252y0313 9 29 03 ECO252 QBA2 FIRST HOUR EXAM October 2 2003 Name KEY Hour of class registered Show your work Make Diagrams I 8 points Do all the following Normal distribution problems Before you take another exam please read Things that you should never do on an exam or anywhere else in your syllabus supplement 1 The owner of a fish market determined that the average weight for a catfish is 3 2 pounds with a standard deviation of 0 8 pounds Assuming the weights of catfish are normally distributed the probability that a randomly selected catfish will weigh more than 4 4 pounds is the x N 3 2 0 8 Make a diagram Either show a Normal curve with a mean at zero and shade area above 1 5 or show a Normal curve with a mean at 3 2 and shade the area above 4 4 4 4 3 2 P x 4 4 P z P z 1 5 P z 0 P 0 z 1 5 0 8 5 4332 0668 2 The owner of a fish market determined that the average weight for a catfish is 3 2 pounds with a standard deviation of 0 8 pounds Assuming the weights of catfish are normally distributed the probability that a randomly selected catfish will weigh between 3 and 5 pounds is the x N 3 2 0 8 Make a diagram Either show a Normal curve with a mean at zero and shade area between 0 25 and 1 00 or show a Normal curve with a mean at 3 2 and shade the area between 3 and 5 5 3 2 3 3 2 P 3 x 5 P z P 0 25 z 1 00 0 8 0 8 P 0 25 z 0 P 0 z 2 25 0987 4878 5865 3 The owner of a fish market determined that the average weight for a catfish is 3 2 pounds with a standard deviation of 0 8 pounds A citation catfish should be one of the top 2 in weight Assuming the weights of catfish are normally distributed at what weight in pounds should the citation designation be established Find x 02 Make a diagram Draw a Normal curve with a mean at zero By definition z 02 is the point with 2 above it and 98 below it Since 50 is below zero it must be P 0 z z 02 4800 Show this on your diagram If you look at true that the Normal table the closest you can come to 4800 is P 0 z 2 05 4798 though P 0 z 2 06 4803 is almost as good So z 02 is 2 05 or 2 06 Use the formula x 02 z 02 3 2 2 05 0 8 3 2 1 64 4 84 or x 02 z 02 3 2 2 06 0 8 3 2 1 648 fine An exact answer should be between these two maybe 4 848 Both of these are 4 843 252y0313 9 29 03 4 The owner of a fish market determined that the average weight for a catfish is 3 2 pounds with a standard deviation of 0 8 pounds Assuming the weights of catfish are normally distributed the probability that a randomly selected catfish will weigh less than 2 2 pounds is the x N 3 2 0 8 Make a diagram Either show a Normal curve with a mean at zero and shade area below 1 25 or show a Normal curve with a mean at 3 2 and shade the area below 2 2 2 2 3 2 P x 2 2 F 2 2 P z P z 1 25 P z 0 P 1 25 z 0 0 8 5 3944 1056 2 252y0313 9 29 03 II 5 points 2 point penalty for not trying part a A random sample is taken of the number of automobiles involved in fog related accidents on the New Jersey Turnpike The following data is found Accident 1 2 3 4 5 No of cars 30 3 2 5 4 a Compute the sample standard deviation s of the number of vehicles involved Show your work 3 b A turnpike spokesperson says that the mean number of automobiles involved in this type of pile up is at most 6 6 Use the mean and standard deviation that you computed above to evaluate this statement at the 95 confidence level 3 c Extra credit A turnpike spokesperson says that the median number of automobiles in one of these accidents lies between 3 and 5 Treating this as a confidence interval for the median what is the confidence level What about these data makes me suspect that locating the median is a better idea than the hypothesis test in b 4 Solution a x x2 index x in order x 44 x 8 80 30 900 1 2 n 5 3 9 2 3 x 2 nx 2 2 4 3 4 s 5 25 4 5 n 1 4 16 5 900 954 5 8 80 2 44 954 141 70 2 4 So x 44 x 954 n 5 s 141 70 11 904 b H 0 6 H 1 6 From the formula table Interval for Confidence Interval Mean x t 2 s x unknown DF n 1 H 0 0 H1 0 Test Ratio t x 0 sx Critical Value x cv t 2 s x 141 70 28 34 5 3235 5 n You have not done a hypothesis test unless you have stated your hypotheses run the numbers and stated your conclusion You must do only one of the following Note that s x s Hypotheses i Test ratio for a test of the mean The test ratio is t calc x 0 8 80 6 0 5260 Since this is sx 5 3235 4 2 132 The reject zone is the area above 2 132 Since a one sided right tail test pick t n 1 t 05 0 526 is not in the reject zone do not reject the null hypothesis 3 252y0313 9 29 03 ii A critical value for the sample mean Because the alternate hypothesis is 6 we need a critical value for x above 6 Since this is a one sided test we use t n 1 t 4 2 132 The two sided n 1 formula x cv 0 t 2 05 s x becomes x cv 0 t n 1 s x 6 2 132 5 3235 6 11 350 17 350 Make a diagram showing an almost Normal curve with a mean at 6 and a shaded reject zone above 17 350 Since x 8 80 is not above 17 350 we do not reject H 0 iii A confidence interval for the population mean Because the alternate hypothesis is 6 we need a confidence interval x cv t 2 s x becomes x cv t s x 8 80 2 132 5 3235 8 8 11 350 or 2 550 Make a diagram showing an almost Normal curve with a mean …
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