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Chapter 17: Problem 16

NCAA collected data on graduation rates of athletes in Division I in the mid-1980s. Among 2,332 men, 1,343 had not graduated from college, and among 959 women, 441 had not graduated. a. Set up a two-way table to examine the relationship between gender and graduation. b. Identify a test procedure that would be appropriate for analyzing the relationship between gender and graduation. Carry out the procedure and state your conclusion

Short Answer

Expert verified
Construct a two-way table and use a Chi-squared test, finding a significant relationship at p < 0.05.

Step by step solution

01

Define Categories and Data

First, identify the categories you will use to set up your two-way table. The categories here are 'Gender' (men and women) and 'Graduation Status' (graduated and not graduated). We have the following data: for men, 1,343 did not graduate, hence 2,332 - 1,343 = 989 did graduate. For women, 441 did not graduate, hence 959 - 441 = 518 did graduate.
02

Construct the Two-Way Table

Using the data from Step 1, construct the table: | | Graduated | Not Graduated | Total | |-----------|-----------|---------------|-------| | Men | 989 | 1,343 | 2,332 | | Women | 518 | 441 | 959 | | Total | 1,507 | 1,784 | 3,291 | Fill in the total numbers for rows and columns to complete the table.
03

Choose the Test Procedure

A Chi-squared test of independence is suitable for determining if there is an association between gender and graduation status in a categorical data context.
04

Calculate Expected Values

For each cell in the table, calculate the expected value using the formula: \[\text{Expected} = \frac{\text{(Row total)} \times \text{(Column total)}}{\text{Grand Total}}\] For instance, for men who graduated, the expected value is: \(\frac{2,332 \times 1,507}{3,291} \approx 1,068.04\) after calculating similarly for all the cells as follows: Graduated Men: 1068.04, Not Graduated Men: 1263.96, Graduated Women: 438.96, Not Graduated Women: 520.04.
05

Conduct the Chi-Squared Test

Compute the Chi-squared statistic using the formula:\[\chi^2 = \sum \frac{(O - E)^2}{E} \]where \(O\) represents the observed value and \(E\) represents the expected value. Calculate for each cell:- For Graduated Men: \(\frac{(989 - 1068.04)^2}{1068.04} = 6.27\)- For Not Graduated Men: \(\frac{(1343 - 1263.96)^2}{1263.96} = 5.22\)- For Graduated Women: \(\frac{(518 - 438.96)^2}{438.96} = 14.61\)- For Not Graduated Women: \(\frac{(441 - 520.04)^2}{520.04} = 11.22\)Calculate the sum: \(\chi^2 = 6.27 + 5.22 + 14.61 + 11.22 = 37.32\).
06

Determine Degrees of Freedom and p-value

The degrees of freedom for a Chi-squared test is calculated by \[( ext{number of rows} - 1) \times ( ext{number of columns} - 1)\] which in this case is (2-1)\(\times\)(2-1) = 1. Use a Chi-squared table or calculator to find the p-value for \(\chi^2 = 37.32\) with 1 degree of freedom.
07

Conclusion

Compare the p-value to the significance level, typically 0.05. If the p-value is less than 0.05, reject the null hypothesis, indicating that there is a significant relationship between gender and graduation. In this case, the p-value is likely to be much less than 0.05, suggesting a significant relationship.

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Key Concepts

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

Two-Way Table
A two-way table is an incredibly useful tool for organizing and summarizing data that involves two categorical variables. In our case, we have 'Gender' and 'Graduation Status'. This allows you to see different combinations of categories and how frequently they occur.

Here's how you can think about the two-way table in this context:
  • The rows represent one category from the data 鈥� Gender (Men and Women).
  • The columns represent the other category - Graduation Status (Graduated and Not Graduated).
  • Each cell in the table gives the count of individuals for a specific combination of row and column categories.
  • The margins of the table show totals, helping to provide context for the individual cell counts.
Creating a two-way table like this offers an illustration of how different groups contribute to the overall data. It is a straightforward way to visualize relationships in categorical data.
Graduation Rates
Graduation rates represent the percentage of students completing their academic program within a specific timeframe, often measured against the total number of enrolled students. In the problem given, the NCAA collected graduation rates for athletes to explore the patterns within different gender groups.

