251y0231 12 02 02 ECO251 QBA1 THIRD EXAM NOVEMBER 20 2001 Name Key Open this document in Print Layout view Section Enrolled Circle MWF 10 MWF 11 TR 11 Since there is no correct answer for the Take home exam a possible solution appears below Many errors you made are in Things that you should never do on a Statistics Exam or Anyplace Else Wake up and read it 251x023a 11 16 01 ECO251 QBA1 THIRD EXAM NOVEMBER 22 23 2002 TAKE HOME SECTION Name Seymour Butz Section Enrolled Circle MWF 10 MWF 11 TR 11 TR 12 30 Social Security Number 234567891 Throughout this exam show your work Please indicate clearly what sections of the problem you are answering If you are following a rule like E ax aE x please state it If you are using a formula state it If you answer a yes or no question explain why Part I Do all the Following 10 Points Show your work My Social Security Number is 265398248 If I write it as below so that for example x3 5 Minitab tells me that the sample mean is 5 222 and the sample standard deviation is 2 682 Please don t do these again Write your Social Security next to it in the same way and call it y x 1 2 3 4 5 6 7 8 9 2 6 5 3 9 8 2 4 8 Compute the following showing the steps clearly as if you had done it by hand Do not tell me That is what my calculator said though you are welcome to use your calculator Excel or Minitab to check your work 1 The sample variance of y 2 2 The sample covariance between x and y 2 3 The sample correlation between x and y 2 4 Interpret the correlation 1 Answers to questions 5 and 6 must be based on the mean standard deviation covariance and correlation that you found in questions 1 4 Do not recompute the answers after changing x or y 5 If all the numbers in y rise by 20 so that if y was 2 3 4 5 6 7 8 9 1 it is now 22 23 24 25 26 27 28 29 21 what will the new mean and standard deviation of y be What will be the correlation between x and y be now 1 5 6 If instead all the numbers in y are multiplied by 3 5 so that if y was 2 3 4 5 6 7 8 9 1 it is now 7 10 5 14 17 5 21 24 5 28 31 5 3 5 what will the new mean and standard deviation of y be What will be the correlation between x and y be now 1 5 251y0231 11 19 02 Worksheet obs x 1 2 3 4 5 6 7 8 9 x2 y y2 xy x 2 4 2 4 4 6 36 3 9 18 5 25 4 16 20 3 9 5 25 15 9 81 6 36 54 8 64 7 49 56 2 4 8 64 16 4 16 9 81 36 8 64 1 1 8 47 303 45 265 227 x 47 5 22222 n 9 2 x nx 2 303 9 5 22222 s n 1 8 7 19444 s x 2 68225 2 x y y 45 5 00000 n 9 y ny 2 285 9 5 00000 n 1 8 7 50000 s y 2 73861 2 s y2 rxy xy xy 227 227 9 5 22222 5 00000 8 1 00000 8 n 1 8 s xy 1 00000 0 136135 sx sy 7 19444 7 50000 nx y 2 n 9 x 47 x 2 303 y 45 y 2 285 and Note that the x and x 2 columns and their sums were not needed s xy 2 so Solution From above 1 The sample variance of y s 2y 7 50000 2 The sample covariance between x and y s xy 1 00000 3 The sample correlation between x and y rxy 0 136135 4 Interpretation The negative sign of the covariance and correlation indicates that x and y tend to move in opposite directions The correlation squared is about 019 On a zero to one scale this is extremely weak 5 If all the numbers in y rise by 20 what will the new mean and standard deviation of y be What will be the correlation between x and y be now E cy d cE y d means that E 1 y 20 1E y 20 works for sample means too so that the sample mean rises by 20 to 25 0000 And Var cy d c 2Var y so Var y 20 1 Var y 1 7 50000 7 50000 The standard deviation remains 2 73861 Also Cov ax b cy d acCov x y so Cov x 0 y 20 1 1 Cov x y 1 1 1 00000 1 00000 2 2 and if w ax b and v cy d wv sign ac xy In this case w 1x 0 and v 1 y 20 sign ac sign 1 1 and we started with a 2 251y0231 11 19 02 correlation of rxy 0 136135 so the correlation between the two is 0 136135 In short adding 20 to y has no effect 6 If instead all the numbers in y are multiplied by 3 5 what will the new mean and standard deviation of y be What will be the correlation between x and y be now 1 5 E cy d cE y d means that E 3 5 y 0 3 5 E y 0 works for sample means too so that the sample mean which was 5 00000 is multiplied by 3 5 to become 17 5 And Var cy d c 2Var y so if our original standard deviation was s y2 7 50000 we find Var 3 5 y 0 3 5 2 Var y 3 5 2 7 50000 19 75 Also Cov ax b cy d acCov x y and s xy 1 00000 so Cov x 0 3 5 y 0 1 3 5 Cov x y 1 3 5 1 00000 3 5000 and if w ax b and v cy d wv sign ac xy sign ac sign 1 3 5 so the correlation between the two is unchanged at rxy 0 136135 Part II Take your Social Security number again let g 1 be the 2nd and 3rd digits of the number g 2 be the 2nd 3rd and 4th digits and g 3 2g1 and g 4 3g1 For example my Social Security number is 265398248 so g 1 65 g 2 653 g 3 2 65 130 g 4 3 g 1 3 65 195 A jorcillator s lifespan failure time can be represented by a continuous uniform distribution between g 1 and g 2 years My jorcillator has a lifespan between 65 and 653 years 1 What is the probability that it lasts between g 3 and g 4 years …
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