STA 6207 – Exam 3 – Fall 2011
PRINT Name _____________________
Q.1. A scientist is interested in comparing mean purity of chemicals for the 2 suppliers of her lab. She chooses
to fit a simple linear regression model (where Xi = 1 if supplier 1, 0 if supplier 2):
Yi   0  1 X i   i
i  1,..., n  i ~ N  0,  2 
She measures the purity for 4 units from each supplier. For supplier 1, she observes: (12,14,12,10), and for
supplier 2, she observes: (18,20,22,20).
p.1.a. Write out X, Y, X’X, X’Y, (X’X)-1
^
^
^
^
p.1.b. Compute β, Y, e, s 2 , V  β 
 
Q.2. A large electronics retailer is interested in the relationship between net revenue of plasma TV sales (Y,
$1000s) , and the following 4 predictors: X1= shipping costs ($/unit), X2= print advertising ($1000s),
X3= electronic media ads ($1000s), and X4= rebate rate (% of retail price). A sample of n=50 stores is selected
and the resulting (partial) regression output is obtained:
ANOVA
df
Regression
Residual
Total
Intercept
ShipCost
PrintAds
WebAds
Rebate%
INV(X'X)
1.005224
-0.029489
-0.006808
-0.002514
-0.019146
49
Coefficients
Standard Error
4.31
70.82
-0.08
4.68
2.27
1.05
2.50
0.85
16.70
3.57
-0.029489
0.004386
-0.000011
-0.000282
0.000021
-0.006808
-0.000011
0.000221
-0.000031
-0.000228
SS
259411.8
224539.0
483950.8
MS
F
F(0.05)
t Stat
P-value
0.0608
0.9518
-0.0175
0.9861
2.1562
0.0364
2.9535
0.0050
4.6766
0.0000
-0.002514
-0.000282
-0.000031
0.000143
-0.000002
-0.019146
0.000021
-0.000228
-0.000002
0.002555
p.2.a. Complete the ANOVA table.
p.2.b. Give the prediction for net revenue, when ShipCost=10, PrintAds=50, WebAds=40, Rebate%=15.
p.2.c. Controlling for all other factors, give a 95% confidence interval for the change in expected net revenue
($1000s) when Rebate% is increased by 1.
p.2.d. Test H0: PrintAds - WebAds = 0 vs HA: PrintAds - WebAds ≠ 0
p.2.d.i. Test Statistic:
at  = 0.05 significance level:
p.2.d.ii. Rejection Region
p.2.e. What proportion of variation in revenues is “explained” by the regression model?
Q.3. When the functional relationship between the variance is known. Bartlett devised a means of obtaining a
transformation to make the error variance approximately constant:
 2  ( )

f ( )  
1
( )1 / 2
d
Give the variance-stabilizing transformation if ( )  
Q.4. A regression model is fit, relating cockpit noise (Y, decibels) to the following predictors:


Flight Phase (CLIMB, Cruise, DESCENT), with 2 coded dummy variables for climb and Descent
Speed, Altitude, Speed2, and Alt2
We fit the following models, and obtain the following Residual Sums of Squares (n=61)
Independent Variables
Climb,Descent,Speed,Alt,SpeedSqr,AltSqr
Climb,Descent,Speed,Alt
Climb,Descent
Speed,Alt
Speed,Alt,SpeedSqr,AltSqr
SS(Residual)
75.05
87.09
863.34
96.23
88.31
p.4.a. Test H0: No quadratic speed or altitude effects, controlling for flight phase, speed, altitude
Test Statistic _________________________ Rejection Region ______________________
p.4.b. Test H0: No flight phase effects, controlling for speed, altitude, speed2, and alt2
Test Statistic _________________________ Rejection Region ______________________
Q.5. You are given results of the sample means based on ni observations from levels Xi from a simple linear
regression with normal and independent errors. The following table gives the published results.
ni
25
4
16
9
Ybari
34.0
48.0
60.0
75.0
Xi
0
4
8
12
p.5.a. Obtain the weighted least squares estimate of   W   X *' X * X *'Y * X *  WX , Y *  WY 
^
1


