251y0321 4 18 03 ECO251 QBA1 THIRD HOUR EXAM April 21 2003 Name KEY Social Security Number Part I 30 points Do all the following All questions are 2 points each except as marked Exam is normed on 50 points including take home Showing your work can give partial credit on some problems 1 Engineers designed a monorail so that the amount of time that people have to wait at airport concourse B has a continuous uniform distribution ranging from 0 to 5 minutes The mean waiting time for some one who just arrived at concourse B is a 2 5 minutes b 3 minutes c 4 minutes d 5 minutes 2 A recent study on teen age pregnancy indicates that 5 of the female population will get pregnant during their teenage years and that teenage pregnancies occur independently of one another Suppose that ten female teen agers are selected at random Find the approximate probability that at least two get pregnant some time in their teenage years a 914 b 988 c 012 d 086 Explanation This is a binomial problem with n 10 and p 05 P x 2 1 P x 1 1 91386 08614 3 Assume that x is a random variable with a mean of 200 and a standard deviation of 2 Assume that u 2 x 3 The standard deviation of u is a b c d Explanation If 1 4 7 4 u ax b Var u a 2Var x So here a 2 and Var u 2 2 2 2 16 So the standard deviation is 16 4 4 Let x and y represent the quantity of two goods that you sell and assume that the correlation between their sales is xy 6 Assume that the first good sells for 0 90 so that the amount you make on it is u 90 x and the second good sells for 0 80 so that the amount that you make on it is v 80 y What is the correlation between u and v 6 Explanation If u ax b and v cy d Corr u v Sign ac Corr x y Since a 90 and c 80 b d 0 Sign ac and Corr u v Sign ac 6 6 251y0321 4 18 03 5 What type of probability distribution will most likely be used to analyze the number of chocolate chip parts per cookie in the following problem The quality control manager of Marilyn s Cookies is inspecting a batch of chocolate chip cookies When the production process is in control the average number of chocolate chip parts per cookie is 6 0 The manager is interested in analyzing the probability that any particular cookie being inspected has fewer than 5 0 chip parts a binomial distribution b Poisson distribution c normal distribution d Hypergeometric distribution Explanation Probability of the number of successes when you know the mean 6 The number of power outages at a nuclear power plant has a Poisson distribution with a mean of 6 outages per year The probability that there will be between 1 and 3 power outages in a year is 14872 Explanation P 1 x 3 P x 3 P x 0 15120 00248 14872 I went through this several times in class and it s in the outline Is the problem simply that you didn t believe me 7 A debate team of 4 members for a high school will be chosen randomly from a potential group of 15 students Ten of the 15 students have no prior competition experience while the others have some degree of experience What is the probability that at most 1 of the members chosen for the team have some prior competition experience Solution Hypergeometric N 15 M 5 have experience and n 4 P x 1 P 0 P 1 P 0 C 5 C 10 0 154 C4 P 1 C15 C 310 C 415 P x 10 6 4 1 C xM C nN xM C nN C rn n n r r 10 9 8 7 4 3 2 1 10 9 8 7 15385 15 14 13 12 15 14 13 12 4 3 2 1 15 11 4 5 10 4 1 7 3 10 9 8 5 10 9 8 4 3 2 1 43956 15 14 13 12 15 14 13 12 4 3 2 1 5 15 11 4 P x 1 P 0 P 1 15385 43956 59341 2 251y0321 4 18 03 TABLE 13 2 A candy bar manufacturer is interested in trying to estimate how sales are influenced by the price of their product To do this the company randomly chooses 6 small cities and offers the candy bar at different prices Note that there are no probabilities here please don t invent them City Price Sales River Falls 1 30 100 Row price sales Hudson 1 60 90 y x y2 Ellsworth 1 80 90 x2 xy Prescott 2 00 40 Rock Elm 2 40 38 1 1 3 100 1 69 10000 130 0 2 1 6 90 2 56 8100 144 0 Stillwater 2 90 32 Note For price x 2 25 66 3 4 5 6 x 12 x 2 and s x2 0 3320 1 8 2 0 2 4 2 9 12 0 90 40 38 32 390 3 24 4 00 5 76 8 41 25 66 8100 1600 1444 1024 30268 162 0 80 0 91 2 92 8 700 0 8 Compute a sample variance for sales 3 Answer 983 6 see below 9 Compute a sample correlation between sales and price 3 Answer 8854 see below Computations 2 Given x 12 x 25 66 x 2 and s x2 0 3320 y 390 y So x 2 s x2 y 2 s 2y nx 2 n 1 was done for you s y ny 2 n 1 2 30268 rxy xy nx y n 1 s xy sx s y and y 390 65 6 s 25 66 6 2 2 25 66 24 0 3320 5 5 30268 6 65 2 30268 25350 983 6 5 5 983 60 31 3624 s xy xy 700 x 0 3320 5762 This 700 6 2 65 700 780 16 00 5 5 16 00 0 3220 983 6 16 2 0 3320 983 6 78394 8854 10 Suppose z has a standard normal distribution with a mean of 0 and standard deviation of 1 Make a diagram The probability that z is less than 2 20 is 0139 Explanation Make a diagram of a Normal curve with zero in the middle Shade the area below 2 20 P z 2 2 P z 0 P 2 20 z 0 5 4861 0139 11 Suppose z has a standard normal distribution with a mean of 0 and standard deviation of 1 Make a diagram The probability that z is less than 2 20 is 9861 Explanation Make a diagram of a Normal curve with zero in the middle Shade the area below 2 20 This will be both the area below zero and the area between zero and …
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