1、第 PAGE13 頁 共 NUMPAGES13 頁商務與經濟統計習題答案（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 condu

2、cting 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 th

3、e 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

4、 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

5、 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, 3

6、00 (.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) (

7、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 cha

8、nged. 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 t

9、he 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

10、= 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 frequenci

11、es 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 -

12、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 3

13、0.6250 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 fligh

14、t 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、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

16、 independent of the type of flight. 12.a.Observed Frequency (fij) Domestic European Asian Total Same 125 55 68 248 Different 140 105 107 352 Total 265 160 175 600 E_pected Frequency (eij) Domestic European Asian Total Same 109.53 66.13 72.33 248 Different 155.47 93.87 102.67 352 Total 265 160 175 60

17、0 Chi Square (fij - eij)2 / eij Domestic European Asian Total Same 2.18 1.87 0.26 4.32 Different 1.54 1.32 0.18 3.04 c2 = 7.36 Degrees of freedom = 2 = 5.99 Reject H0; conclude brand loyalty is not independent of manufacturer. b.Brand Loyalty Domestic 125/265 = .472 (47.2%) Highest European 55/160 =

18、 .344 (34.4%) Asian 68/175 = .389 (38.9%) 13. Industry Major Oil Chemical Electrical puter Business 30 22.5 17.5 30 Engineering 30 22.5 17.5 30 Note: Values shown above are the e_pected frequencies. = 11.3449 (3 degrees of freedom: 1 _ 3 = 3) c2 = 12.39 Reject H0; conclude that major and industry no

19、t independent.14. E_pected Frequencies: e11 = 31.0 e12 = 31.0 e21 = 29.5 e22 = 29.5 e31 = 13.0 e32 = 13.0 e41 = 5.5 e42 = 5.5 e51 = 7.0 e52 = 7.0 e61 = 14.0 e62 = 14.0 Observed E_pected Frequency Frequency Most Difficult Gender (fi) (ei) (fi - ei)2 / ei Spouse Men 37 31.0 1.16 Spouse Women 25 31.0 1

20、.16 Parents Men 28 29.5 0.08 Parents Women 31 29.5 0.08 Children Men 7 13.0 2.77 Children Women 19 13.0 2.77 Siblings Men 8 5.5 1.14 Siblings Women 3 5.5 1.14 In-Laws Men 4 7.0 1.29 In-Laws Women 10 7.0 1.29 Other Relatives Men 16 14.0 0.29 Other Relatives Women 12 14.0 0.29 Totals: 20_13.43 = 11.07

21、05 with (6 - 1) (2 - 1) = 5 degrees of freedom Since 13.43 11.0705.we conclude that gender is not independent of the most difficult person to buy for. 15. E_pected Frequencies: e11 = 17.16 e12 = 12.84 e21 = 14.88 e22 = 11.12 e31 = 28.03 e32 = 20.97 e41 = 22.31 e42 = 16.69 e51 = 17.16 e52 = 12.84 e61

22、 = 15.45 e62 = 11.55 Observed E_pected Frequency Frequency Magazine Appeal (fi) (ei) (fi - ei)2 / ei News Guilt 20 17.16 0.47 News Fear 10 12.84 0.63 General Guilt 15 14.88 0.00 General Fear 11 11.12 0.00 Family Guilt 30 28.03 0.14 Family Fear 19 20.97 0.18 Business Guilt 22 22.31 0.00 Business Fear

23、 17 16.69 0.01 Female Guilt 16 17.16 0.08 Female Fear 14 12.84 0.11 African-American Guilt 12 15.45 0.77 African-American Fear 15 11.55 1.03 Totals: 201 3.41 = 15.09 with (6 - 1) (2 - 1) = 5 degrees of freedom Since 3.41 13.28, we conclude that the ratings are not independent. 20. First estimate m f

24、rom the sle data. Sle size = 120.Therefore, we use Poisson probabilities with m = 1.3 to pute e_pected frequencies. _ Observed Frequency Poisson Probability E_pected Frequency Difference (fi - ei) 0 39 .2725 32.700 6.300 1 30 .3543 42.516 -12.516 2 30 .2303 27.636 2.364 3 18 .0998 11.976 6.024 4 or

25、more 3 .0430 5.160 - 2.160 = 7.81473 with 5 - 1 - 1 = 3 degrees of freedom Since c2 = 9.0348 7.81473 Reject H0 Conclude that the data do not follow a Poisson probability distribution. 21. With n = 30 we will use si_ classes with 16 2/3% of the probability associated with each class. = 22.80 s = 6.26

