Lesson 12 - R

1. Lesson 12 - R Chapter 12 Review

2. Objectives • Summarize the chapter • Define the vocabulary used • Complete all objectives • Successfully answer any of the review exercises • Use the technology to compute required objectives

3. Key Concepts • Expected Counts in a Goodness of Fit Test: Ei = μi = npi for i = 1, 2, …, k • Chi-Square Test Statistic: (Oi – Ei)2 χ2 = Σ ------------ for i = 1, 2, …, k Ei • Expected Frequencies in a Test for Independence: (row total)(column total) Expected Frequency = ------------------------------------ table total

4. Marginal Distributions • Marginal Distributions are along the end of the rows or the columns of a contingency table • Marginal Distributions effectively take out the other variable in the table • Marginal Distributions help calculated Expected values for Independence test using (through use of Multiplication Law of Probability)

5. Requirements • Goodness-of-Fit Test: • all E(xi) ≥ 1 • no more than 20% of E(xi) < 5 • Independence • same as Goodness-of-Fit Test • Homogeneity • same as Goodness-of-Fit Test

6. Problem 1 Which probability distribution do we use when we want to test the counts of a categorical variable? • The normal distribution • The chi-square distribution • The t-distribution • The categorical distribution

7. Problem 2 In the test of a categorical variable, to compare the observed value O to the expected value E, we use the quantity • O – E • E – O • E2 – O2 • (E –O)2 / E

8. Problem 3 A contingency table has what types of marginal distributions? • A row marginal distribution and a column marginal distribution • A marginal distribution for each combination of row and column value • One marginal distribution that summarizes the entire set of data • A different marginal distribution for each different relative frequency

9. Problem 4 If a contingency table has variables “Gender” and “Color of Eyes”, then which of the following is a conditional distribution? • The number of males with blue eyes • The number of females who have either brown eyes or green eyes • The proportion of the population who are male • The proportion of females who have blue eyes

10. Problem 5 In a contingency table where one variable is “Day of Week” and the other variable is “Rainy or Sunny”, a test for independence would test • Whether rainy days are independent of sunny days • Whether rainy or sunny days are independent of the day of the week • Whether Sundays are independent of Saturdays • Whether weekdays are independent of weekends

11. Problem 6 For a study with row variable “Color of Car” and column variable “Gender”, if 18% of males have blue cars, then the null hypothesis for the test for homogeneity would assume that • 18% of males have white cars • 18% of males do not have blue cars • 18% of females do not have white cars • 18% of females have blue cars

12. Summary and Homework • Summary • We can use a chi-square test to analyze the frequencies from categorical data • For the analysis of one categorical variable, we can use the chi-square goodness-of-fit test • For the analysis of two categorical variables in a contingency table, we can • Use the test for independence to analyze whether the two variables are independent • Use the test for homogeneity to analyze whether the proportions are equal • Homework • pg 662 - 667: 1, 4, 5, 11, 12, 16

13. Even Homework Answers • 4: a) Reject H0; school crime has become more violent sum of chi-sq is 1448.31 (way out in the tail!) • 12:a) FTR H0, not enough evidence to support the claim that martial status and gender are independent p=value = 0.1811596, Chi-Sq Test = 4.8753 • 16: a) Mohr in both positionsb) Erstadc) Mohr’s more at-bats with runners in scoring position dragged his overall average down more than Erstads.