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SolutionsToReviewProblems

SolutionsToReviewProblems - Solutions to the Review...

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Solutions to the Review Problems Problem 1 Let { X i } N i =1 be the list of original numbers. We know that ¯ X 1 N N i X i = 50 and s 2 x 1 N - 1 N i ( X i ¯ X ) 2 = 10 (a) Let Y i = X i + 20 ¯ Y 1 N N summationdisplay i Y i = 1 N N summationdisplay i ( X i + 20) = 1 N N summationdisplay i X i + 1 N N summationdisplay i 20 = ¯ X + 20 s 2 y 1 N 1 N summationdisplay i [ Y i ¯ Y ] 2 = 1 N 1 N summationdisplay i [( X i + 20) ( ¯ X + 20)] 2 = 1 N 1 N summationdisplay i [ X i ¯ X ] 2 = s 2 x Since the variances of x and y are the same, the standard deviations of x and y are also the same. (b) Adding a consant does not change the variance or standard deviation, which are the measures of how spread the observations are. (c) Let Z i = 2 X i ¯ Z 1 N N summationdisplay i Z i = 1 N N summationdisplay i 2 X i = 2 1 N N summationdisplay i X i = 2 ¯ X 1

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s 2 z 1 N 1 N summationdisplay i [ Z i ¯ Z ] 2 = 1 N 1 N summationdisplay i [2 X i 2 ¯ X ] 2 = 1 N 1 N summationdisplay i 4[ X i ¯ X ] 2 = 4 1 N 1 N
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SolutionsToReviewProblems - Solutions to the Review...

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