MS-Sol-EE-C18

# MS-Sol-EE-C18 - CHAPTER 18 COMPARISON OF EMPIRICAL FREQUENCY DISTRIBUTIONS WITH FITTED DISTRIBUTIONS EXERCISE 18.2 1 Section 18.2 Fitting a

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CHAPTER 18 COMPARISON OF EMPIRICAL FREQUENCY DISTRIBUTIONS WITH FITTED DISTRIBUTIONS EXERCISE 18.2 Section 18.2 Fitting a discrete uniform distribution (page 504) 1. If the 4 models are equally popular, the expected frequency of each cell = 280 × 1 4 = 70 Model f o f e f o - f e 2-door Coupe 65 70 - 5 3-door Liftback 60 70 - 10 4-door Sedan 80 70 10 5-door Liftback 75 70 5 The observed frequencies do not deviate very much from the expected frequencies. Therefore, there is no evidence to believe that the 4 models are not equally popular. 2. If the absences occur equally likely on the five weekdays, the probability of an absence should be 5 1 for each weekday. Hence, the expected number of absences should be 100 × 5 1 = 20 for each day. We tabulate the results as follows: Day of the week Observed number of absences, f o Expected number of absences, f e Absolute discrepancy, | f o f e | Mon 29 20 9 Tue 12 20 8 Wed 9 20 11 Thu 23 20 3 Fri 27 20 7 We see that the absolute values of the discrepancies are quite large (ranging from 20 3 to 20 11 , i.e. 15% to 55% of the expected frequencies). Hence, we conclude that the absences do not occur equally likely on the five weekdays. 3. The total number of customers = 378 + 331 + 446 + 439 + 417 = 2 011 If the 5 movies attract the same proportion of audience, the number of customers in each studio should be 2 011 × 5 1 = 402.2. We calculate the discrepancies of the observed and expected data and tabulate them as follows: 66

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HAPTER 18 C OMPARISON OF E MPIRICAL F REQUENCY D ISTRIBUTIONS WITH F ITTED D ISTRIBUTIONS Studio Observed number of customers, f o Expected number of customers, f e Discrepancy, f o f e A 378 402.2 –24.2 B 331 402.2 –71.2 C 446 402.2 43.8 D 439 402.2 36.8 E 417 402.2 14.8 Total 2 011 2 011.0 0 As the discrepancies are numerically fairly large compared with the expected number of customers, the manager would not consider each movie as equally popular. 4. If the die is fair, the probability that each point will occur = 6 1 . Hence, the expected frequency of each cell = 300 × 6 1 = 50 We compare the observed frequencies and the expected frequencies in the table below. Score Observed frequency, f o Expected frequency, f e Discrepancy, f o - f e 1 47 50 - 3 2 52 50 2 3 48 50 - 2 4 57 50 7 5 56 50 6 6 40 50 - 10 Since the discrepancies are not unreasonably large, the die cannot be considered as biased. 5. The total number of students, N = 164 + 159 + 140 + 150 + 87 = 700 If the data follow the stated uniform distribution, the expected frequency of each cell = 700 × 10.5 5 . 20 2 - = 140 Age (Years) f o f e f o - f e 11 - 12 164 140 24 13 - 14 159 140 19 15 - 16 140 140 0 17 - 18 150 140 10 19 - 20 87 140 - 53 Since the discrepancies of the first 4 cells are non-negative while the discrepancy of the last cell is a negative and numerically large, the data do not indicate that they follow the stated uniform distribution. 67
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## This note was uploaded on 10/17/2011 for the course IELM 3010 taught by Professor Fugee during the Winter '11 term at HKUST.

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MS-Sol-EE-C18 - CHAPTER 18 COMPARISON OF EMPIRICAL FREQUENCY DISTRIBUTIONS WITH FITTED DISTRIBUTIONS EXERCISE 18.2 1 Section 18.2 Fitting a

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