2011_Midterm 1_soln

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

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Unformatted text preview: STAT 410/510 F-2011 Midterm I NAME: ________________________________________ 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) (̂ ) (∑ ) (∑ ⏟ ∑ ( ) )∑ ∑ ∑ () ( ) CI for c. Find a Is this interval narrower than the estimator for the case where both slope and intercept are unknown? Why? (8 points) Answer) Since (̂ ) ∑ both are unknown. (uncertainties). and ̂ , so we get ̂ (̂ ) ( )√ ∑ , which is narrower than when Intuitively, the CI will be wider if there exist more unknown parameters 2. Consider the simple linear regression model uncorrelated. Show that ̅ . (30 points total) (̂ ̂ ) with () () and Answer) (̂ ̂) ̂ ̅ ̂) (̅ ⏟ (̅ ̂ ) () ∑( ) ˆ where 1 k Y , where k ii i ⏟ (̂ ̅ ̂ ) ∑ ⏟ ̅ (̂ ̂ ) ̅ ̅ (̂ ) Xi X X X 2 i 3. A person’s muscle mass is expected to decrease with age. To explore this relationship in women, a nutritionist randomly selected 15 women from each 10-year age group beginning with age 40 and ending with age 79. The response variable is a measure of muscle mass. The independent variable is age. (40 points total) DF 1 58 59 Sum of Squares 11627 3874.44750 15502 Root MSE Dependent Mean Coeff Var 8.17318 84.96667 9.61927 Source Model Error Corrected Total Parameter Mean Square 11627 66.80082 R-Square Adj R-Sq F Value 174.06 Pr > F <.0001 0.7501 0.7458 Standard 1 STAT 410/510 F-2011 Midterm I Variable Intercept age DF 1 1 NAME: ________________________________________ Estimate 156.34656 -1.19000 Error 5.51226 0.09020 Variable: Mean 59.98333 Median 60.00000 Mode 78.00000 Interquartile Range t Value 28.36 -13.19 Pr > |t| <.0001 <.0001 age Std Deviation Variance Range 20.50000 11.79700 139.16921 37.00000 Analysis Variable : xdevsq Sum = 8210.98 a. Obtain a point estimate of the mean muscle mass for women aged 60 years. (3 points) Answer) ̂ . b. Obtain a point estimate of . (3 points) Answer) MSE = 66.80082 c. Construct a test to decide whether or not there is a negative linear association between amount of muscle mass and age. Control the risk of Type I error at .05. State the alternatives, decision rule and conclusion. What is the P-value of the test? (5 points) Answer) and t(.05; 58) = −1.67155 (̂ ) If ≥ −1.67155 conclude H0, otherwise Ha. Conclude Ha. P-value= 0+ d. Estimate with a 95% CI the difference in expected muscle mass for women whose ages differ by one year. (5 points) Answer) t(.975; 58) = 2.00172, −1.19 ± 2.00172(.090197), −1.3705 ≤ β1 ≤ −1.0095 e. Obtain a 95% CI for the mean muscle mass for women of age 60. (5 points) Answer) ̂ , ̅) ( ( ) [ [ (̂ ) (̂ ) ̅) ∑( t(.975; 58) = 2.00172, 84.9468 ± 2.00172(1.05515), 82.835 ≤ E{Yh} ≤ 87.059 f. Obtain a 95% prediction interval for the muscle mass for women of age 60. (5 points) Answer) ( ) [ (̂ ) (̂ ) 84.9468 ± 2.00172(8.24101), 68.451 ≤ Yh(new) ≤ 101.443 g. Determine the boundary values of the 95% confidence band for the regression line for women of age 60. (5 points) Answer) W 2 2F2, n 2 (1 ) 2F2,58 (.95) 2 * 3.15593 6.31186 W 2.512342 84.9468±2.512342(1.05515), 82.296 ≤ β0 + β1Xh ≤ 87.598 h. Obtain the correlation coefficient . (4 points) Answer) R2 = 0.750067, r = −0.866064 2 STAT 410/510 F-2011 Midterm I NAME: ________________________________________ i. Test whether muscle mass and age are statistically independent in the population; use 5% significance level. State the alternatives, decision rule, and conclusion. (5 points) Answer) 3 ...
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This document was uploaded on 12/14/2011.

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