# Chapter 10: Chi-Square Tests and the F-Distribution Chapter 10 Chi-Square Tests and the FDistribution 10.1 Goodness of Fit Multinomial Experiments A multinomial experiment is a probability experiment consisting of a fixed number of trials in which there are more than two possible outcomes for each independent trial. (Unlike the binomial experiment in which there were only two possible outcomes.) Example: A researcher claims that the distribution of favorite pizza toppings among teenagers is as shown below. Topping Each outcome is Cheese Pepperoni classified into Sausage categories. Mushrooms Onions Frequency, f 41%

25% 15% 10% 9% The probability for each possible outcome is fixed. Larson & Farber, Elementary Statistics: Picturing the World, 3e 3 Chi-Square Goodness-of-Fit Test A Chi-Square Goodness-of-Fit Test is used to test whether a frequency distribution fits an expected distribution. To calculate the test statistic for the chi-square goodness-offit test, the observed frequencies and the expected frequencies are used. The observed frequency O of a category is the frequency for the category observed in the sample data. The expected frequency E of a category is the calculated frequency for the category. Expected frequencies are obtained assuming the specified (or hypothesized) distribution. The expected frequency for the ith category is Ei = npi where n is the number of trials (the sample size) and pi is the assumed probability of the ith category. Larson & Farber, Elementary Statistics: Picturing the World, 3e 4

Observed and Expected Frequencies Example: 200 teenagers are randomly selected and asked what their favorite pizza topping is. The results are shown below. Find the observed frequencies and the expected frequencies. Topping Cheese Pepperoni Sausage Mushroom s Onions Results (n = 200) 78 52 30 25 % of teenager s 41% 25% 15%

10% 15 9% Observed Frequenc y 78 52 30 25 15 Expected Frequency 200(0.41) = 82 200(0.25) = 50 200(0.15) = 30 200(0.10) = 20 200(0.09) = 18 Larson & Farber, Elementary Statistics: Picturing the World, 3e 5 Chi-Square Goodness-of-Fit Test For the chi-square goodness-of-fit test to be used, the following

must be true. 1. 2. The observed frequencies must be obtained by using a random sample. Each expected frequency must be greater than or equal to 5. The Chi-Square Goodness-of-Fit Test If the conditions listed above are satisfied, then the sampling distribution for the goodness-of-fit test is approximated by a chisquare distribution with k 1 degrees of freedom, where k is the number of categories. The test statistic for the chi-square goodness-of-fit test is 2 2 (O E ) E The test is always a right-tailed test. where O represents the observed frequency of each category and E represents the expected frequency of each category. Larson & Farber, Elementary Statistics: Picturing the World, 3e 6 Chi-Square Goodness-of-Fit Test

Performing a Chi-Square Goodness-of-Fit Test In Words 1. Identify the claim. State the null and alternative hypotheses. In Symbols State H0 and Ha. Identify . 2. Specify the level of significance. d.f. = k 1 3. Identify the degrees of freedom. Use Table 6 in Appendix B. 4. Determine the critical value. Continued. 5. Determine the rejection region. Larson & Farber,

Elementary Statistics: Picturing the World, 3e 7 Chi-Square Goodness-of-Fit Test Performing a Chi-Square Goodness-of-Fit Test In Words In Symbols 6. Calculate the test statistic. (O E )2 E 7. Make a decision to reject or fail to reject the null hypothesis. If 2 is in the rejection region, reject H0. Otherwise, fail to reject H0. 2 8. Interpret the decision in the context of the original claim.

Larson & Farber, Elementary Statistics: Picturing the World, 3e 8 Chi-Square Goodness-of-Fit Test Example: A researcher claims that the distribution of favorite pizza toppings among teenagers is as shown below. 200 randomly selected teenagers are surveyed. Topping Cheese Pepperoni Sausage Mushrooms Onions Frequency, f 39% 26% 15% 12.5% 7.5% Using = 0.01, and the observed and expected values previously calculated, test the surveyors claim using a chi-square goodness-of-fit test. Continued. Larson & Farber, Elementary Statistics: Picturing the World, 3e

9 Chi-Square Goodness-of-Fit Test Example continued: H0: The distribution of pizza toppings is 39% cheese, 26% pepperoni, 15% sausage, 12.5% mushrooms, and 7.5% onions. (Claim) Ha: The distribution of pizza toppings differs from the claimed or expected distribution. Because there are 5 categories, the chi-square distribution has k 1 = 5 1 = 4 degrees of freedom. With d.f. = 4 and = 0.01, the critical value is 20 = 13.277. Continued. Larson & Farber, Elementary Statistics: Picturing the World, 3e 10 Chi-Square Goodness-of-Fit Test Example continued: Topping Rejectio n region Observed Frequenc y

78 52 30 25 Expected Frequenc y 82 50 30 20 Cheese Pepperoni X2 Sausage 20 = 13.277 Mushroom s 2 2 Onions 18 (30 30)2 (2515 20)2 (15 18) (O E )2 (78 82)2 (52 50) 2

0.01 E 82 50 30 20 18 2.025 Fail to reject H0. There is not enough evidence at the 1% level to reject the surveyors claim. Larson & Farber, Elementary Statistics: Picturing the World, 3e 11