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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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