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2011_Midterm 1_soln - STAT 410/510 F-2011 Midterm I NAME 1...

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STAT 410/510 F-2011 Midterm I NAME: ________________________________________ 1 1. Consider the simple linear regression model where the intercept is known. (30 points total) a. Find the LS estimator of for this model. (9 points) Answer) ( ) ∑( ) where is known. After taking the derivative of ( ) w. r. t. , ( ̂ ) ̂ ( ) ̂ ( ) b. What is the variance of the slope ̂ for the LS estimator found in part a? (8 points) Answer) ( ̂ ) (∑ ) (∑ ) ( ) ( ) c. Find a ( ) CI for Is this interval narrower than the estimator for the case where both slope and intercept are unknown? Why? (8 points) Answer) Since ( ̂ ) and ̂ ( ̂ ) , so we get ̂ ( ) , which is narrower than when both are unknown. Intuitively, the CI will be wider if there exist more unknown parameters (uncertainties). 2. Consider the simple linear regression model with ( ) ( ) and uncorrelated. Show that ( ̂ ̂ ) ̅ . (30 points total) Answer) ( ̂ ̂ ) ( ̅ ̂ ̅ ̂ ) ( ̅ ̂ ) ∑( ) ( ) ( ̂ ̅ ̂ ) ̅ ( ̂ ̂ ) ̅ ( ̂ ) ̅ where , ˆ 1 i i Y k where 2 X X X X k i i i 3. A person’s muscle mass is expected to decrease with age.
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