The Chi-Square Tests
Lecture 8

Contingency Tables

1. If you take a variable with a normal distribution and square it, then you get a chi-square distribution

2. Pearson Chi-Square Test

   1. Example: You survey 80 students from a university

      1. Event A: 30 students are men

      2. Event B: 50 students are women

   2. Mutually exclusive – all events have to add up to the total

      • In our case, a male cannot be a woman at the same time, or vice versa

   3. We can test the hypothesis that men and women occur 50/50 in a group

      1. Expected value for men is 40

      2. Expected value for women is 40

   4. Compute the X² statistic

5. Notation

   1. O_i is the observed data

   2. E_i is the expected

   3. n is the outcomes of each event

6. The degrees of freedom are df = n – 1 = 2 – 1 = 1

7. Chi-square test statistic is

   1. Reject the H₀

   2. Test statistic is calculated in Excel using =chiinv(a, df)

8. Note – Chi-squares are one-tail tests, because negative numbers are converted to positive when they are squared

3. Contingency tables

   1. The simplest is called a 2 X 2

   2. Example: Students took an exam

3. You have to use the number of occurrences

   • Do not use percents, proportions, etc.

4. Have to calculate the expected occurrences from the marginals

Expected	Failed	Passed	Marginal Total
Male			50
Female			30
Marginal Total	30	50	80

5. The degrees of freedom are df = (columns – 1)(rows – 1) = 1 (1) = 1

   1. Chi-square test statistic is

   2. Fail to reject the H₀ hypothesis and conclude males and females are equal when taking the exam

6. Note – There is a fast way to calculate X² for a 2 X 2 Contingency Table

7. Re-doing the example using the fast method

8. Note – You can build contingency tables with any dimensions

9. The Chi-square test may be poor if

   1. Grand total is less than 100

   2. Or a cell total is less than 10

   3. Then you should use Yate’s Correction

Yate's Correction

1. Yate’s Correction

1. Lowers the Chi-square making the statistic more conservative

2. A simpler formula for the 2 X 2 Contingency Table

3. Applying the Yate’s Correction to the example

4. Fail to reject

2. Expanding the contingency table

   1. An example where the instructor teaches a Forecasting Class

   2. Are the classes different?

3. Perfect for an Excel spreadsheet

4. The degrees of freedom, df = (5 – 1)(2 – 1) = 4

   1. Chi-square test statistic is

   2. Fail to reject the H₀ and conclude the classes are the same

   3. Without the Yate’s Correction, then X² = 3.97

5. Note – the grades have a ranking. This ranking is not included in the statistical test. However, there are other tests that can incorporate the ranking of the grades into the Chi-Square statistic