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University of Toronto St. George

Statistical Sciences

STA302H1

Hadas Moshonov

Fall

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STA 302 H1F / 1001 HF { Fall 2007
Test 1
October 24, 2007
LAST NAME: SOLUTIONS FIRST NAME:
STUDENT NUMBER:
ENROLLED IN: (circle one) STA 302 STA 1001
INSTRUCTIONS:
▯ Time: 90 minutes
▯ Aids allowed: calculator.
▯ A table of values from the t distribution is on the last page (page 8).
▯ Total points: 50
Some formulae:
P P
(X X)(Y ▯Y ) X Y ▯nXY
b1= P i i = P ii 2 b0= Y ▯ b1X
(Xi▯X)2 X inX
▯ ▯
▯2 2 1 X2
Var(b1) =P 2 Var(b0) = ▯ n+ P 2
(Xi▯X) (Xi▯)
2 P
Cov(b ;b ) = ▯ P ▯ X SSTO = (Y ▯ Y )2
0 1 (Xi▯X)2 i
P ^ 2 2P 2 P ^ 2
SSE = (Yi▯ Yi) SSR = b 1 (X i X) = (Yi▯ Y )
▯ ▯ ▯ ▯
2 ^ ^ 2 1 P(Xh▯X) 2 2 ^ 2 1 PXh▯X) 2
▯ fY h = Var(Y h = ▯ n + (Xi▯X)2 ▯ fpredg = Var(Yh▯ Y h = ▯ 1 + n+ (Xi▯)2
P (X ▯X)(Y ▯Y ) p
r = p P i P i Working-Hotelling coe▯cient: W = 2F 2;n▯2; ▯
(Xi▯X)2 (Yi▯Y )
1 2a 2bcdef 2ghi 2j 3
1 1. The following questions require derivations of results for the simple linear regression model.
(a) (2 marks) In lecture we showed that e = 0 andP n e X = 0. Given these results,
P i=1 i i=1 i i
what is i=1ei i? Justify your answer.
Xn X
^
ei i = ei(b0+ b1X i
i=1 i=1
X Xn
= b 0 ei+ b1 eiX i
i=1 i=1
= 0
(b) (5 marks) Show that the total Sum of Squares in a regression can be decomposed as
X ▯ ▯ Xn ▯ ▯
^ 2 ^ 2
Yi▯ Y + Yi▯ Yi
i=1 i=1
You may use any results that were derived in lecture.
Xn
SSTO = (Y ▯ Y )
i
i=1
Xn
^ ^ 2
= (Yi▯ Yi+ Yi▯ Y )
i=1
Xn Xn Xn
^ 2 ^ 2 ^ ^
= (Yi▯ Y ) + (Yi▯ Yi) + 2 (Yi▯ Y )(i ▯ i )
i=1 i=1 i=1
and
Xn X
(Yi▯ Y )(i ▯ i ) = (Yi▯ Y )i
i=1 i=1
X Xn
= i i▯ Y ei= 0
i=1 i=1
▯ ▯ ▯ ▯
Pn ^ 2 Pn ^ 2
So SSTO = i=1 Y i Y + i=1 Y i Y i
(c) (5 marks) Assume that the X are non-random. Derive the formula for Cov(b ;b ) given
i 0 1
on the ▯rst page. You may use any other formulae from the ▯rst page that you require
except the formulae whose derivations require knowing0Co1(b ;b ).
(Hint: You may want to start with the formula for the estimated intercept.)
From the formula for0b := b0+ b1X
So
2
Var(Y ) = Var(b 0 + X Var(b 1 + 2XCov(b 0b1)
2 !
▯2 2 1 X 2 ▯2
= ▯ + + X + 2XCov(b 0b 1
n n SXX SXX
▯ X P n 2
Rearranging gives Cov0b1;b ) S ▯ where XX = i=1X i X)
XX
2 2. The SAS output that follows was produced to examine the relationship between full-scale
IQ (FSIQ) and brain size as measured by MRI (MRIcount). Measurements were taken on 20
university students chosen because their full-scale IQ was at least 130.
The REG Procedure
Descriptive Statistics
Uncorrected Standard
Variable Sum Mean SS Variance Deviation
Intercept 20.00000 1.00000 20.00000 0 0
MRIcount 18518961 925948 1.725648E13 5730703420 75701
FSIQ 2728.00000 136.40000 372396 15.62105 3.95235
Dependent Variable: FSIQ
Analysis of Variance
Sum of Mean
Source DF Squares Square F Value Pr > F
Model 1 89.22306 89.22306 7.74 0.0123
Error 18 207.57694 11.53205
Corrected Total 19 296.80000
Root MSE 3.39589 R-Square 0.3006
Dependent Mean 136.40000 Adj R-Sq 0.2618
Coeff Var 2.48965
Parameter Estimates
Parameter

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