STA 6207 – Exam 1 – Fall 2009
PRINT Name _____________________
All Questions are based on the following 2 regression models, where SIMPLE REGRESSION refers to
the case where p=1, and X is of full column rank (no linear dependencies among predictor variables).
Model 1 : Yi   0  1 X i1     p X ip   i

i  1,..., n  i ~ NID 0,  2

Model 2 : Y  Xβ  ε X  n  p' β  p'1 ε ~ N 0,  2 I


d a' x 
d x' Ax 
a
 2Ax ( A symmetric)
E Y' AY   trAVY   μ Y ' Aμ Y
dx
dx
Cochran’s Theorem: Suppose Y is distributed as follows with nonsingular matrix V:
Y ~ N μ,  2 V
r V   n
the n if AV is idempotent :
Given:


1
 1 
1. Y'  2 A Y is distribute d non - central  2 with : (a) df  r ( A) and (b) Noncentral ity parameter :  
μ' Aμ
2 2


__________________________________________________________________________________________
1. For model 2, derive the least squares estimate for 
2
n
^ 
2. For model 2, obtain SS(Model)    Y i  and SS(Residua l)   ei2 in matrix form (quadratic forms in

i 1 
i 1
Y). Obtain the distributions of the two sum of squares (be specific with regard to their family of
distributions, degrees of freedom, and non-centrality parameters).
n
3. A firm has 2 types of expenditures that can varied in their marketing plan: advertising and in-store
promotion. A regression model is fit, relating Y=weekly sales to levels of these expense variables
(X1=advertising, X2=in-store promotion). The model fit is: E(Y) = X1+2X2. Set up the K’ matrix
and m vector for testing: (a) whether mean sales are 500 when no advertising or in-store promotion is
conducted, and (b) the effects of increasing X1 and X2 by 1 unit have the same effect on mean sales.
That is, H0A: 0=500 H0B: .
4. For simple regression with model 1, we get:
n 
n
^
X X
^

Yi and Y    1 Yi
 1    i
COV
  1 , Y   ???

S XX 


i 1 
i 1  n 
5. Use the following output to obtain the quantities given below:
X
1
1
1
1
1
1
0
5
10
0
5
10
P
0.5833
0.3333
0.0833
0.2500
0.0000
-0.2500
Y
4
6
9
7
10
12
2
2
2
8
8
8
0.3333
0.3333
0.3333
0.0000
0.0000
0.0000
0.0833
0.3333
0.5833
-0.2500
0.0000
0.2500
0.2500
0.0000
-0.2500
0.5833
0.3333
0.0833
(X'X)^-1
0.8796
-0.0500
-0.0926
0.0000
0.0000
0.0000
0.3333
0.3333
0.3333
-0.0500
0.0100
0.0000
-0.2500
0.0000
0.2500
0.0833
0.3333
0.5833
-0.0926
0.0000
0.0185
Beta-hat
2.7222
0.5000
0.5556
X'Y
48
290
270
Y'Y
Y'PY
Y'(I-P)Y
Y'(J/n)Y
Y'(P-J/n)Y
426.00
425.67
0.33
384.00
41.67
Total Corrected:
Sum Of Squares ________________ Degrees of Freedom ___________
Regression:
Sum Of Squares ________________ Degrees of Freedom ___________
Residual:
Sum Of Squares ________________ Degrees of Freedom ___________
S2 ____________________
^ 
s  1  _________________
 
Testing H0: F-stat Num df _______ Den df _________
Predicted Value for Y2: Based on:
^
X  ___________________________
PY____________________________________
^ 
s Y 2   ________________________
 
se2   ________________________