STA 6207 – Exam 3 – Fall 2010
PRINT Name _____________________
Q.1. A researcher fits a multiple linear regression model, relating yield (Y) of a chemical process to
temperature (X1), and the amounts of 2 additives (X2 and X3, respectively). She fits the following model:
E Y   0  1 X1  2 X 2  3 X 3
She wishes to test the following three hypotheses simultaneously:



The mean response when X1=70, X2=10, X3=10 is 80
The average yield increases by 4 units when temperature increases by 1ºF, controlling for X2 and X3
The partial effect of increasing each additive is the same (controlling for all other factors)
p.1.a. Fill in the following matrix and vectors that she is testing (this is her null hypothesis):












0

 

 

H 0 : K '   m  0   
 

 




0



 
 
 

 
 
 




0

 

 
  0

 

 



0

p.1.b. She obtains the following results from fitting the regression based on n = 24 measurements while
conducting the experiment:
 K '   m   K '  X ' X  K   K '   m   1800
'
1
1
Y '  I  P  Y  7800
Conduct her test at the  = 0.05 significance level.

Test Statistic:
Reject H0 if the Test Statistic falls in the range: ____________________________________
Q.2. For each of the following models, give the variance-covariance matrix of Y:
p.2.a. Yt  0  1 X t   t

2 
t  1,...,3 1 ~  0,
 t   t   t 1  t ~ 0,  2  vt   1,...,  t 1
2 
 1  
Y   i  i X ij   ij
p.2.b. ij
i  1, 2
2
 i       


j  1, 2,3   ~   , 
 i       

   


2 
  
Q.3. A regression model is fit, where X is the dose individuals receive and Y is a measure of therapeutic
response. The following table gives the sample size, mean, and variance for each level of dose (the overall mean
is 25). Consider the simple linear regression model Y=X
X
0
2
4
8
16
Sample Size
9
16
4
16
9
Mean
10
15
27
30
48
p.3.a. Obtain the weighted least squares estimate of   W   X *' X * X *'Y * X *  WX , Y *  WY 
^
1


p.3.a.1W =




p.3.a.ii. X* =
p.3.a.iii. Y* =
p.3.a.iv.  W =
^
^
p.3.b. Obtain the fitted values in the original scale: Y W  X  W
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
Q.5. A nonlinear regression model is fit, relating available chlorine (Y, proportion) to the length of time since it
was produced (X, in weeks). Based on data considerations, the model fit is of the form with output below:
yi     0.49    exp     xi  8    i
Nonlinear Regression Summary Statistics
Dependent Variable CHLOR
Source
DF
Sum of Squares
Regression
Residual
Uncorrected Total
2
42
44
7.98200
5.001680E-03
7.98700
(Corrected Total)
43
.03950
Mean Square
3.99100
1.190876E-04
R squared = 1 - Residual SS / Corrected SS =
Parameter
Estimate
ALPHA
BETA
.390140032
.101632757
Asymptotic
Std. Error
.005044840
.013360412
.87338
Asymptotic 95 %
Confidence Interval
Lower
Upper
.379959134
.400320931
q.5.a. Give the predicted proportion of chlorine remaining at each of the following times since production:
q.5.a.i. 10 weeks:
q.5.a.ii. 20 weeks:
q.5.a.iii. 30 weeks:
q.5.b. Compute a 95% Confidence Interval for 
q.5.c. Give the estimated rate of change in chlorine remaining as a function of age (using point estimates of
parameters in the derivative Y X
Q.6. You are given the following results from a statistical software package fit of a simple linear regression.
ANOVA
df
Source
MS
SS
F
13.709
Regression
Residual
44
16.544
Coef
0.5598
0.00102418
SE(Coef)
0.1983
0.00007103
Total
Predictor
Intercept
X
t
p.6.a. Complete the relevant cells in the table.
p.6.b. Obtain the following quantities from the given values.
p.6.b.i.
 X  X 
p.6.b.ii.
 X  X Y  Y 
p.6.b.iii.
 Y  Y 
2
p.6.b.iv. Y
2
p.6.b.v..
X
t(.025)
F(.05)
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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