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Statistics

This chapter explores methods for making inferences about population proportions and differences between proportions using categorical data. It covers one-way and two-way analyses, chi-square hypothesis tests, and multinomial experiments.

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Statistics

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  1. Statistics Chapter 13: Categorical Data Analysis

  2. Where We’ve Been • Presented methods for making inferences about the population proportion associated with a two-level qualitative variable (i.e., a binomial variable) • Presented methods for making inferences about the difference between two binomial proportions McClave, Statistics, 11th ed. Chapter 13: Categorical Data Analysis

  3. Where We’re Going • Discuss qualitative (categorical) data with more than two outcomes • Present a chi-square hypothesis test for comparing the category proportions associated with a single qualitative variable – called a one-way analysis • Present a chi-square hypothesis test relating two qualitative variables – called a two-way analysis McClave, Statistics, 11th ed. Chapter 13: Categorical Data Analysis

  4. 13.1: Categorical Data and the Multinomial Experiment • Properties of the Multinomial Experiment • The experiment consists of n identical trials. • There are k possible outcomes (called classes, categories or cells) to each trial. • The probabilities of the k outcomes, denoted by p1, p2, …, pk, where p1+ p2+ … + pk = 1, remain the same from trial to trial. • The trials are independent. • The random variables of interest are the cell counts n1, n2, …, nk of the number of observations that fall into each of the k categories. McClave, Statistics, 11th ed. Chapter 13: Categorical Data Analysis

  5. 13.2: Testing Categorical Probabilities: One-Way Table • Suppose three candidates are running for office, and 150 voters are asked their preferences. • Candidate 1 is the choice of 61 voters. • Candidate 2 is the choice of 53 voters. • Candidate 3 is the choice of 36 voters. • Do these data suggest the population may prefer one candidate over the others? McClave, Statistics, 11th ed. Chapter 13: Categorical Data Analysis

  6. 13.2: Testing Categorical Probabilities: One-Way Table Candidate 1 is the choice of 61 voters. Candidate 2 is the choice of 53 voters. Candidate 3 is the choice of 36 voters. n =150 McClave, Statistics, 11th ed. Chapter 13: Categorical Data Analysis

  7. 13.2: Testing Categorical Probabilities: One-Way Table Reject the null hypothesis McClave, Statistics, 11th ed. Chapter 13: Categorical Data Analysis

  8. 13.2: Testing Categorical Probabilities: One-Way Table Test of a Hypothesis about Multinomial Probabilities: One-Way Table H0: p1= p1,0, p2= p2,0, … , pk= pk,0 where p1,0, p2,0, …, pk,0 represent the hypothesized values of the multinomial probabilities Ha: At least one of the multinomial probabilities does not equal its hypothesized value where Ei = np1,0, is the expected cell count given the null hypothesis. McClave, Statistics, 11th ed. Chapter 13: Categorical Data Analysis

  9. 13.2: Testing Categorical Probabilities: One-Way Table Conditions Required for a Valid 2 Test: One-Way Table • A multinomial experiment has been conducted. • The sample size n will be large enough so that, for every cell, the expected cell count E(ni) will be equal to 5 or more. McClave, Statistics, 11th ed. Chapter 13: Categorical Data Analysis

  10. 13.2: Testing Categorical Probabilities: One-Way Table Example 13.2: Distribution of Opinions About Marijuana Possession Before Television Series has Aired Table 13.2: Distribution of Opinions About Marijuana Possession After Television Series has Aired McClave, Statistics, 11th ed. Chapter 13: Categorical Data Analysis

  11. 13.2: Testing Categorical Probabilities: One-Way Table McClave, Statistics, 11th ed. Chapter 13: Categorical Data Analysis

  12. 13.2: Testing Categorical Probabilities: One-Way Table Expected Distribution of 500 Opinions About Marijuana Possession After Television Series has Aired McClave, Statistics, 11th ed. Chapter 13: Categorical Data Analysis

  13. 13.2: Testing Categorical Probabilities: One-Way Table Expected Distribution of 500 Opinions About Marijuana Possession After Television Series has Aired McClave, Statistics, 11th ed. Chapter 13: Categorical Data Analysis

  14. 13.2: Testing Categorical Probabilities: One-Way Table Expected Distribution of 500 Opinions About Marijuana Possession After Television Series has Aired Reject the null hypothesis McClave, Statistics, 11th ed. Chapter 13: Categorical Data Analysis

  15. 13.2: Testing Categorical Probabilities: One-Way Table • Inferences can be made on any single proportion as well: • 95% confidence interval on the proportion of citizens in the viewing area with no opinion is McClave, Statistics, 11th ed. Chapter 13: Categorical Data Analysis

  16. 13.3: Testing Categorical Probabilities: Two-Way Table • Chi-square analysis can also be used to investigate studies based on qualitative factors. • Does having one characteristic make it more/less likely to exhibit another characteristic? McClave, Statistics, 11th ed. Chapter 13: Categorical Data Analysis

  17. 13.3: Testing Categorical Probabilities: Two-Way Table The columns are divided according to the subcategories for one qualitative variable and the rows for the other qualitative variable. McClave, Statistics, 11th ed. Chapter 13: Categorical Data Analysis

  18. 13.3: Testing Categorical Probabilities: Two-Way Table McClave, Statistics, 11th ed. Chapter 13: Categorical Data Analysis

  19. 13.3: Testing Categorical Probabilities: Two-Way Table • The results of a survey regarding marital status and religious affiliation are reported below (Example 13.3 in the text). Religious Affiliation Marital Status H0: Marital status and religious affiliation are independent Ha: Marital status and religious affiliation are dependent McClave, Statistics, 11th ed. Chapter 13: Categorical Data Analysis

  20. 13.3: Testing Categorical Probabilities: Two-Way Table • The expected frequencies (see Figure 13.4) are included below: Religious Affiliation Marital Status The chi-square value computed with SAS is 7.1355, with p-value = .1289. Even at the  = .10 level, we cannot reject the null hypothesis. McClave, Statistics, 11th ed. Chapter 13: Categorical Data Analysis

  21. 13.3: Testing Categorical Probabilities: Two-Way Table McClave, Statistics, 11th ed. Chapter 13: Categorical Data Analysis

  22. 13.4: A Word of Caution About Chi-Square Tests McClave, Statistics, 11th ed. Chapter 13: Categorical Data Analysis

  23. 13.4: A Word of Caution About Chi-Square Tests Be sure McClave, Statistics, 11th ed. Chapter 13: Categorical Data Analysis

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