Chapter 13

# Chapter 13 - STAT 3000 Chapter 13 Inference for Counts...

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STAT 3000 Chapter 13: Inference for Counts: Chi-Square Tests So far, we’ve done inference mostly with quantitative variables. (The exception was confidence intervals and hypothesis tests for one- proportion.) Most recently, we looked at inference when building regression models, usually using only quantitative models. Does this mean there are no more inferential procedures for categorical variables? No, there are 3 useful hypothesis tests. We’ll look at two of these: 1. Goodness-of-Fit Tests 2. Tests for Independence 23

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STAT 3000 Section 13.1: Goodness-of-Fit Tests Given the following… Counts of items in each of several categories A model that predicts the distribution of the relative frequencies …this question naturally arises: “Does the actual distribution differ from the model because of random error , or do the differences mean that the model does not fit the data ?” In other words, “ How good is the fit ?” 24
Assumptions and Conditions for the χ 2 Goodness of Fit Test Counted Data Condition – The data must be counts for the categories of a categorical variable. • Independence Assumption – The counts should be independent of each other. Think about whether this is reasonable. • Randomization Condition – The counted individuals should be a random sample of the population. • Expected Cell Frequency Condition – Expect at least 5 individuals per cell. 25

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STAT 3000 Section 13.2: Interpreting Chi-Square Values The Chi-Square Distribution The distribution is right-skewed and becomes broader with increasing degrees of freedom: The test is a one-sided test. The larger the value, the greater the evidence against the null hypothesis and in support of the alternative hypothesis 26 2 χ 2 2
Ex: An randomized experiment measuring one categorical variable with 3 categories and n = 320 produced the data shown. Are these data consistent with the following theoretical model? Use α = .05 Category 1 contains = 15% (of the population) Category 2 contains = 35%, and Category 3 contains = 50%? np 1st category = 320( ) = np 2nd category = 320( ) = np 3rd category = 320( ) = 27 Cell 1 2 3 Total n i 50 108 162 320

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1. H o : the model is a good fit of the data 2. H a : at least one category contradicts the model 3. 2 χ = 4. Using a chi-square table with df = 2, the p-value is 5. Conclusion: 28 Cell 1 2 3 Total Observed Counts 50 108 162 320 Expected Counts
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Ex : Zodiac Signs for Executives Suppose you suspect that some zodiac signs produce superior personality qualities that affect leadership in business. You categorize the zodiac sign for a random selection of 256 business executives. Are birth signs distributed equally among business executives or not? Hypotheses:
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Chapter 13 - STAT 3000 Chapter 13 Inference for Counts...

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