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AP Statistics Chapter 14: The Chi-Square Procedures

14.1 – Chi-Square Goodness of Fit Test

Goodness of Fit

A goodness of fit test is used to help determine whether a populaion has a certain hypothesized distribution, expressed as proportions of individuals in the population falling into various outcome categories. There are two types of goodness of fit tests

1. Equal Proportions (all proportions are expected to be the same)
2. Fixed or Given Proportions (proportions are expected to follow given values)

Hypotheses for the Goodness of Fit Test

Ho: The actual population proportions are equal to the hypothesized proportions
Ha: The actual population proportions are different from the hypothesized proportions

The Chi-Square Statistic

The formula is

Expected Counts

The expected counts for the given proportions GOF test are NOT all the same. They are found by multiplying the total of the counts by each given percentage.

Conditions for the Chi-Square Test

• We have an SRS, as always, from the population of interest.
• None of the expected counts are 0
• No more than 20% of the expected counts are less than 5

Degrees of Freedom for the Goodness of Fit Test

The degrees of freedom for the GOF test are n-1 where n is the number of categories.

14.2 – Test of Association for Two-Way Tables

Two-Way Tables

When there are two categorical variables, data can be arranged in a row and column format, called a Two-Way Table (we first saw these in Chapter 4). Here is an example:

Test for Association between Two Categorical Variables

Ho: There is no association between the variables
Ha: There is an assocation

OR

Ho: The variables are independent (there is no association)
Ha: The variables are NOT independent (there is an association)

Degrees of Freedom for the Goodness of Fit Test

The degrees of freedom for the Test of Association test are (r-1) x (c-1) where r is the number of rows and c is the number of columns in the table.

The Chi-Square Statistic

The formula is the same as in the goodness of fit test.

Expected Counts

The expected counts for this test can be found as follows:

For example, for the expected count for 2nd grade/green in the table above, we would use the calculation