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q2solutions - Statistics 20 Quiz 2 Solutions 1 The variance...

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Statistics 20: Quiz 2 Solutions September 27, 2006 1. The variance of data X is defined as V ar ( X ) = [ SD ( X )] 2 . Starting from this definition, prove that V ar ( X ) = 1 n n i =1 X 2 i - ( ¯ X ) 2 . ( Hint: Start by writing V ar ( X ) in terms of the standard deviation, then expand the sums to get the result. Going in the other direction is more difficult.) V ar ( X ) = [ SD ( X )] 2 = 1 n n i =1 ( X i - ¯ X ) 2 2 = 1 n n i =1 ( X i - ¯ X ) 2 = 1 n (∑ n i =1 X 2 i ) - 2 (∑ n i =1 X i ¯ X ) + (∑ n i =1 ( ¯ X ) 2 ) = 1 n n i =1 X 2 i - 2 ¯ X 1 n n i =1 X i + n n ( ¯ X ) 2 = 1 n n i =1 X 2 i - 2 ¯ X ( ¯ X ) + ( ¯ X ) 2 = 1 n n i =1 X 2 i - ( ¯ X ) 2 . 2. Suppose the university conducted a study attempting to predict Y , a student’s first year GPA, based upon X , the student’s SAT math score. Students at the university had an average SAT math score of 500 with a standard deviation of 100, and the average first year GPA is 2.5 with a standard deviation of 0.5. A scatter plot of the data is football shaped. The correlation of X and Y is r = 0 . 8. A particular student chosen at random has an SAT math score of 800.
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