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## introducing Χ 2 Inference for Distribution of Categorical Variables

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**A new test statistic**introducingΧ2 Inference for Distribution of Categorical Variables**ThreeTests**1 3 2 Goodness Of Fit Association/ Independence Homogeneity Of Populations Conditions?**1**Goodness of Fit Are the hypothesized proportions true? Proportion of Colors in a bag of M&M’s**Tech Toolbox pp. 843 and 863**All individual EXPECTED counts at least 1 and no more than 20% are less than 5. Some books say each cell must have at least 5 expected counts. Ho: the actual population proportions are equal to the hypothesized proportions Ha: at least two of the actual population proportions differ from their hypothesized proportions**Put data into lists as described on page 843**What is the largest component affecting our X2 statistic? Remember that Ho: pred = .13 pbrown = .13 pyellow = .14 pgreen = .16 porange = .20 pblue = .24 Let’s try our m&m data Look at 14.2 on page 846…let’s do some inference.**For Goodness of Fit –**Single categorical variable that takes values on a single population. Should Marijuana be legalized for medical purposes? Activity 14B p. 849 For Test of Homogeneity – Either two categorical variables taking values on a single population Or A single categorical variable taking values on two or more populations. • Explain what a two-waytable is. • Compute row or column conditional distributions. • Identify the null hypothesis for a X2 test for homogeneity of populations.**2**Test for Homogeneity Our example What type of 2 way table do we have?**Ho: The distribution of the response variable is the same**in all c populations. What is the null hypothesis?**Expected Cell counts**Let’s find our expected cell counts. Remember, the alternative hypothesis is that the response variable distribution is not the same over the explanatory variables.**Any questions still for Homogeneity of Proportions?**Remember – Just because we find a X2 statistic that is significant and reject our null; we only know that not all of the population proportions were equal!**Other ways to state hypotheses**Ho: the two categorical variables are not related. Ha: the two categorical variables are related. Ho: the two categorical variables are independent. Ha: the two categorical variables are not independent (or, are dependent).**Remember:**Homogeneity: data must come from a comparative randomized experiment or from independent random samples from the populations of interest. Association/Independence: data should be obtained by classifying a single sample according to both categorical variables. Heath Care: Canada vs. USLife after being treated for a heart attack Let’s find the conditional distributions of the quality of life given the country where treated.**What are our hypotheses?**Ho: There is no relationship between nationality and quality of life.Ha: There is some relationship between nationality and quality of life.**?**Conditions? Expected Counts large enough – let’s calculate! SRS the data came from independent random samples from the two populations (US and Canada, eh?)**Medical Care and Age**• The type of medical care that a patient receives sometimes varies with the age of the patient. • Here are data from a study that asked whether women received a mammogram and biopsy after noticing a suspicious lump in their breast. • In this study, a single sample was classified two ways: by age and by whether or not the tests were done.**Now you can work on:**ExercisesSpecial ProblemCase Closed We can also watch a video on Inference for Two Way Tables**Spread the Message!**For a media-rich presentation, you can Optimize for Compatibility orCompress Media to share your presentation reliably without exploding your inbox Package your presentation for easy sharing**?**But wait…There’s More! View your slides from anywhere!**AccessAnywhere**• Check out the • PowerPoint Web App • Access slides wherever you are**POWERPOINT 2010**What’s Your Message?