Chi-Square Analysis for Program Value Ratings by Gender

Math Problem Statement

A leading CEO mentoring organization offers a program for chief executives, presidents, and business owners with a focus on developing entrepreneurial skills. Women and men executives that recently completed the program rated its value and the results are given in the accompanying table (marked Original Data). The chi-square procedures are not appropriate for the Original Data table because the expected counts are too small. However, by changing the table, an analysis can still be performed. We can often proceed after combining cells in some way that makes sense and also produce a table in which the conditions are satisfied. Both accompanying tables display the same data, but the second combines "Marginal" and "Poor" into a new category called "Below Average". Are the perceptions of the program's value the same for men and women? Complete parts a through c below.

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Part 1

a) Find the expected counts for each cell in the new table (Combined Data table).

(Round to two decimal places as needed.)

MenMen

WomenWomen

Excellent

enter your response here

Above Average

enter your response here

Average

enter your response here

Below AverageBelow Average

enter your response here

Click the icon to view the data tables.Click the icon to view the data tables.Click the icon to view the data tables.Click the icon to view the data tables.Click the icon to view the data tables.Click the icon to view the data tables.Click the icon to view the data tables.Click the icon to view the data tables.

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Data table

Original Data

MenMen

WomenWomen Excellent

1010

Above Average

1212

Average

1515

Marginal

Poor

Combined Data

MenMen

WomenWomen Excellent

1010

Above Average

1212

Average

1515

Below Average

1010

pop-up content ends

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Solution

To calculate the expected counts for the Combined Data table, we use the formula for expected frequency in a contingency table:

$E = \frac{(Row \, Total) \times (Column \, Total)}{Grand \, Total}$

Step 1: Extract the totals

From the Combined Data table:

Category	Men	Women	Row Totals
Excellent	33	10	43
Above Average	9	12	21
Average	15	8	23
Below Average	10	3	13
Column Totals	67	33	100

Step 2: Compute expected counts

Using the formula $E = \frac{(Row \, Total) \times (Column \, Total)}{Grand \, Total}$ , we compute the expected count for each cell.

Excellent:

Men: $\frac{43 \times 67}{100} = 28.81$
Women: $\frac{43 \times 33}{100} = 14.19$

Above Average:

Men: $\frac{21 \times 67}{100} = 14.07$
Women: $\frac{21 \times 33}{100} = 6.93$

Average:

Men: $\frac{23 \times 67}{100} = 15.41$
Women: $\frac{23 \times 33}{100} = 7.59$

Below Average:

Men: $\frac{13 \times 67}{100} = 8.71$
Women: $\frac{13 \times 33}{100} = 4.29$

Step 3: Organize the results in a table

Category	Men (Expected)	Women (Expected)
Excellent	28.81	14.19
Above Average	14.07	6.93
Average	15.41	7.59
Below Average	8.71	4.29

Let me know if you want help calculating the $\chi^2$ statistic or interpreting the results! Here are some follow-up questions and a tip:

Follow-Up Questions:

What is the formula to compute the $\chi^2$ statistic?
How do you interpret the degrees of freedom in this scenario?
What conditions must be met for the $\chi^2$ test to be valid?
How would combining categories affect the statistical power of this test?
What are the implications if the null hypothesis is rejected?

Tip:

Always ensure that each expected cell count is greater than 5 to meet the assumptions of the $\chi^2$ test.

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Math Problem Analysis

Mathematical Concepts

Chi-Square Test of Independence
Contingency Tables
Expected Frequency Calculation

Formulas

Expected frequency formula: E = (Row Total × Column Total) / Grand Total
Chi-Square statistic formula: χ² = Σ((Observed - Expected)² / Expected)

Theorems

Chi-Square Test Assumptions

Suitable Grade Level

Undergraduate Statistics or Advanced High School Statistics

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