STA 4210 – Exam 2 – Fall 2012 – PRINT Name _________________________
For all significance tests, use  = 0.05 significance level.
S.1. A multiple linear regression model is fit, relating household weekly food expenditures (Y, in $100s) to weekly
income (X1, in $100s) and the number of people living in the household (X2). Assuming the model has an intercept, and is
based on a sample of n=40 households. Give the appropriate degrees of freedom.
dfTotal = ________________ dfRegression = ___________________________ dfError = ____________________________
S.2. In a multiple linear regression model with 2 predictors (X1 and X2), if X1 and X2 are uncorrelated, SSR(X1) =
SSR(X1|X2).
TRUE or FALSE
S.3. In simple linear regression, the hat-matrix is 2x2. TRUE
or
FALSE
S.4. In a multiple linear regression model with 2 predictors (X1 and X2), SSR(X1) + SSR(X2|X1) = SSR(X2) + SSR(X1|X2).
TRUE or FALSE
S.5. In a multiple linear regression model with 3 predictors (X1, X2, and X3 (where each has 3 levels)), a researcher wishes
to include all 2-variable interactions, and all squared terms. Give the appropriate degrees of freedom, assuming there is 1
observation at all combinations of the levels of each predictor, and 3 extra points when each factor is at its mid-level.
dfTotal = ___________________ dfRegression = ___________________________ dfError = ____________________________
S.6. A researcher reports that for a linear regression model, the regression sum of squares is twice as large as the error sum
of squares. Compute R2 for this model
R2 = ____________________________________________________
S.7. A simple linear regression model is fit, based on a sample of n=10 observations. You obtain the following estimates:
b0 = 12.0 s{b0} = 3.0
b1 = 5.0 s{b1} = 2.0. Use Bonferroni’s method to obtain simultaneous 90% Confidence
Intervals for 0 and 

0: ______________________________________________ 1: ___________________________________________
Q.1. A simple linear regression model is to be fit: Yi = 0 +  1Xi + I . The data are as follows: Complete the following parts
in matrix form (Note: SSTO=70):
X
0
0
2
2
4
4
Y
0
2
4
6
10
8
p.1.a. X=
Y=
p.1.b. X’X =
X’Y =
p.1.c. (X’X)-1 =
b=
^
p.1.d. Y 
e
p.1.e. MSE =
s2{b} =
p.1.f. Complete the following tables:
ANOVA
df
Regression
Residual
Total
SS
MS
F
CoefficientsStandard Error t Stat
Intercept
X
Q.2. A regression model is fit, relating a response (Y) to 3 predictors (X1, X2, X3) based on n=30 individuals. Two models
are fit:
Model1: E(Y) = 0 + 1X1 + 2X2 +3X3
SSE1 = 1800
Model2: E(Y)=0 + 1X1 + 2X2 + 3X3 + 11X12 + 22X22 +33X32 +  12X1X2 + 13X1X3 +3X2X3 SSE2=1400
Test whether none of the quadratic terms or interaction terms contributes above and beyond the effects
of X1, X2, and X3.
p.2.a. H0: _________________________________________
p.2.b. Test Statistic:
p.2.c. Reject H0 if the test statistic falls above / below ______________________________________
p.2.d. Compute RY2, X 2 , X 2 , X 2 , X X
1
2
3
1
2 , X1 X 3 , X 2 X 3 | X1 , X 2 , X 3
Q.3. Regression models are fit, relating total revenues (Y, in millions of dollars) for n=155 films released in the 1930s to
production costs (X1, in millions of dollars) and distribution costs (X2, in millions of dollars).
Model 1: E Y    0  1 X 1
Model 2: E Y    0  1 X 1   2 X 2
Model 3: E Y    0  1 X 1   2 X 2  3 X 1 X 2
ANOVA
Regression
Residual
Total
Intercept
prodcostc
ANOVA
Regression
Residual
Total
Intercept
prodcostc
distcostc
Model 4: E Y    0  1  X 1  X 2 
Model 1
df
SS
MS
F
1 3344.973 3344.973 166.5618
153 3072.619 20.08247
154 6417.591
Coefficients
Standard Error t Stat
1.807034 0.631288 2.862457
1.392948 0.107931 12.90588
Model 2
df
SS
MS
F
2 6078.419 3039.209 1362.018
152 339.173 2.231401
154 6417.591
Coefficients
Standard Error t Stat
0.164311 0.215601 0.762109
0.101986 0.051525 1.979341
2.482252 0.070922 34.99987
df
Regression
Residual
Total
Intercept
prodcostc
distcostc
prodxdist
ANOVA
Regression
Residual
Total
Intercept
totcostc
ANOVA
Model 3
SS
MS
F
3 6078.609 2026.203 902.5746
151 338.9822 2.244915
154 6417.591
Coefficients
Standard Error t Stat
0.105335 0.296108 0.355733
0.115236 0.068821 1.674438
2.500326 0.094358 26.4983
-0.00295 0.010111 -0.29156
Model 4
df
SS
MS
F
1 5099.65 5099.65 592.0188
153 1317.942 8.613999
154 6417.591
Coefficients
Standard Error t Stat
-0.04937 0.423133 -0.11667
1.073282 0.044111 24.33144
Complete the following parts.
p.3.a. Based on model 3, test whether there is an interaction between production (X1) and distribution (X2) costs.
H0: _______________ HA: ________________ Test Statistic: ___________________ Rejection Region: _____________
p.3.b. Compute
SSR  X1   ____________ SSR  X 2 | X1   ______________________ RY22|1  ________________________
p.3.c. Based on models 2 and 4, test whether the effects of increasing production and distribution costs by 1 unit are
equal. H0:  HA: ≠
Test Statistic: ____________________________________ Rejection Region: ______________________________
p.3.d. Based on model 2, give the predicted total revenues (in dollars) for a movie that had production costs of $500,000
and distribution costs of $250,000
p.3.e. Conduct the Brown-Forsyth test, to determine whether the variances for the low predicted values (group=1) and
~
the high predicted values (group 2) are equal. dij  eij  ei
Group
1
2
Mean(D)
0.56
1.40
Var(D)
0.64
1.46
i  1, 2
j  1,..., ni

~
ei  median ei1 ,..., eini

n
77
78
p.3.e.i. Test Statistic:
p.3.e.ii. Rejection Region _____________________________________________________
^
Q.4. You obtain the following spreadsheet from a regression model. The fitted equation is Y  4.67  4.00 X
Conduct the F-test for Lack-of-Fit. n = ______________ c = _______________
X
2
2
4
4
6
6
Source
Lack-of-Fit
Pure Error
Y
3
5
8
12
18
22
Ybar
Y-hat
Pure Error
Lack of Fit
df
SS
MS
F
F(0.05)
Critical Values for t, 2, and F Distributions
F Distributions Indexed by Numerator Degrees of Freedom
CDF - Lower tail probabilities
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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