Chapter 13: Chi-Square Procedures 13.1 Test for Goodness of Fit 13.2 Inference for Two-Way Tables

M&Ms Example Sometimes we want to examine the distribution of proportions in a single population. As opposed to comparing distributions from two populations, as in Chapter 12. Does the distribution of colors in your bags match up with expected values? We can use a chi-square goodness of fit test. Χ2 We would not want to do multiple one-proportion z-tests. Why?

Performing a X2 Test 1. H0: the color distribution of our M&Ms is as advertised: Pbrown=0.30, Pyellow=Pred=0.20, and Porange=Pgreen=Pblue=0.10 Ha: the color distribution of our M&Ms is not as advertised. Conditions: All individual expected counts are at least 1. No more than 20% of expected counts are less than 5. Chi-square statistic:

X2 Family of Distribution Curves Used to assess the evidence against H0 represented in the value of X2. The member of the family we choose is determined only by the degrees of freedom. P-value is the probability of observing a value X2 at least as extreme as the one actually observed.

X2 Family of Distribution Curves (Figure 13.2, p. 732)

Performing a test with our calculators Enter data: L1: observed values (not percentages) L2: expected values L3: (L1-L2)2/L2 LIST/MATH Sum (L3) X2cdf(ans,1099, df)

Practice Exercise 13.10, p. 743

More Practice Exercise 13.4, p. 737

13.4 summary

13.4 Follow-Up So we reject the null, but so what? What does this mean in the context of the problem? Where did the differences occur?

Graph for Problem 13.4

Section 13.2: Inference for Two-Way Tables

Example 13.4, pp. 744-748 Relapse? Treatment No Yes Total Desipramine 14 10 24 Lithium 6 18 Placebo 4 20 48 72 Is there a difference between proportion of successes? At left is a two-way table for use in studying this question. Explanatory Variable: Type of Treatment Response Variable: Proportion of no relapses

Example 13.4, pp. 744-748 Relapse? Treatment No Yes Total Desipramine 14 10 24 Lithium 6 18 Placebo 4 20 48 72 H0: p1=p2=p3 Ha: at least one proportion is not equal to the others.

Example 13.4, pp. 744-748 We need expected counts for each cell: For our example, 2/3 relapsed (48/72). So, if Ho is true, then 24(2/3) of those taking Desipramine would relapse. Write expected counts for each cell in your table at the bottom of page 747.

Chi-Square Test for Homogeneity of Populations In this example, we are comparing the proportion of relapses in three populations: addicts who take desipramine, addicts who take lithium, and addicts who take a placebo. Our question is this: Are the populations homogeneous in terms of the proportion of relapsed addicts? We use a chi-square test for homogeneity of populations.

Conditions All individual expected counts are at least one, and No more than 20% of expected counts are less than 5.

Calculations for Example 13.4 Note: df=(#r-1)(#c-1)=(3-1)(2-1)=2

Full Analysis of Example 13.4 See Example 13.7, p. 752

Practice 13.14, pp. 748-749

Let’s begin with a practice problem … 13.15, p. 749 + 13.17, p. 756

Two Settings for Chi-Square Tests for Two-Way Tables Yesterday we studied the problem of treatments for cocaine addicts. The cocaine addicts study is an experiment that assigned 24 addicts to each of three groups. Each group is a sample from a separate population corresponding to a separate treatment. We used a chi-square test for homogeneity of populations. (H0: p1=p2=p3 vs. Ha: at least one not the same) Today, we look at problems where subjects from a single sample are classified with respect to a categorical variable. We will use a chi-square test of association/independence. Notes: See bottom paragraph, p. 763. The analysis for today’s problem will be essentially identical to the analysis from yesterday.

Example 13.9, 13.10, and 13.11, pp. 758-761 Hypotheses: H0: there is no relationship between smoking status and SES (two categorical variables). Ha: there is a relationship between smoking status and SES.

Expected Counts and Conditions All expected counts are at least 1, no more than 20% less than 5.

Practice Problem 13.20, p. 762 Make a bar graph to show the data graphically before beginning your calculations for the chi-square test.

Graph for 13.20

Practice 13.32, p. 770 Chapter 13 Test on Tuesday.