Dawson College - Fall 2004 Mathematics Department Final Examination Statistics (201-257-DW) No. Score Out of 1 8 2 10 3 8 Date: Thursday, December 16, 2004 Time: 9:30 12:30 Instructors: Kourosh A. Zarabi and Rodney Acteson Student s Name: Student s ID Number: Teacher s Name: INSTRUCTIONS: 1. This package has 9 pages. Check now to make sure that you have all 9 pages. 2. Do not remove any pages from this package. Any loose papers will be removed from your work area! 3. Attempt all 12 questions for a total of 100 points. Show all your work. 4. Reverse sides of pages may be used for rough work and/or to complete a solution as long as this is clearly indicated by the student (ie. please turn over ). 5. A calculator with statistical mode is required and allowed. 6. If necessary, consult an invigilator by raising your hand. You are not permitted to speak to any other person for any reason. 4 8 5 9 6 8 7 8 8 8 9 8 10 8 11 8 12 8 13 8 Total 100
2 QUESTION 1 [8 Marks] The number of times a person gets a busy signal when calling the customer service number at Dell is a random variable with the following distribution. (a) What is P( x )? Ans. 0.11 x 0 1 2 3 4 P( x ).23.34.17.15? (c) What is the probability that the person does not get a busy signal? Ans. 0.23 What is the probability that a person gets at least one busy signal? Ans. 0.77 (d) What is the expected number of busy signals? Ans. 1.57 QUESTION 2 [10 Marks] A 2000 randomly selected adults were asked whether or not they have ever shopped on the Internet. The following table gives their responses. Male Female Have Shopped 300 200 Have Never Shopped 900 600 (8) (a) Suppose one adult is selected find the probabilities i. P (has never shopped on the Internet and is a male). Ans. 0.45 ii. P (has shopped on Internet and is a female). Ans. 0.1 iii. P (is male). Ans. 0.6 iv. P (is a female given that they have shopped on the Internet). Ans. 0.4 (2) Are the events shopping and gender independent? Ans. Yes.
3 QUESTION 3 [8 Marks] Consider the frequency distribution of the hourly rates (in dollars) charged by 50 professional consultants. Frequency Rate 80-84 6 85-89 15 90-94 14 95-99 10 100-104 5 (6) (a) Estimate x and s. Ans. x 91.3 s 5.89 (2) Of the professional consultants charging $90 or over, what is the probability that they charge $100 or more? Ans. 5 29 QUESTION 4 [8 Marks] If a National Basketball league finalist is 75% certain to win each game it plays, what is the probability that it will win the championship being the first team to win 3 games in a possible 5 game series? Ans. 0.896484 QUESTION 5 [9 Marks] The time taken by auto mechanics to replace a faulty fuel pump is normally distributed with a mean of 90 minutes and a standard deviation of 12 minutes. (a) If you bring your car into a service station to have the fuel pump replaced, what is the probability that it will take more than 2 hours? Ans. 0.0062 (c) If you wish to have only a 5% probability that your car will not be ready when you go to pick it up, how long after the work is started should you return to the service station? Ans. 109.74 If a service station replaces 16 fuel pumps in a month, what is the probability that the mean time taken to replace these pumps is between 84 minutes and 93 minutes. Ans. 0.8187 QUESTION 6 [8 Marks] (a) Nine cars of the same model were driven in an identical manner, using one gallon of unleaded gasoline. The mean distance travelled by the 9 cars was 19 miles, with a standard deviation of 2.7 miles. Establish a 0.95 confidence interval estimate of the average mileage per gallon for this car model. Ans. 16.9246 to 21.0754 The management of a manufacturing concern wishes to determine the average time required to complete a certain manual operation. There should be 0.95 confidence that the error in the estimate will not exceed 2 minutes. What sample size is required if the standard deviation of the time needed to complete the manual operation is estimated by a time and motion expert as 10 minutes. Ans. 97
4 QUESTION 7 [8 Marks] Air Canada finds that only 90% of all persons making reservations actually show up for their flight. Hence, if they take 400 reservations for a flight on a 375 seat plane, what is the probability that there will be more passengers than seats for the flight? Ans. 0.0049 QUESTION 8 [8 Marks] A trash company claims that the average weight of any of its fully loaded garbage trucks is 11000 pounds with a standard deviation of 800 pounds. A highway department inspector decides to check on this claim. She randomly checks 16 trucks and finds that the average weight of these trucks is 11400 pounds. Does this indicate that the average weight of a garbage truck is more than 11000 pounds? Use 0.05 level of significance. Ans. t0 = 2> t c = 1.753 accept H a QUESTION 9 [8 Marks] A survey of 1000 Montrealers and 1000 Torontonians revealed that 34% of the Montrealers were hockey fans, while only 29% of the Torontonians were hockey fans. Is there a significant difference. Use a level of significance of 0.05. Ans. z0 = 2.41 > z c = 1.96 Yes, significant diff. QUESTION 10 [8 Marks] A candidate for an office believes that he will receive at least 40% of the votes. An advance poll of 2400 voters indicated that 918 will vote for the candidate. Does this poll support the belief? Use a 0.01 level of significance to test the belief. Ans. z0 = 1.75 < z c = 1.645 does not support. QUESTION 11 [8 Marks] The number of points scored by half for a random sample of 10 games by a college basketball team is recorded below. 1 st half 29 26 25 25 25 24 26 26 30 31 2 nd half 25 26 25 25 24 23 27 25 29 30 Use the 0.05 level of significance to test there is a significant difference in scoring between halves. Ans. t0 = 1.92 > t c = 2.262. Not significant diff.
5 ANSWER EITHER 12 OR 13. QUESTION 12 [8 Marks] Suppose that we want to investigate whether there is a relationship between the test scores of persons who have gone through a certain job training program and their subsequent performance on the job. A random sample of 400 cases yielded the following results. Performance Poor Fair Good Below average 67 64 25 Test Average 42 76 56 Scores Above average 10 23 37 Test at the 0.01 level of significance whether on the job performance of persons who have gone through the training program is independent of their test score. 2 2 Ans. x0 = 41.013 > x ( df = 4) = 13.277 depends on test score. QUESTION 13 [8 Marks] In the past an NHL player scored 30% of his goals in January, 20% in each of December and March, and 10% in each of October, November and February. Last year he scored 50 goals as follows: MONTH OCT. NOV. DEC. JAN. FEB. MAR. Qi 6 6 6 20 7 5 Was there a significant change in scoring distribution last year? Use α = 0.10. 2 2 Ans. x0 = 6.97 > x c = 9.23636. Not significant change.