251y0431d 5 03 04 ECO251 QBA1 THIRD HOUR EXAM Apr 20 2004 Name KEY Student Number Class Time Circle 1pm 2pm Part I 10 points IQs are supposedly Normally distributed with a population mean of 100 and a standard deviation of 16 x N 100 16 Find the following Make diagrams Add a vertical line at zero to the diagrams below 78 4 100 P x 78 4 P z 16 P z 1 35 P 1 35 z 0 P z 0 4115 5 9115 Colors in diagram are reversed 0 4 0 3 f 1 0 2 0 1 0 0 4 3 2 1 0 1 2 3 4 1 2 3 4 1 2 3 4 x 0 4 2 f 121 6 100 78 4 100 P 78 4 x 121 6 P z 16 16 P 1 35 z 1 35 2 P 0 z 1 35 2 4115 8230 0 3 0 2 0 1 0 0 4 3 2 1 0 x P 47 52 x 84 2 84 2 100 47 52 100 P z 16 16 P 3 28 z 0 99 P 3 28 z 0 P 0 99 z 0 4995 3389 1606 The value for 3 28 comes from the lower part of the Normal table 3 0 4 f 0 3 0 2 0 1 0 0 4 3 2 1 0 x 251y0431d 4 20 04 4 P 111 04 x 121 6 0 4 0 3 f 121 6 100 111 04 100 P z 16 16 P 0 69 z 1 35 P 0 z 1 35 P 0 z 0 69 4155 2459 1566 0 2 0 1 0 0 4 3 2 1 0 1 2 3 4 x 5 To get into Mensa you must be in the top 2 0f the population What IQ do you need Hint find x 02 the 98th percentile Make a diagram Show a Normal curve with a mean at zero Label the area below zero 50 By definition z 02 is the point with a probability of 2 above it which means it must have 98 below it and 98 50 48 between it and zero We can try to find a point of the Normal table with P 0 z z 02 4800 The closest we can come to 4800 is P 0 z 2 05 4798 or P 0 z 2 06 4803 Either is acceptable but a good compromise might be z 02 2 054 So if you use the formula x z your answer could be x 02 100 2 05 16 132 8 x 02 100 2 054 16 132 9 or x 02 100 2 06 16 133 0 Check 132 9 100 P x 132 9 P z P z 2 06 16 P z 0 P 0 z 2 06 5 4803 0197 02 2 251y0431 4 20 04 Part II 20 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 In open ended questions it is expected Please indicate clearly what sections of the problem you are answering and what formulas you are using Neatness counts 1 Thirty six of the staff of 80 teachers at a local intermediate school are certified in CardioPulmonary Resuscitation CPR In 180 days of school about how many days can we expect that the teacher on bus duty will likely be certified in CPR a 5 days b 45 days c 65 days d 81 days 36 Explanation If you want to get technical this is a Binomial distribution with p 80 45 and n 180 So np 45 180 81 2 What type of probability distribution will most likely be used to analyze the number of cars with defective radios in the following problem From an inventory of 48 new cars being shipped to local dealerships corporate reports indicate that 12 have defective radios installed The sales manager of one dealership wants to predict the probability that out of the 8 new cars it just received when each is tested no more than 2 of the cars have defective radios a binomial distribution b Poisson distribution c hypergeometric distribution d none of the above Explanation This is hypergeometric and not binomial because there are only 12 cars with defective radios On the first pick the probability of getting a defective radio is p 12 48 but on the second pick p 11 47 or 12 47 Without a constant value for p we cannot have a binomial distribution A company has 125 personal computers The probability that any one of them will require repair on a given day is 0 025 To find the probability that exactly 20 of the computers will require repair on a given day one will use what type of probability distribution a binomial distribution b Poisson distribution c hypergeometric distribution d none of the above Explanation This is binomial and not hypergeometric because the probability that any pc needs repair is a constant 025 Though binomial is the correct answer the binomial n 125 distribution can be approximated by the Poisson because p 025 5000 500 3 4 The probability that a particular type of smoke alarm will function properly and sound an alarm in the presence of smoke is 0 8 You have 2 such alarms in your home and they operate independently The probability that both sound an alarm in the presence of smoke is 64 Explanation If you want to get technical this is a Binomial distribution with p 8 and n 2 So P x C xn p x q n x and P 2 C 22 8 2 2 0 64 Alternatively let A1 be First alarm works and A2 be Second alarm works If the events are independent P A1 A2 P A1 P A2 8 2 64 3 251y0431 4 20 04 5 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 at least 1 power outage in a year is 0 9975 Explanation P x 1 1 P 0 1 00248 99752 6 An Undergraduate Study Committee of 6 members at a major university is to be formed from a pool of faculty of 18 men and 6 women If the committee members are chosen randomly what is the probability that all of the members will be men 3 Solution This is hypergeometric because we are taking a sample of 6 that is more than 5 of a population of 18 6 24 The formula is P x C nN xM C xM C nN which gives the probability of x 6 successes in a sample of n 6 taken from a population of N 24 in which there are M 18 successes 18 C 06 C 618 12 6 18 17 16 15 14 13 P 6 1379 but you could also 24 24 24 23 22 21 20 19 …
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