COURSE: JUST 3900 INTRODUCTORY STATISTICS FOR CRIMINAL JUSTICE Chapter 17: Chi Square Peer Tutor Slides Instructor: Mr. Ethan W. Cooper, Lead Tutor © PLEASE DO NOT CITE, QUOTE, OR REPRODUCE WITHOUT THE WRITTEN PERMISSION OF THE AUTHOR. FOR PERMISSION OR QUESTIONS, PLEASE MR. COOPER AT THE FOLLWING:

Key Terms: Don’t Forget Notecards Nonparametric tests (p. 593) Nonparametric tests (p. 593) Chi-square Test for Goodness of Fit (p. 594) Chi-square Test for Goodness of Fit (p. 594) Observed Frequency (p. 597) Observed Frequency (p. 597) Expected Frequency (p. 597) Expected Frequency (p. 597) Chi-square Test for Independence (p. 605) Chi-square Test for Independence (p. 605) Independent (p. 607) Independent (p. 607)

Formulas

Introduction to Nonparametric Tests Question 1: What type of data do nonparametric tests use? Question 1: What type of data do nonparametric tests use?

Introduction to Nonparametric Tests Question 1 Answer: Question 1 Answer: Nonparametric tests use nominal or ordinal data. Nonparametric tests use nominal or ordinal data.

Chi-square Test for Goodness of Fit Question 2: A researcher has developed three different designs for a computer keyboard. A sample of n = 60 participants is obtained, and each individual tests all three keyboards and identifies his or her favorite. Were any of the designs selected more (or less) often than would be expected by chance? Use the Chi-square test for goodness of fit with an alpha level of α = The frequency distribution of preferences is as follows: Question 2: A researcher has developed three different designs for a computer keyboard. A sample of n = 60 participants is obtained, and each individual tests all three keyboards and identifies his or her favorite. Were any of the designs selected more (or less) often than would be expected by chance? Use the Chi-square test for goodness of fit with an alpha level of α = The frequency distribution of preferences is as follows: Design ADesign BDesign C n = 60

Chi-square Test for Goodness of Fit Question 2 Answer: Question 2 Answer: Step 1: State the hypothesis Step 1: State the hypothesis H 0 : There is no preference between keyboard designs. H 0 : There is no preference between keyboard designs. H 1 : There is a preference. H 1 : There is a preference. Design ADesign BDesign C Design ADesign BDesign C 33.3%

Chi-square Test for Goodness of Fit Question 2 Answer: Question 2 Answer: Step 2: Locate the critical region Step 2: Locate the critical region Find df. Find df. df = C – 1 = 3 – 1 = 2 df = C – 1 = 3 – 1 = 2 Use df and alpha level (α = 0.05) to find the critical Χ 2 value for the test. Use df and alpha level (α = 0.05) to find the critical Χ 2 value for the test. Χ 2 crit = 5.99 Χ 2 crit = 5.99

Chi-square Test for Goodness of Fit Question 2 Answer: Question 2 Answer: Step 3: Calculate the Chi-Square Statistic Step 3: Calculate the Chi-Square Statistic Identify proportions required to compute expected frequencies (f e ). Identify proportions required to compute expected frequencies (f e ). The null specifies the proportion for each cell. With a sample of 60 participants, the expected frequencies for each category (Designs A, B & C) are equal in proportion (33.3 % or 1/3). The null specifies the proportion for each cell. With a sample of 60 participants, the expected frequencies for each category (Designs A, B & C) are equal in proportion (33.3 % or 1/3). Calculate the expected frequencies with proportions from H 0. Calculate the expected frequencies with proportions from H 0. f e = pn = (1/3) * (60) = 20 participants f e = pn = (1/3) * (60) = 20 participants Calculate the Chi-Square Statistic Calculate the Chi-Square Statistic Design ADesign BDesign C

Chi-square Test for Goodness of Fit

Question 2 Answer: Question 2 Answer: Step 4: Make a Decision Step 4: Make a Decision If X 2 ≤ 5.99, fail to reject H 0 If X 2 ≤ 5.99, fail to reject H 0 If X 2 > 5.99, reject H 0 If X 2 > 5.99, reject H < 5.99, thus, we reject H 0. It does not appear any of the keyboard designs were preferred to the others < 5.99, thus, we reject H 0. It does not appear any of the keyboard designs were preferred to the others.

Chi-square Test for Independence Question 3: A researcher would like to know which factors are most important to people who are buying a new car. A sample of n = 200 customers are asked to identify the most important factor in the decision process: Performance, Reliability, or Style. The researcher would like to know whether there is a difference between the factors identified by women compared to those identified by men. Use the chi-square test for independence with an alpha level of α = The data are as follows: Question 3: A researcher would like to know which factors are most important to people who are buying a new car. A sample of n = 200 customers are asked to identify the most important factor in the decision process: Performance, Reliability, or Style. The researcher would like to know whether there is a difference between the factors identified by women compared to those identified by men. Use the chi-square test for independence with an alpha level of α = The data are as follows: PerformanceReliabilityStyleTotals Male Female Totals

Chi-square Test for Independence Question 3 Answer: Question 3 Answer: Step 1: State the hypothesis Step 1: State the hypothesis H 0 : In the population, the distribution of preferred factors for men has the same proportions as the distribution for women. H 0 : In the population, the distribution of preferred factors for men has the same proportions as the distribution for women. H 1 : In the population, the distribution of preferred factors for men has different proportions than the distribution for women. H 1 : In the population, the distribution of preferred factors for men has different proportions than the distribution for women.

Chi-square Test for Independence Question 3 Answer: Question 3 Answer: Step 2: Locate the critical region. Step 2: Locate the critical region. Find df. Find df. df = (R -1)*(C – 1) = (2 – 1)*(3 – 1) = (1)*(2) = 2 df = (R -1)*(C – 1) = (2 – 1)*(3 – 1) = (1)*(2) = 2 Use df (2) and alpha level (0.05) to find the critical X 2 value. Use df (2) and alpha level (0.05) to find the critical X 2 value. X 2 crit = 5.99 X 2 crit = 5.99

Chi-square Test for Independence PerformanceReliabilityStyleTotals Male80 Female120 Totals

Chi-square Test for Independence PerformanceReliabilityStyle Male(16)21(40)33(24)26 Female(24)19(60)67(36)34

Chi-square Test for Independence Question 3 Answer: Question 3 Answer: Step 4: Make a Decision Step 4: Make a Decision If X 2 ≤ 5.99, fail to reject H 0 If X 2 ≤ 5.99, fail to reject H 0 If X 2 > 5.99, reject H 0 If X 2 > 5.99, reject H > 5.99, thus, we fail to reject H 0, which means the distribution of preferred factors for men has the same proportions as the distribution for women > 5.99, thus, we fail to reject H 0, which means the distribution of preferred factors for men has the same proportions as the distribution for women.

Effect Size for the Chi Square Test for Independence

Medium effect Small effect