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Why men and women play sports 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. \({ }^{13}\) 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-L.M), low social comparison-high mastery (LSC-HM), and low social comparison-low mastery (LSC-L.M). 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: $$ \begin{array}{lcc} {\text { Gender }} \\ \text { Goal } & \text { Female } & \text { Male } \\ \text { HSC-HM } & 14 & 31 \\ \text { HSC-LM } & 7 & 18 \\ \text { LSC-HM } & 21 & 5 \\ \text { LSC-LM } & 25 & 13 \\ \hline \end{array} $$ (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

Understanding Conditional Distribution

Conditional distribution refers to the probability distribution of a subset of data given a particular condition. Here, we calculate the proportion of each sports goal category within each gender group. To do so, we determine how each goal category relates to the total number of sports participants of that gender.

02

Computing Conditional Distribution for Females

For females, we calculate the proportion for each goal category. There are 67 female participants in total.- HSC-HM: \( \frac{14}{67} \approx 0.209 \) or 20.9%- HSC-LM: \( \frac{7}{67} \approx 0.104 \) or 10.4%- LSC-HM: \( \frac{21}{67} \approx 0.313 \) or 31.3%- LSC-LM: \( \frac{25}{67} \approx 0.373 \) or 37.3%.

03

Computing Conditional Distribution for Males

For males, similar calculations apply since there are also 67 male participants in total.- HSC-HM: \( \frac{31}{67} \approx 0.463 \) or 46.3%- HSC-LM: \( \frac{18}{67} \approx 0.269 \) or 26.9%- LSC-HM: \( \frac{5}{67} \approx 0.075 \) or 7.5%- LSC-LM: \( \frac{13}{67} \approx 0.194 \) or 19.4%.

04

Creating a Bar Graph for Comparison

Construct a side-by-side bar graph to visually represent the conditional distribution for each category by gender. Each bar displays the proportion of participants with a given sports goal, categorizing by gender to facilitate comparison.

05

Analyzing the Graph and Data

Review the graph and data to determine patterns and insights. Males are more associated with high social comparison (HSC-HM and HSC-LM), indicating a stronger desire to compete with others compared to females. On the other hand, females tend to fall within the low social comparison categories (LSC-HM and LSC-LM), suggesting a greater focus on self-improvement and mastery rather than competition.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Conditional Distribution

Conditional distribution helps in understanding how data is distributed within a sub-group when a specific condition is applied. In this exercise, the condition is the gender of the sports participants. Each participant is classified into specific categories based on sports goals, and we calculate the proportion of each category within male and female groups. The focus is to see how different sports goals are prioritized within each gender.

To compute the conditional distribution, take each sports goal category and divide the number of participants in that category by the total number of participants of that gender. For instance, with 67 females, the conditional distribution for the HSC-HM category is calculated as \( \frac{14}{67} \approx 0.209 \), or 20.9%. These calculations give us a clear view of the distribution of sports goals for males and females when considering gender as a condition. This is a powerful method to reveal patterns in data subgroups.

Two-Way Table

A two-way table is a tool used to organize data into categories across two different dimensions. In our sports participation analysis, the table presents data on sports goals for two dimensions: gender (male and female) and the type of sports goal. This arrangement makes it easy to see how each group is distributed across these categories.

Looking at the data visually displayed this way, one can easily read and compare the number of male and female participants within each sports goal category. It facilitates the understanding of how particular sports goals are prevalent among males and females. Two-way tables are particularly useful because they allow for efficient computation of distributions and enable straightforward comparisons, which can be critical in revealing deeper insights into the dataset.

Gender Comparison

Gender comparison involves analyzing the differences or similarities in behavior or proportions between males and females. In the context of this exercise, it explores how genders differ in prioritizing sports goals. The data reveals that males predominantly fit into the high social comparison categories, indicating a competitive nature. In contrast, females are mainly in the low social comparison categories, suggesting a greater emphasis on mastery and self-improvement.

This comparison not only reflects different motivational factors but can also highlight broader social or cultural dynamics at play. Recognizing these differences is critical when tailoring programs or interventions in sports or other fields, to address unique motivational factors effectively. Gender comparison is an essential analytical tool for understanding the role of gender in diverse activities.

Proportional Analysis

Proportional analysis breaks down the data further by looking at the relative sizes of different categories within groups. This means comparing the relative numbers, rather than just the absolute counts. In our sport participation case, proportional analysis helps us focus on the fractions of each category relative to the total within each gender.

For example, even if males and females had equal numbers in a particular goal category, the proportions may reveal a different narrative about preferences or motivation. Proportional analysis allows us to make more direct and meaningful comparisons between groups, as it normalizes data differences due to group size. It's an insightful approach that directs our attention to relevance and significance instead of mere numbers.