商务与经济统计习题答案第8版，中文版SBE8Chapter 12 Tests of Goodness of Fit and Independence Learning Objectives 1. Know how to conduct a goodness of fit test. 2.Know how to use sle data to test for independence of two variables. 3.Understand the role of the chi-square distribution in conducting tests of goodness of fit and independence. 4.Be able to conduct a goodness of fit test for cases where the population is hypothesized to have either a multinomial, a Poisson, or a normal probability distribution. 5.For a test of independence, be able to set up a contingency table, determine the observed and e_pected frequencies, and determine if the two variables are independent. Solutions: 1. E_pected frequencies: e1 = 20_(.40) = 80, e2 = 20_(.40) = 80 e3 = 20_(.20) = 40 Actual frequencies: f1 = 60, f2 = 120, f3 = 20 = 9.21034 with k - 1 = 3 - 1 = 2 degrees of freedom Since = 35 9.21034 reject the null hypothesis. The population proportions are not as stated in the null hypothesis. 2. E_pected frequencies: e1 = 300 (.25) = 75, e2 = 300 (.25) = 75 e3 = 300 (.25) = 75, e4 = 300 (.25) = 75 Actual frequencies: f1 = 85, f2 = 95, f3 = 50, f4 = 70 = 7.81473 with k - 1 = 4 - 1 = 3 degrees of freedom Since c2 = 15.33 7.81473 reject H0 We conclude that the proportions are not all equal. 3. H0 = pABC = .29, pCBS = .28, pNBC = .25, pIND = .18 Ha = The proportions are not pABC = .29, pCBS = .28, pNBC = .25, pIND = .18 E_pected frequencies: 300 (.29) = 87, 300 (.28) = 84 300 (.25) = 75, 300 (.18) = 54 e1 = 87, e2 = 84, e3 = 75, e4 = 54 Actual frequencies: f1 = 95, f2 = 70, f3 = 89, f4 = 46 = 7.81 (3 degrees of freedom) Do not reject H0; there is no significant change in the viewing audience proportions. 4. Observed E_pected Hypothesized Frequency Frequency Category Proportion (fi) (ei) (fi - ei)2 / ei Brown 0.30 177 151.8 4.18 Yellow 0.20 135 .2 11.29 Red 0.20 79 .2 4.87 Orange 0.10 41 50.6 1.82 Green 0.10 36 50.6 4.21 Blue 0.10 38 50.6 3.14 Totals: 506 29.51 = 11.07 (5 degrees of freedom) Since 29.51 11.07, we conclude that the percentage figures reported by the pany have changed. 5. Observed E_pected Hypothesized Frequency Frequency Category Proportion (fi) (ei) (fi - ei)2 / ei Full Service 1/3 264 249.33 0.86 Discount 1/3 255 249.33 0.13 Both 1/3 229 249.33 1.66 Totals: 748 2.65 = 4.61 (2 degrees of freedom) Since 2.65 9.24, we conclude that there is a difference in the proportion of ads with guilt appeals among the si_ types of magazines. 7. E_pected frequencies: ei = (1 / 3) (135) = 45 With 2 degrees of freedom, = 5.99 Do not reject H0; there is no justification for concluding a difference in preference e_ists. 8. H0: p1 = .03, p2 = .28, p3 = .45, p4 = .24 df = 3 = 11.34 Reject H0 if c2 11.34 Rating Observed E_pected (fi - ei)2 / ei E_cellent 24 .03(400) = 12 12.00 Good 124 .28(400) = 112 1.29 Fair 172 .45(400) = 180 .36 Poor 80 .24(400) = 96 2.67 400 400 c2 = 16.31 Reject H0; conclude that the ratings differ. A parison of observed and e_pected frequencies show telephone service is slightly better with more e_cellent and good ratings. 9. H0 = The column variable is independent of the row variable Ha = The column variable is not independent of the row variable E_pected Frequencies: A B C P 28.5 39.9 45.6 Q 21.5 30.1 34.4 = 7.37776 with (2 - 1) (3 - 1)= 2 degrees of freedom Since c2 = 7.86 7.37776 Reject H0 Conclude that the column variable is not independent of the row variable. 10. H0 = The column variable is independent of the row variable Ha = The column variable is not independent of the row variable E_pected Frequencies: A B C P 17.5000 30.6250 21.8750 Q 28.7500 50.3125 35.9375 R 13.7500 24.0625 17.1875 = 9.48773 with (3 - 1) (3 - 1)= 4 degrees of freedom Since c2 = 19.78 9.48773 Reject H0 Conclude that the column variable is not independent of f the row variable. 11. H0 : Type of ticket purchased is independent of the type of flight Ha: Type of ticket purchased is not independent of the type of flight. E_pected Frequencies: e11 = 35.59 e12 = 15.41 e21 = 150.73 e22 = 65.27 e31 = 455.68 e32 = 197.32 Observed E_pected Frequency Frequency Ticket Flight (fi) (ei) (fi - ei)2 / ei First Domestic 29 35.59 1.22 First International 22 15.41 2.82 Business Domestic 95 150.73 20.61 Business International 121 65.27 47.59 Full Fare Domestic 518 455.68 8.52 Full Fare International 135 197.32 19.68 Totals: 920 100.43 = 5.99 with (3 - 1)(2 - 1) = 2 degrees of freedom Since 100.43 5.99, we conclude that the type of ticket purchased is not independent of the type of flight. 12.a.Observed Frequency (fij) Domestic European Asian Total Same 125 55 68 248 Different