# But give 8 points for more intuitive explanations on

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BUT GIVE 8 POINTS FOR MORE “INTUITIVE” EXPLANATIONS ON THIS PROBLEM.] a) If independent, then C is the “X” variable in the regression.
Now, if we let * denote the betahat value after the transformation, * 1 1 2 2 (10 (10 ))( ) ( ))( ) ˆ ˆ (10 (10 )) ( ) i i i i i i X X Y Y X X Y Y X X X X   The slope is equal but of opposite sign as the original. Makes sense since all we did was turn the X variable on its head, so the relationship should be the same but backwards. For this problem, that means the slope is .000177. The intercept will be * * 0 1 1 1 1 0 1 ˆ ˆ ˆ ˆ ˆ ˆ ˆ (10 ) (10 ) 10 10 Y X Y X Y X , which here is -9780.89. This change is a little tricky to decipher. Essentially the intercept term is shifting to account for the flip in the sign of the regression line. The direction of the shift will depend on whether beta1hat is positive or negative and its size. b) If instead, corruption was the “Y” variable, the effect on the slope is exactly the same as above.
2) (20 points) The last lessons have spent a lot of time describing the slope and intercept terms (and their variances) of the one- variable sample regression function. We also know that for any particular value of the independent variable (call it X 0 ), that the predicted value of Y 0 is 0 0 1 0 ˆ ˆ ˆ Y X . (This is sometimes called a “point prediction.”) If we haven’t covered it yet in class, take as given that the estimators 0 1 ˆ ˆ and are unbiased. a) (10) Prove that 0 ˆ Y is an unbiased estimator of Y 0 . 0 0 1 0 0 1 0 0 1 0 ˆ ˆ ˆ ˆ ˆ [ ] [ ] [ ] [ ] E Y E X E E X X