STA 6207 – Exam 1 – Fall 2010
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


__________________________________________________________________________________________
Q.1. A researcher fits a simple linear regression model, relating yield of a chemical process to temperature
when all inputs beside temperature are at a specific level. She wishes to test the following two hypotheses
simultaneously (the temperature range the experiment was conducted was: 55ºF – 85ºF):


The average yield increases by 2 units when temperature increases by 1ºF
The average yield is 400 when the temperature is set to 70ºF
p.1.a. For model 2, fill in the following matrix and vectors that she is testing (this is her null hypothesis):
0
H0 : K '   m   
 

0













  
 0
    
  
 


 
 

 0

p.1.b. She obtains the following results from fitting the regression based on n = 18 measurements while
conducting the experiment:

K '  m

'
K ' X ' X  K
1
  K '   m   640
1
Y '  I  P  Y  3200
Conduct her test at the  = 0.05 significance level.

Test Statistic:

Reject H0 if the Test Statistic falls in the range: ____________________________________
Q.2. For model 1) with simple linear regression, derive the least squares estimates of .
Q.3. For model 2:
p.3.a. Derive the Covariance of the vector of sample means and the vector of fitted values:
Y  Y 1n 
1
JnY
n
Y  Xβ  X  X'X  X'Y  PY
p.3.b. Obtain the sampling distribution of SS() = Y'Y :
p.3.c. Are SS() and SS(Residual) independent? Why?
-1
Q.4. Use the following output to obtain the quantities given below:
X
1
1
1
1
1
1
P
0.5833
0.3333
0.0833
0.2500
0.0000
-0.2500
0
2
4
0
2
4
Y
10
16
22
22
25
25
10
10
10
20
20
20
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
0.0000
0.0000
0.0000
0.3333
0.3333
0.3333
(X'X)^-1
1.9167
-0.1250
-0.1000
-0.2500
0.0000
0.2500
0.0833
0.3333
0.5833
-0.1250
0.0625
0.0000
Y'Y
Y'PY
Y'(J/n)Y
-0.1000
0.0000
0.0067
X'Y
120
270
1920
2574.00
2552.25
2400.00
Complete the following elements of the regression model:
ANOVA
Source
Total (Uncorr)
Model
Mu
Regression
Residual
Total (Corr)
X'X
40
90
S2 = ________________
df
SS
 
s  2  ________________
Tests of (TS=Test statistic, RR=Rejection Region) each based on  = 0.05 significance level:
H 0 :  2  0 H A :  2  0 TS: ____________ RR:________
H 0 : 1   2  0 H A : 1  0 and/or  2  0 TS: ____________ RR:___________
Predicted Value for Y4 based on each of these two forms and its residual (show work):
Xβ : Y4  _____________________ PY : Y4  ____________
e4  ________________
 
s Y4  ___________________
s e4   _______________
Beta-hat
4.250
1.875
0.800