why men and women play sports Do men and women participate in sports for the same reasons? One goal for sports participants is social comparision—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 under-graduates at a large university 15 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: Observed Counts for Sports Goals

(a) Check that the conditions for performing the chi-square test are met.

(b) Use Table C to find the P-value. Then use your calculator’s C2cdf command.

(c) Interpret the P-value from the calculator in context.

(d) What conclusion would you draw? Justify your answer.

Given that

The expected table is

The following conditions must be met in order to use the chi-square test: - Two categorical variables.

- Each variable has two or more categories.

- Observation independence - Expected frequency for each cell is at least 1 or should be at least 5 for the majority (80%) of the cells.

Because the given information satisfies each condition. As a result, the chi-square test can be applied to the given dataset.

$\text{The test statistics is}24.8978.$

$df=(n-1)(m-1)$

$p-\text{value}=P\left({\chi }^2>{\chi }_{\text{text}}^2\right)$

$df=(4-1)(2-1)$

$df=(4-1)(2-1)$

= 3

There is the following commend to find p-value on TI83,

$"{\chi }^{2}cdf(24.8978,99,3)"$

The output is,

localid="1653293946609" $\text{Sothe p-value is}0.1621759222\mathrm{E}-5\text{.}$

$H_0:\left(\begin{array}{l}\text{There is no difference in the distribution sports goals}\\ \text{for male and female undergraduates at the university}\end{array}\right)$

$H_1:\left(\begin{array}{l}\text{There is a difference in the distribution sports goals}\\ \text{for male and female undergraduates at the university}\end{array}\right)$

If the null hypothesis is true, the p-value represents the probability of obtaining results that are as extreme as the test statistic value. If there is no relationship between gender and sports goal, the likelihood of getting a sample as extreme or more extreme than the given sample is$0.1621759222E-5.$

When the $p$value is less than the level of significance, the null hypothesis is rejected.

In this case, the$p$value is less than the level of significance. As a result, the null hypothesis is rejected.

As a result, there is enough evidence to conclude that there is a relationship between the variables goal and sex.