STA 6207 – Exam 1 – Fall 2013
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 multiple regression model is fit, with p predictors and an intercept. Define the projection matrix based on
-1
Model 2 as: P = X  X'X  X' .
p.1.a. Show that P is symmetric and idempotent. (Hint: (X’X)-1 is symmetric)
n
p.1.b. What does
P
j 1
equal? Where Pij is the element of P in the ith row and jth column
ij
n
p.1.c. What does
P
j 1
2
ij
equal?
 P  P 
n
p.1.d. What does
j 1
ij
i
2
equal?
Q.2. A multiple linear regression model is fit, relating height (Y, mm) to hand length (X1, mm) and foot length
(X2, mm), for a sample of n = 20 adult males. The following partial computer output is obtained, for model 1
with 2 predictors.
ANOVA
df
Regression
Residual
Total
SS
37497
19772
57269
MS
F
F(0.05)
#N/A
#N/A
#N/A
#N/A
#N/A
Coefficients
Standard Error t Stat
P-value
Intercept
1055.78
132.86
7.95
0.0000
Hand
1.26
0.55
2.28
0.0357
Foot
1.71
0.39
4.42
0.0004
p.2.a Complete the table. Do you reject the null hypothesis H0: ?
Yes
or No
p.2.b. Give the predicted height of a man with a hand length of 210mm and a foot length of (260mm). Just give
the point estimate, not confidence interval for the mean or a prediction interval.
p.2.c. Give an unbiased estimate of the error variance 2
p.2.d. The coefficient of determination represents the proportion of variation in heights “explained” by the
model with hand and foot length as predictors. What is the proportion explained for this model?
Q.3. The total salaries (X, in millions of pounds) and the number of points earned (Y) for the n = 20 English
Premier League teams in 1995/6 are used to fit a simple linear regression model (Model 2, with p = 1). For this
problem, we will treat this as a sample from a population of all possible league teams.
Team
Arsenal
Aston Villa
Blackburn
Bolton
Chelsea
Coventry
Everton
Leeds
Liverpool
Manchester City
Manchester United
Middlesbrough
Newcastle
Nottingham Forest
Queens Park
Sheffield
Southampton
Tottenham
West Ham United
Wimbledon
p.3.a. Compute
Int
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
 X'X
-1
WageK
Points(Y)
10.1
63
7.7
63
10.8
61
3.3
29
9.2
50
5.8
38
10.1
61
10.1
43
13.2
71
6.4
38
13.3
82
6.5
43
19.7
78
8.5
58
5
33
6.4
40
4.1
38
9.8
61
6.2
51
4.7
41
X'X
X'Y
20
170.9
170.9
1745.15
^
β
and
p.3.b. Compute the fitted value and residual for Arsenal.
p.3.c. Given Y  52.1 and
 Y  Y 
20
i 1
i
2
 4367.8
p.3.d. Obtain a 95% Confidence Interval for 1
Compute SSRegression and SSResidual
1042
9870.3
Q.4. For the simple regression model (scalar form): Yi   0  1 X i   i
X X
1   i
S XX
i 1 
^
we get:
p.4.a. Derive
n
i  1,..., n  i ~ NID  0,  2 
n
^
^

1
Yi , Y    Yi ,  0  Y   1 X
i 1  n 

 
^ 
^ 
E  1 , E Y , E   0 
 
 
 
^ 
^

^ 
^ ^ 
p.4.b. Derive V   1  , V Y , Cov   1 , Y  , V   0  , Cov   1 ,  0 
 


 


Q.5. For the Analysis of Variance in model 2, with n observations and p predictors, complete the following
parts.
p.5.a. Write the Regression and Residual sums of squares as quadratic forms.
p.5.b. Derive the distributions of SSRegression/ and SSResidual/
p.5.c. Show that SSRegression/ and SSResidual/ are independent
p.5.d. What is the sampling distribution of MSRegression/MSResidual when p = 0?
df |
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2
3
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7
8
9
10
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13
14
15
16
17
18
19
20
21
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26
27
28
29
30
40
50
60
70
80
90
100
110
120
130
140
150
160
170
180
190
200

