1. Warm-up Researchers want to cross two yellow- green tobacco plants with genetic makeup (Gg). See the Punnett square below. When the researchers perform the experiment, the resulting offspring are 22 green (GG), 50 yellow-green (Gg), and 12 albino (gg) seedlings. Use a chi-square goodness of fit test to assess the validity of the researchers’ genetic model. Gg GGGGg g gg

2. Section 11.2 The Chi-Squared Test for Homogeneity of Proportions

3. Review… Recall that there are three types of χ 2 tests: – Goodness of fit test Tests whether a sample distribution matches a hypothesized distribution – Homogeneity of proportions Tests whether p 1 = p 2 = p 3 = … – Association/Independence Tests whether two categorical variables are related Today’s task Tomorrow’s task

4. Homogeneity of Proportions We’ve already performed hypothesis tests to determine if p 1 = p 2. Which type of test did we use? Now, we’ll look to compare more than two proportions using a χ 2 test statistic. Two-proportion z- test

5. Example Chronic users of cocaine need the drug to feel pleasure. Perhaps giving cocaine addicts anti- depressants will help them stay off cocaine. A three- year study compared the anti-depressant desipramine with lithium (which was already being used to treat cocaine addiction). A placebo was also used in the experiment. The subjects were 72 chronic users of cocaine. Twenty-four of the subjects were randomly assigned to each of the three treatment groups. The variable of interest is the proportion of users who did not experience a relapse.

6. The data TreatmentSubjectsNo relapse Desipramine2414 Lithium246 Placebo244

7. First Step The first step is to arrange the data in a two- way table. This table must account for everyone (so our example would list success and failure). This is called a 3 x 2 table because it has 3 rows and 2 columns. You’ll need to know this to be able to use your calculator for a χ 2 test. What does this remind you of?

8. The hypotheses… H 0 : p 1 = p 2 = p 3 (Be sure you have defined 1, 2, and 3) H a : Not all of the proportions are equal.

9. Using your calculator Once you have the two-way table created, you will need to enter the data in Matrix A. – Press 2 nd MATRX. – Choose EDIT. – Hit ENTER. – Type the dimensions of the matrix (3 x 2 in this case). – Enter the data.

10. Performing the test Once you have the data entered, perform the χ 2 test on your calculator. – Press STAT. – Choose TESTS. – Choose C: χ 2 test – The calculator asks you where you have the observed data. You should always have it in Matrix A so that you don’t have to change this screen. – The calculator calculates the expected counts (yay!) and puts them in Matrix B. IN OTHER WORDS, DON’T CHANGE ANYTHING ON THIS SCREEN!

11. Information Received You can look at your graph or you can just “calculate”. If you conclude that there ARE differences (reject null hypothesis), you need to look at where the major differences are. View Matrix B to find expected values. Compare these to your observed values. You MUST copy down your expected values!

12. The assumptions Recall the assumptions for a χ 2 test. – Independent SRSs – All of the expected counts are at least 1. – No more than 20% of the expected counts are less than 5. Since this is step 2, leave space to check your conditions to fill in after you have completed your test in your calculator and note your expected values.

13. Finishing up the test Recall the 4 steps – State your hypotheses. Define any subscripts! – State your assumptions/conditions. – Name the test statistic and calculate its value. Graph the test statistic. State the p-value in symbols and calculate its value. – State the conclusion in the context of the problem.

14. Final notes… Degrees of freedom for the homogeneity of proportions: (r – 1)(c – 1)

15. Example The nonprofit group Public Agenda conducted telephone interviews with 3 randomly selected groups of parents of high school children. There were 202 black parents, 202 Hispanic parents, and 201 white parents. One question asked was, “Are the high schools in your state doing an excellent, good, fair, or poor job, or don’t you know enough to say.” You want to know if there is evidence that there is a difference in response based on race.

16. Results Black Parents Hispanic Parents White Parents Excellent123422 Good695581 Fair756160 Poor24 Don’t Know222814

17. Homework Chapter 11 # 27, 29, 31, 38, 40, 42