p.5.a.iW =




p.5.a.ii. X* =
p.5.a.iii. Y* =
p.5.a.iv.  W =
^
^
p.5.b. Obtain the fitted values in the original scale: Y W  X  W
Q.6. A nonlinear regression model is to be fit, relating Area (Y, in m2) of palm trees to age (X, in years) by the
Gompertz model: E(Y) = +exp[-exp(-X)] for 
p.6.a. What is E(Y), in terms of the model parameters when X=0? When X →∞?
p.6.b. What is the instantaneous growth rate of Area in terms of the model parameters and X?
Q.7. An enzyme kinetics study of the velocity of reaction (Y) is expected to be related to the concentration of
the chemical (X) by the following model (based on n=18 observations):
 X
Yi  0 i   i  i ~ N  0,  2 
1  X i
The following results are obtained.
The NLIN Procedure
Parameter
b0
b1
Estimate
28.1
12.6
Approx
Std Error
0.73
0.76
p.7.a. Give a 95% Confidence Interval for the Maximum Velocity of Reaction
p.7.b. . Give a 95% Confidence Interval for the dose needed to reach 50% of Maximum Velocity of Reaction
p.7.c. Give the predicted velocity when X=0, 10, 20and difference between each
Y0 
Y 10 - Y 0 
Y 10 
Y 20 
Y 20  Y 10 
Model 2: Y  Xβ  ε X  n  p ' β  p '1 ε ~ N  0,  2I 
d a ' x
a
dx
d  x'Ax 
 2Ax (A symmetric)
dx
Q.8. Consider model 2.
p.8.a. Derive the least squares estimator for . Show all work.
p.8.b. Derive the mean and variance of the least squares estimator. Show all work.
Q.4. A simple linear regression model is fit, based on n=5 individuals. The data and the projection matrix are
given below:
X
1
1
1
1
1
0
2
4
6
18
Y
10
14
17
21
88
P
0.38
0.32
0.26
0.20
-0.16
0.32
0.28
0.24
0.20
-0.04
0.26
0.24
0.22
0.20
0.08
0.20
0.20
0.20
0.20
0.20
-0.16
-0.04
0.08
0.20
0.92
p.4.a. Give the leverage values for each observation. Do any exceed twice the average of the leverage values?
Observation 1 ______
Obs 2 ___________
Obs 3 _________
Obs 4 _________ Obs 5 __________
p.4.b. Give β , based on the following results
X'X
5
30
30
380
INV(X'X)
0.380
-0.030
-0.030
0.005
X'Y
150
1806
p.4.c. Compute SSE and Se (Note: Y'Y  8770 Y'PY  8604.18 )
p.4.c. The following table contains the fitted values with and without each observation, residual standard
deviation when that observation was not included in the regression, and the regression coefficients when that
observation was not included in the regression.
Y-Hat
2.82
11.88
20.94
30.00
84.36
Y-hat(-i)
-1.58
11.06
22.05
32.25
42.50
S_i
6.43
8.93
8.54
5.68
0.32
beta1_i
4.88
4.59
4.48
4.53
1.80
beta0_i
-1.58
1.88
4.13
5.07
10.10
p.4.c.i. Compute DFFITS for the fifth observation
p.4.c.ii. Compute DFBETAS0 for the first observation
p.4.c.iii. Compute DFBETAS1 for the third observation
Critical Values for t, 2, and F Distributions
F Distributions Indexed by Numerator Degrees of Freedom
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
130
140
150
160
170
180
190
200

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t.95
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
t.975
 .295
F.95,1
F.95,2
F.95,3
F.95,4
F.95,5
F.95,6
F.95,7
F.95,8
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
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
---
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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