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t.05
t.025
6.314
2.920
2.353
2.132
2.015
1.943
1.895
1.860
1.833
1.812
1.796
1.782
1.771
1.761
1.753
1.746
1.740
1.734
1.729
1.725
1.721
1.717
1.714
1.711
1.708
1.706
1.703
1.701
1.699
1.697
1.684
1.676
1.671
1.667
1.664
1.662
1.660
1.659
1.658
1.657
1.656
1.655
1.654
1.654
1.653
1.653
1.653
1.645
12.706
4.303
3.182
2.776
2.571
2.447
2.365
2.306
2.262
2.228
2.201
2.179
2.160
2.145
2.131
2.120
2.110
2.101
2.093
2.086
2.080
2.074
2.069
2.064
2.060
2.056
2.052
2.048
2.045
2.042
2.021
2.009
2.000
1.994
1.990
1.987
1.984
1.982
1.980
1.978
1.977
1.976
1.975
1.974
1.973
1.973
1.972
1.960
2
 .05
3.841
5.991
7.815
9.488
11.070
12.592
14.067
15.507
16.919
18.307
19.675
21.026
22.362
23.685
24.996
26.296
27.587
28.869
30.144
31.410
32.671
33.924
35.172
36.415
37.652
38.885
40.113
41.337
42.557
43.773
55.758
67.505
79.082
90.531
101.879
113.145
124.342
135.480
146.567
157.610
168.613
179.581
190.516
201.423
212.304
223.160
233.994
---
F.05,1
F.05,2
F.05,3
F.05,4
F.05,5
F.05,6
F.05,7
F.05,8
161.448
18.513
10.128
7.709
6.608
5.987
5.591
5.318
5.117
4.965
4.844
4.747
4.667
4.600
4.543
4.494
4.451
4.414
4.381
4.351
4.325
4.301
4.279
4.260
4.242
4.225
4.210
4.196
4.183
4.171
4.085
4.034
4.001
3.978
3.960
3.947
3.936
3.927
3.920
3.914
3.909
3.904
3.900
3.897
3.894
3.891
3.888
3.841
199.500
19.000
9.552
6.944
5.786
5.143
4.737
4.459
4.256
4.103
3.982
3.885
3.806
3.739
3.682
3.634
3.592
3.555
3.522
3.493
3.467
3.443
3.422
3.403
3.385
3.369
3.354
3.340
3.328
3.316
3.232
3.183
3.150
3.128
3.111
3.098
3.087
3.079
3.072
3.066
3.061
3.056
3.053
3.049
3.046
3.043
3.041
2.995
215.707
19.164
9.277
6.591
5.409
4.757
4.347
4.066
3.863
3.708
3.587
3.490
3.411
3.344
3.287
3.239
3.197
3.160
3.127
3.098
3.072
3.049
3.028
3.009
2.991
2.975
2.960
2.947
2.934
2.922
2.839
2.790
2.758
2.736
2.719
2.706
2.696
2.687
2.680
2.674
2.669
2.665
2.661
2.658
2.655
2.652
2.650
2.605
224.583
19.247
9.117
6.388
5.192
4.534
4.120
3.838
3.633
3.478
3.357
3.259
3.179
3.112
3.056
3.007
2.965
2.928
2.895
2.866
2.840
2.817
2.796
2.776
2.759
2.743
2.728
2.714
2.701
2.690
2.606
2.557
2.525
2.503
2.486
2.473
2.463
2.454
2.447
2.441
2.436
2.432
2.428
2.425
2.422
2.419
2.417
2.372
230.162
19.296
9.013
6.256
5.050
4.387
3.972
3.687
3.482
3.326
3.204
3.106
3.025
2.958
2.901
2.852
2.810
2.773
2.740
2.711
2.685
2.661
2.640
2.621
2.603
2.587
2.572
2.558
2.545
2.534
2.449
2.400
2.368
2.346
2.329
2.316
2.305
2.297
2.290
2.284
2.279
2.274
2.271
2.267
2.264
2.262
2.259
2.214
233.986
19.330
8.941
6.163
4.950
4.284
3.866
3.581
3.374
3.217
3.095
2.996
2.915
2.848
2.790
2.741
2.699
2.661
2.628
2.599
2.573
2.549
2.528
2.508
2.490
2.474
2.459
2.445
2.432
2.421
2.336
2.286
2.254
2.231
2.214
2.201
2.191
2.182
2.175
2.169
2.164
2.160
2.156
2.152
2.149
2.147
2.144
2.099
236.768
19.353
8.887
6.094
4.876
4.207
3.787
3.500
3.293
3.135
3.012
2.913
2.832
2.764
2.707
2.657
2.614
2.577
2.544
2.514
2.488
2.464
2.442
2.423
2.405
2.388
2.373
2.359
2.346
2.334
2.249
2.199
2.167
2.143
2.126
2.113
2.103
2.094
2.087
2.081
2.076
2.071
2.067
2.064
2.061
2.058
2.056
2.010
238.883
19.371
8.845
6.041
4.818
4.147
3.726
3.438
3.230
3.072
2.948
2.849
2.767
2.699
2.641
2.591
2.548
2.510
2.477
2.447
2.420
2.397
2.375
2.355
2.337
2.321
2.305
2.291
2.278
2.266
2.180
2.130
2.097
2.074
2.056
2.043
2.032
2.024
2.016
2.010
2.005
2.001
1.997
1.993
1.990
1.987
1.985
1.938
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