商務與經(jīng)濟統(tǒng)計習題答案（第8版中文版）SBE9.doc

1、商務與經(jīng)濟統(tǒng)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 conducting tests of goodness o

2、f 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 epected fr

3、equencies, and determine if the two variables are independent. Solutions: 1. Epected 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 hypothes

4、is. The population proportions are not as stated in the null hypothesis. 2. Epected 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.

5、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 Epected frequencies: 300 (.29) = 87, 300 (.28) = 84 300 (.25) = 75, 300 (.18) = 54 e1 = 87,

6、 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 signifant change in the vieg audience proportions. 4. Observed Epected Hypothesized Frequency Frequency Category Proportion (fi) (ei) (fi - ei)2 / ei Brown 0.

7、30 177 151.8 4.18 Yellow 0.20 135 101.2 11.29 Red 0.20 79 101.2 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 company have changed. 5. Obser

8、ved Epected Hypothesized Frequency Frequency Category Proportion (fi) (ei) (fi - ei)2 / ei Full Serve 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

9、ads with guilt eals among the si types of magazines. 7. Epected frequencies: ei = (1 / 3) (135) = 45 With 2 degrees of freedom, = 5.99 Do not reject H0; there is no justifation for concluding a difference in preference eists. 8. H0: p1 = .03, p2 = .28, p3 = .45, p4 = .24 df = 3 = 11.34 Reject H0 if

10、c2 > 11.34 Rating Observed Epected (fi - ei)2 / ei Ecellent 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 comparison of observed and epected frequencies show telephone se

11、rve is slightly better with more ecellent 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 Epected 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

12、 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 Epected Frequencies: A B C P 17.5000 30.6250 21.8750 Q 28.

13、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 tket purchased is independent of the type of flight Ha: Type of tke

14、t purchased is not independent of the type of flight. Epected Frequencies: e11 = 35.59 e12 = 15.41 e21 = 150.73 e22 = 65.27 e31 = 455.68 e32 = 197.32 Observed Epected Frequency Frequency Tket Flight (fi) (ei) (fi - ei)2 / ei First Domest 29 35.59 1.22 First International 22 15.41 2.82 Business Domes

15、t 95 150.73 20.61 Business International 121 65.27 47.59 Full Fare Domest 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 tket purchased is not independent of the type o

16、f flight. 12.a.Observed Frequency (fij) Domest European Asian Total Same 125 55 68 248 Different 140 105 107 352 Total 265 160 175 600 Epected Frequency (eij) Domest European Asian Total Same 109.53 66.13 72.33 248 Different 155.47 93.87 102.67 352 Total 265 160 175 600 Chi Square (fij - eij)2 / eij

17、 Domest 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 Domest 125/265 = .472 (47.2) ¬ Highest European 55/160 = .344 (34.4) Asian 68/175 = .

18、389 (38.9) 13. Industry Major Oil Chemal Electral Computer Business 30 22.5 17.5 30 Enneering 30 22.5 17.5 30 Note: Values shown above are the epected frequencies. = 11.3449 (3 degrees of freedom: 1 3 = 3) c2 = 12.39 Reject H0; conclude that major and industry not independent.14. Epected Frequencies

19、: 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 Epected Frequency Frequency Most Diffult Gender (fi) (ei) (fi - ei)2 / ei Spouse Men 37 31.0 1.16 Spouse Women 25 31.0 1.16 Parents Men 28 29.5 0.08 Parents Wome

20、n 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: 2013.43 = 11.0705 with (6 - 1) (2 - 1) = 5 degrees of fre

21、edom Since 13.43 > 11.0705.we conclude that gender is not independent of the most diffult person to buy for. 15. Epected 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 = 15.45 e62 = 11.55 Observed Epected Fr

22、equency Frequency Magazine eal (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 17 16.69 0.01 Female Guilt 16 17.16 0.08 Fe

23、male Fear 14 12.84 0.11 Afran-Ameran Guilt 12 15.45 0.77 Afran-Ameran 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 from the sle data. Sle size = 120.Therefore, we use P

24、oisson probabilities with m = 1.3 to compute epected frequencies. Observed Frequency Poisson Probability Epected 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 more 3 .0430 5.160 - 2.160 = 7.81473 with 5 - 1 - 1 =

25、 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 = 6.2665 The z values that create 6 intervals, each with

26、 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 Frequency Epected Frequency Difference less than 16.66 3 5

27、 -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 comes from a normal distribution cannot be rejected. 22.Use Poisson probabili

28、ties 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 Epected 0 15 .1353 13.53 1 31 .2707 27.07 2 20 .2707 27.07 3 15 .1804 18.04 4 13 .0902 9.02 5 or more 6 .0527 5.27 c2 = 4.98 =

29、 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 Epected Frequency less than 21.56 5 5 21.56 - 23.21 4 5 23.21 - 24.50 3 5 24.50 - 25.79 7 5 25.79 - 27.44 7 5 27.41 up 4 5

30、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 Epected Frequency less than 56.7 7 5 56.7 - 66.5 7 5 66.5 - 74.6 1 5 74.6 - 84.5 1 5 84.5 up 9 5 c2

31、= 11.2 = 9.21034 (2 degrees of freedom) Reject H0; conclude the distribution is not a normal distribution. 26. Observed 60 45 59 36 Epected 50 50 50 50 c2 = 8.04 = 7.81473 (3 degrees of freedom) Reject H0; conclude that the order potentials are not the same in each sales territory. 27. Observed 48 3

32、23 79 16 63 Epected 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 Epected 35 35 35 35 Since 31.43 > 7.81473, reject H0. The park manager shoul

33、d 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 Epected 18 18 18 18 18 c2 = 7.44 = 9.48773 Do not reject H0; the assumption that the number of riders is uniformly distributed cannot be rejected.30. Observed Epected Hypothe

34、sized 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 = 9.49 (4 degrees of freedom) Since 42.53 &

35、gt; 9.49, we conclude that the job satisfaction for computer programmers is different than the job satisfaction for IS managers. 31. Epected 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 of freedom) Reject H0; conclude tha

36、t shift and quality are not independent. 32. Epected 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 Epected Frequency Frequency Employment Reon (fi) (ei) (fi - ei)2 / ei Full-Time Eastern 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 w