1. AP STATISTICS LESSON 13 -2 (DAY 1) INFERENCE FOR TWO – WAY TABLES

2. ESSENTIAL QUESTION: How is Chi-square Used to Test for Homogeneity of Populations? Objectives: To create two-way tables. To use techniques to test data for homogeneity of populations.

3. Inference for Two-way Tables We want to compare more than two groups. The test we will use starts by presenting the data in a new way, as a two-way table. Two-way tables have more general uses than comparing the proportions of success in several groups. They describe relationships between any two categorical variables.

4. Example 13.4 Page 744 Treating Cocaine Addiction The subjects were 72 chronic users of cocaine who wanted to break their drug habit. Twenty-four of the subjects were randomly assigned to each treatment.

5. The Problem of Multiple Comparisons Call the population proportions of successes in the three groups p 1, p 2, p 3. Test H o : p 1 = p 2 to see if the success rate of desipramine differs from that of lithium. Test H o : p 1 = p 2 to see if the success rate of desipramine differs the placebo. Test H o : p 1 = p 3 to see if the success rate of lithium differs the placebo. The weakness of doing three tests is that we get three P-values.

6. Statistical Methods for Dealing with Many Comparisons Usually Have Two Parts 1.An overall test to see if there is good evidence of any differences among the parameters that we want to compare. 2.A detailed follow-up analysis to decide which of the parameters differ and to estimate how large the differences are. The overall test is one with which we are familiar – the chi-square test – but in this new setting it will be used for comparing several population proportions.

7. Two-way Tables The first step in the overall test for comparing several proportions is to arrange the data in a two- way table that gives counts for both successes and failures. This is a 3 x 2 table because it has 3 rows and 2 columns. A table with r rows and c columns is an r x c table. Relapse NOYES Desipramine1410 Lithium618 Placebo420

8. Two-way Tables (continued…) The table shows the relationship between two categorical variables. The explanatory variable is the treatment (one of three drugs). The response variable is success (no relapse) or failure ( relapse). The two-way table gives the counts for all 6 combinations of values of these variables. Each of the 6 counts occupies a cell of the table.

9. Expected Counts We want to test the null hypothesis that there are no differences among the proportions of successes for addicts given the three treatments: H 0 : p 1 = p 2 = p 3 The alternative hypothesis is that there is some difference, that not all three proportions are equal: H a : not all of p 1, p 2, p 3 are equal The alternative hypothesis is no longer one-sided or two-sided. It is many-sided, because it allows any relationship other than all three equal.

10. Expected Counts (continued…) The expected count in any cell of a two-way table when H o is true is expected count = row total x column total table total

11. Example 13.5 Page 747 Free Throws If we have n independent trials and the probability f a success oneach trila is p, we expect np successes. If we draw and SRS of n individuals from a population in whiich the proportion of successes is p, we expect np successes in the sample.

12. Example 13.6 Page 748 Comparing Observed and Expected Counts ObservedExpected NoYesNoYes Desipramine1410816 Lithium618816 Placebo420816