26、65 The z values that create 6 intervals, each with probability .1667 are -.98, -.43, 0, .43, .98 z Cut off value of _ -.98 22.8 - .98 (6.2665) = 16.66 -.43 22.8 - .43 (6.2665) = 20.11 0 22.8 + 0 (6.2665) = 22.80 .43 22.8 + .43 (6.2665) = 25.49 .98 22.8 + .98 (6.2665) = 28.94 Interval Observed Freque

27、ncy E_pected Frequency Difference less than 16.66 3 5 -2 16.66 - 20.11 7 5 2 20.11 - 22.80 5 5 0 22.80 - 25.49 7 5 2 25.49- 28.94 3 5 -2 28.94 and up 5 5 0 = 9.34840 with 6 - 2 - 1 = 3 degrees of freedom Since c2 = 3.20 9.34840 Do not reject H0 The claim that the data es from a normal distribution c

28、annot be rejected. 22.Use Poisson probabilities with m = 1. c2 = 4.30 = 5.99147 (2 degrees of freedom) Do not reject H0; the assumption of a Poisson distribution cannot be rejected. 23. _ Observed Poisson Probabilities E_pected 0 15 .1353 13.53 1 31 .2707 27.07 2 20 .2707 27.07 3 15 .1804 18.04 4 13

29、 .0902 9.02 5 or more 6 .0527 5.27 c2 = 4.98 = 7.77944 (4 degrees of freedom) Do not reject H0; the assumption of a Poisson distribution cannot be rejected. 24. = 24.5 s = 3 n = 30 Use 6 classes Interval Observed Frequency E_pected Frequency less than 21.56 5 5 21.56 - 23.21 4 5 23.21 - 24.50 3 5 24

30、.50 - 25.79 7 5 25.79 - 27.44 7 5 27.41 up 4 5 c2 = 2.8 = 6.25139 (3 degrees of freedom: 6 - 2 - 1 = 3) Do not reject H0; the assumption of a normal distribution cannot be rejected. 25. = 71 s = 17 n = 25 Use 5 classes Interval Observed Frequency E_pected Frequency less than 56.7 7 5 56.7 - 66.5 7 5

31、 66.5 - 74.6 1 5 74.6 - 84.5 1 5 84.5 up 9 5 c2 = 11.2 = 9.21034 (2 degrees of freedom) Reject H0; conclude the distribution is not a normal distribution. 26. Observed 60 45 59 36 E_pected 50 50 50 50 c2 = 8.04 = 7.81473 (3 degrees of freedom) Reject H0; conclude that the order potentials are not th

32、e same in each sales territory. 27. Observed 48 323 79 16 63 E_pected 37.03 306.82 126.96 21.16 37.03 Since 41.69 13.2767, reject H0. Mutual fund investors attitudes toward corporate bonds differ from their attitudes toward corporate stock. 28. Observed 20 20 40 60 E_pected 35 35 35 35 Since 31.43 7

33、.81473, reject H0. The park manager should not plan on the same number attending each day.Plan on a larger staff for Sundays and holidays. 29. Observed 13 16 28 17 16 E_pected 18 18 18 18 18 c2 = 7.44 = 9.48773 Do not reject H0; the assumption that the number of riders is uniformly distributed canno

34、t be rejected.30. Observed E_pected Hypothesized Frequency Frequency Category Proportion (fi) (ei) (fi - ei)2 / ei Very Satisfied 0.28 105 140 8.75 Somewhat Satisfied 0.46 235 230 0.11 Neither 0.12 55 60 0.42 Somewhat Dissatisfied 0.10 90 50 32.00 Very Dissatisfied 0.04 15 20 1.25 Totals: 500 42.53

35、= 9.49 (4 degrees of freedom) Since 42.53 9.49, we conclude that the job satisfaction for puter programmers is different than the job satisfaction for IS managers. 31. E_pected Frequencies: Quality Shift Good Defective 1st 368.44 31.56 2nd 276.33 23.67 3rd 184.22 15.78 c2 = 8.11 = 5.99147 (2 degrees

36、 of freedom) Reject H0; conclude that shift and quality are not independent. 32. E_pected Frequencies: e11 = 1046.19 e12 = 632.81 e21 = 28.66 e22 = 17.34 e31 = 258.59 e32 = 156.41 e41 = 516.55 e42 = 312.45 Observed E_pected Frequency Frequency Employment Region (fi) (ei) (fi - ei)2 / ei Full-Time Ea

37、stern 1105 1046.19 3.31 Full-time Western 574 632.81 5.46 Part-Time Eastern 31 28.66 0.19 Part-Time Western 15 17.34 0.32 Self-Employed Eastern 229 258.59 3.39 Self-Employed Western 186 156.41 5.60 Not Employed Eastern 485 516.55 1.93 Not Employed Western 344 312.45 3.19 Totals: 2969 23.37 = 7.81 with