ST371 Final Exam Practice Name Solutions The following are some additional practice problems to help you prepare for the midterm exam Note this is not a practice exam and this does not necessarily represent an exhaustive list of the concepts we have covered 1 The Batmobile is to be kept in pristine condition Part of this upkeep is making sure the headlights since Batman does the lion share of his work at night are operating at appropriate intensity The Gotham City legislature has mandated that all vehicle headlights must operate between 2 and 3 kilocandelas Batman likes to keep his headlights operating at 2700 candelas of intensity If we let X represent the intensity of the left headlight and Y be the intensity of the right headlight both in kilocandelas then X and Y have the following joint pdf f x y 38 x2 y2 0 2 x 3 2 y 3 otherwise a What is the covariance between the light intensity of the left and right headlights cid 40 3 E XY xy 3 38 x2 y2 dxdy x3y xy3dxdy y y3dy 5 2 cid 90 3 cid 90 3 cid 90 3 cid 90 3 cid 90 3 2 2 2 65 4 2 3 38 3 38 975 152 2 cid 90 3 cid 90 3 x 3 38 2 3 38 385 152 EX x2 dx 19 3 x3 xdx 19 3 2 EX 2 x2 3 38 x2 dx 19 3 x4 x2dx 19 3 cid 90 3 cid 90 3 2 3 38 2 1852 285 From the last homework assignment we have fX x 3 38 x2 19 3 and fY y 3 38 y2 19 3 Thus Similarly EY 385 152 since the pdf is the same Then Cov X Y E XY E X E Y 975 152 385 152 385 152 0010821 b What is the correlation between the light intensity of the left and right headlights 285 cid 16 385 152 cid 17 2 So V ar X 1852 0826899 Similarly V ar Y 0826899 Then Corr X Y Cov X Y X Y 0010821 0826899 0826899 013086 c What is the strength of the relationship between the two light intensities How do you know We would classify the strength of linear relationship between the light intensities as very weak because the correlation value is so close to zero 2 Based on current literature the average incubation period for Covid 19 is 8 days with a standard deviation of 3 days a If we were to take a random sample of 45 patients that were infected with Covid 19 what is the distribution of the sample mean Total If we take a random sample of 45 patients then the Central Limit Theorem would allow us to approximate the distribution of the sample average That is X N 2 n Using the values given in the problem we arrive at X N 8 32 45 Similarly we would have T0 N 45 8 45 32 b What is the approximate probability that the sample average incubation period is between 7 5 and 9 days days P 7 5 X 9 P Z 7 5 8 45 3 9 8 45 3 P 1 1180 Z 2 2361 normalcdf 1 1180 2 2361 0 1 8555 P T0 380 P Z 380 45 8 45 32 P Z 99380799 normalcdf 99380799 0 1 16016 c What is the approximate probability that the total incubation time for all 45 patients exceeds 380 d What effect does the sample size of 45 have on this problem What would happen if we increased this sample size to 500 What would happen if we decrease this sample size to 10 The sample size of 45 allows us to accurately invoke the Central Limit Theorem Note that there is nothing in the central limit theorem that pairs its use with a sample size So theoretically we can use it regardless of sample size But the accuracy of the result will be heavily affected by the choice of sample size Since 45 is larger than the target value of 30 we are confident that the approximation made by the CLT is good If we increased the sample size to 500 our approximation would become In fact at this point with a sample size that large the true distribution of X is even better indistinguishable from the normal If we decreased the sample size to 10 the Central Limit Theorem would lose almost all of its approximation power Meaning any approximating of probabilities we did with the CLT would be relatively useless 3 Suppose your waiting time in line at Starbucks for your morning coffee is uniformly distributed on the interval 0 5 minutes and your waiting time in line at Chick fil A for your evening lemonade is uniformly distributed on the interval 0 9 a Is it reasonable to treat these wait times as independent of one another Explain My wait time at one restaurant should not affect the wait time at another Thus it is reasonable to consider these wait times as independent b If you go to Starbucks each morning and Chick fil A each evening for a full week what is your total expected wait time Hint you might want to look up Chick fil A s schedule Let X1 X7 be my Starbucks wait times for Monday Sunday and Y1 Y6 be my wait times for Monday Saturday If we let T be the total wait time then T X1 X7 Y1 Y6 Then E T E X1 X7 Y1 Y6 EX1 EX2 EX7 EY1 EY6 b c independence 5 2 7 5 2 9 2 5 2 6 9 2 89 2 9 2 44 5 c What is the variance of your total waiting time Var T Var X1 X7 Y1 Y6 Var X1 Var X7 Var Y1 Var Y6 b c independence 7 25 12 6 81 12 55 083 661 12 d If you randomly select a day what is the expected difference in wait times The variance of the difference e What are the expected value and variance of the difference between total Starbucks wait time and total Chick fil A wait time for a randomly selected week E X1 Y1 EX1 EY1 5 2 Var X1 Y1 Var X1 Var Y1 9 2 25 12 2 minutes 81 12 106 12 8 8333 E T1 T2 E cid 0 X1 X7 Y1 Y6 cid 1 7 5 2 6 9 2 19 2 9 5 minutes Var T1 T2 Var T1 Var T2 Var X1 X7 Var Y1 Y6 Var X1 Var X7 Var Y1 Var Y6 55 083 from c 4 There are many measures of center in addition to the mean and median There is the trimmed mean which trims off a certain percentage of the ends of a data set before calculating the the arithmetic mean There is the mid range which is the average of the smallest and largest observations and there is the mid fourth which is the average of the 25th and 75th percentiles Which of these five measures of …