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 70 male and 70 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 (HSM-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 datadisplayed in a two-way table:Observed Counts for Sports GoalsGoalHSC-HMHSC LMLSC-HMLSC LMFemale 16 6 23 25Male 33 19 4 14a) 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. d) Find the expected counts and display them in a two-way table similar to the table of observed countse) Do the data provide convincing evidence of a difference in the distributions of sports goals for male and female undergraduates at the university? Carry out an appropriate test at the a=0.05 significance level

Comparing the distributions of sports goals for male and female undergraduates, we can see that a higher proportion of male students reported high social comparison goals (HSC-HM and HSC-LM) compared to female students, while a higher proportion of female students reported low social comparison goals (LSC-HM and LSC-LM) compared to male students.

The conditional distribution (in proportions) of the reported sports goals for each gender are:

Female:

HSC-HM: 16/70 = 0.229

HSC-LM: 6/70 = 0.086

LSC-HM: 23/70 = 0.329

LSC-LM: 25/70 = 0.357

Male:

HSC-HM: 33/70 = 0.471

HSC-LM: 19/70 = 0.271

LSC-HM: 4/70 = 0.057

LSC-LM: 14/70 = 0.2

A stacked bar chart would be an appropriate graph for comparing the conditional distributions.

The chart would have two bars, one for each gender, with each bar split into four segments representing the four categories of sports goals.

Comparing the distributions of sports goals for male and female undergraduates, we can see that a higher proportion of male students reported high social comparison goals (HSC-HM and HSC-LM) compared to female students, while a higher proportion of female students reported low social comparison goals (LSC-HM and LSC-LM) compared to male students.

In terms of mastery goals, the proportions are relatively similar between male and female students.

To find the expected counts, we need to calculate the marginal totals for each row and column, and then use these to calculate the expected counts based on the assumption of independence.

The results are displayed in the table below:

Observed Counts and Expected Counts for Sports Goals

Goal HSC-HM HSC-LM LSC-HM LSC-LM Total

Female (Observed) 16 6 23 25 70

Expected 19.1 10.9 23.9 16.1 70

Male (Observed) 33 19 4 14 70

Expected 29.9 17.1 3.1 19.9 70

Total 49 25 27 39 140

To test whether there is a difference in the distributions of sports goals for male and female undergraduates at the university, we can use a chi-squared test of independence.

The null hypothesis is that the distributions are the same for male and female students, and the alternative hypothesis is that they are different. The test statistic is calculated as:

chi-squared = sum((observed - expected)² / expected)

Using the values from the table above, we get:

chi-squared = (16-19.1)²/19.1 + (6-10.9)²/10.9 + (23-23.9)²/23.9 + (25-16.1)²/16.1 + (33-29.9)²/29.9 + (19-17.1)²/17.1 + (4-3.1)²/3.1 + (14-19.9)²/19.9

= 10.32

The degrees of freedom for the test are (number of rows - 1) x (number of columns - 1) = 3 x 3 = 6 (since we have 2 rows and 4 columns).

Using a chi-squared distribution table with 6 degrees of freedom and a significance level of 0.05, the critical value to be 12.59.

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