STA 6207 – Exam 1 – Fall 2014
PRINT Name _____________________
Conduct all tests at =0.05 Significance Level.
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


__________________________________________________________________________________________
df |
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
40
50
60
70
80
90
100
110
120
140
160
180
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
1.656
1.654
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
1.977
1.975
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
168.613
190.516
212.304
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
3.909
3.900
3.894
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
3.061
3.053
3.046
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
2.669
2.661
2.655
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
2.436
2.428
2.422
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
2.279
2.271
2.264
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
2.164
2.156
2.149
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
2.076
2.067
2.061
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
2.005
1.997
1.990
1.985
1.938
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Q.1. A multiple regression model is fit, based on Model 2, with p predictors and an intercept. Define the
-1
projection matrix as: P = X  X'X  X' , where
1 X 11
1 X
21
X


1 X n1
X1 p 
1
1 1


X2p 
1
1
1 1 1
Define J n 
where J n is n  n


n
n



X np 
1
1 1
1
p.1.a. Show that P and J n are symmetric and idempotent. (Hint: (X’X)-1 is symmetric). SHOW ALL WORK.
n
1
p.1.b. Obtain P J n
n
and
1
show that P  J n is idempotent. SHOW ALL WORK
n
1
1
p.1.c. Obtain the rank of P, J n , and P  J n SHOW ALL WORK
n
n
p.1.d. What is the sampling distribution of
p.1.e. Show that
1 

Y ' P  Jn  Y ?

n 

1
2
SHOW ALL WORK
1
1 

Y '  P  J n  Y and 2 Y '  I  P  Y are independent. SHOW ALL WORK.


n 

1
2
Q.2. An Austrian study considered Breath Alcohol Elimination Rates (X, mg/L/hr*100) and Blood Alcohol
Elimination Rates (Y, g/L/hr*100) in a sample of n = 27 adult females. The sample means, standard deviations
and correlations are given below. Complete the following table for the simple linear regression relating Blood
Elimination Rate (Y) to Breath Elimination Rate (X).
X  8.6704 Y  17.8815 sX  1.6522 sY  3.6787 rXY  0.8786
Regression Statistics
R Square
Residual Std Error
Observations
ANOVA
df
SS
MS
F
Regression
Residual
Total
Coefficients
Standard Error t Stat
Intercept
X
t(.025)
F(.05)


Q.3. For the simple regression model (scalar form): Yi  0  1 X i  X   i
p.3.a. Derive least squares estimators of 0 and 1. SHOW ALL WORK
i  1,..., n  i ~ NID  0,  2 
     
^
p.3.b. Derive
^
^
^
 
^
^
E  1 , E  0 , V  1 , V  0 , COV  0 ,  1
SHOW ALL WORK
Q.4. A regression model is fit, relating total team payroll (Y, in millions of £s) to offensive goals scored (X1)
and defensive goals allowed (X2) for the n=20 teams during the 2013 English Premier League season. For this
problem, we will treat this as a sample from a population of all possible league teams.
Rank
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Team
Man City
Chelsea
Manchester United
Arsenal
Liverpool
Tottenham
Aston Villa
Newcastle United
Sunderland
Everton
Fulham
Swansea City
West Brom
Stoke City
Norwich
West Ham
Southampton
QPR
Reading
Wigan
Y
X0
233
179
181
154
132
96
72
62
58
63
67
49
54
60
75
56
47
78
46
44
X1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
X2
66
75
86
72
43
66
47
45
41
55
50
47
53
34
41
45
49
30
43
47
X'X
34
39
43
37
28
46
69
68
54
40
60
51
57
45
58
53
60
60
73
73
X'Y
20
1035
1048
1048
52509
58202
1806
104325
85017
INV(X'X)
2.994125 -0.02661 -0.02991
-0.026608 0.000336 0.000176
-0.029907 0.000176 0.000397
Beta-hat
88.84439
1.953057
-1.90105
Ybar
1035
57445
52509
Y'Y
90
220320
p.4.a. Complete the following ANOVA table.
ANOVA
df
Regression
Residual
Total
SS
MS
F
F(0.05)
p.4.b. Test whether the offensive goals scored and defensive goals allowed effects are of equal magnitude, but
opposite direction: H0: 
Rank
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Team
Man City
Chelsea
Manchester United
Arsenal
Liverpool
Tottenham
Aston Villa
Newcastle United
Sunderland
Everton
Fulham
Swansea City
West Brom
Stoke City
Norwich
West Ham
Southampton
QPR
Reading
Wigan
Y
X0
233
179
181
154
132
96
72
62
58
63
67
49
54
60
75
56
47
78
46
44
X1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1
X2
66
75
86
72
43
66
47
45
41
55
50
47
53
34
41
45
49
30
43
47
X'X
34
39
43
37
28
46
69
68
54
40
60
51
57
45
58
53
60
60
73
73
X'Y
20
1035
1048
1048
52509
58202
1806
104325
85017
INV(X'X)
2.994125 -0.02661 -0.02991
-0.026608 0.000336 0.000176
-0.029907 0.000176 0.000397
Beta-hat
88.84439
1.953057
-1.90105
Ybar
1035
57445
52509
Y'Y
90
220320
Test Statistic: _____________________ Rejection Region: _________________ Reject H0? Yes / No