Understanding graduation rates can provide insights into:
  • The effectiveness of academic programs.
  • The performance and satisfaction levels of students within institutions.
  • Any discrepancies or patterns across various demographic groups, like gender.
In our example, it is clear that understanding gender differences in graduation rates could highlight necessary changes or interventions to support specific groups better.
Categorical Data
Categorical data refers to variables that represent categories rather than numerical values. These are distinct, separate groups or types, such as gender or graduation status in this context.

Some important characteristics of categorical data include:
  • It is typically represented in word form, like "Men" and "Women" or "Graduated" and "Not Graduated".
  • Categorical data can be organized into a limited number of possible values or categories.
  • This type of data is often visualized through bar charts, pie charts, or two-way tables for easy analysis.
In this exercise, categorical data representation helps in performing the Chi-squared test to analyze the relationship between gender and graduation status.
Degrees of Freedom
Degrees of freedom (df) in a statistical context represent the number of values in a calculation that are free to vary. They provide insight into the robustness of a chi-square test.

For a two-way table, the degrees of freedom are calculated using the formula: \[(\text{number of rows} - 1) \times (\text{number of columns} - 1)\]
  • This accounts for the idea that once you know the total observed frequencies, some counts are already determined.
  • In our specific example, with 2 categories for gender and 2 categories for graduation status, the degrees of freedom is \((2-1) \times (2-1) = 1\).
  • The chi-square statistic with its degrees of freedom helps determine the p-value, which indicates if the observed data significantly deviate from the expected distribution under the null hypothesis.
Understanding degrees of freedom is essential in making valid inferences about relationships in categorical data.

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Most popular questions from this chapter

True/false: A Chi Square test is often used to determine if there is a significant relationship between two continuous variables.

Suppose that college students are asked to identify their preferences in political affiliation (Democrat, Republican, or Independent) and in ice cream (chocolate, vanilla, or straw- berry). Suppose that their responses are represented in the following two-way table (with some of the totals left for you to calculate). $$\begin{array}{|l|c|c|c|c|} \hline & \text { Chocolate } & \text { Vanilla } & \text { Strawberry } & \text { Total } \\\\\hline \text { Democrat } & 26 &43 & 13 & 82 \\\\\hline \text { Republican } & 45 & 12 & 8 & 65 \\\\\hline \text { Independent } & 9 & 13 & 4 & 26 \\\\\hline \text { Total } & 80 & 68 & 25 & 173 \\\\\hline\end{array}$$ a. What proportion of the respondents prefer chocolate ice cream? b. What proportion of the respondents are Independents? c. What proportion of Independents prefer chocolate ice cream? d. What proportion of those who prefer chocolate ice cream are Independents? e. Analyze the data to determine if there is a relationship between political party preference and ice cream preference.

A geologist collects hand-specimen sized pieces of limestone from a particular area. A qualitative assessment of both texture and color is made with the following results. Is there evidence of association between color and texture for these limestones? Explain your answer. $$\begin{array}{|l|c|c|c|} \hline & {\text { Colour }} \\\\\hline \text { Texture } & \text { Light } & \text { Medium } & \text { Dark } \\\\\hline \text { Fine } & 4 & 20 & 8 \\\\\text { Medium } & 5 & 23 & 12 \\\\\text { Coarse } & 21 & 23 & 4 \\\\\hline \end{array}$$

Imagine that you believe there is a relationship between a person's eye color and where he or she prefers to sit in a large lecture hall. You decide to collect data from a random sample of individuals and conduct a chi-square test of independence. What would your two-way table look like? Use the information to construct such a table, and be sure to label the different levels of each category.

Some parents of the West Bay little leaguers think that they are noticing a pattern. There seems to be a relationship between the number on the kids' jerseys and their position. These parents decide to record what they see. The hypothetical data appear below. Conduct a Chi Square test to determine if the parents' suspicion that there is a relationship between jersey number and position is right. Report your Chi Square and p values. $$\begin{array}{|l|l|l|l|l|}\hline & \text { Infield } & \text { Outfield } & \text { Pitcher } & \text { Total } \\\\\hline \mathbf{0 - 9} & 12 & 5 & 5 & 22 \\\\\hline \mathbf{1 0 - 1 9} & 5 & 10 & 2 & 17 \\\\\hline \mathbf{2 0 +} & 4 & 4 & 7 & 15 \\\\\hline \text { Total } & 21 & 19 & 14 & 54 \\\\\hline\end{array}$$
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