STA 4210 – Practise set 2b

STA 4210 – Practise set 2b
For all significance tests, use  = 0.05 significance level.
S.1. A linear regression model is fit, relating fish catch (Y, in tons) to the number of vessels (X 1) and fishing pressure (X2)
for a lake over a sample of n=16 years. The model also contains an intercept. Give the appropriate degrees of freedom.
dfTotal = ________________ dfRegression = ___________________________ dfError = ____________________________
S.2. In a multiple linear regression model with 2 predictors (X 1 and X2), then SSR(X1)+SSR(X2|X1) = SSTO–SSE(X1,X2)
TRUE or FALSE
S.3. In simple linear regression, then (X’X) -1 is 2x2.
TRUE
or
FALSE
2
2
2
S.4. In a multiple linear regression model with 2 predictors (X 1 and X2), R ( X 1 ) + RY 2|1 = R ( X 1 , X 2 )
TRUE
or
FALSE
S.5. A multiple regression model is fit, relating Y to X 1, X2, and X3. The regression sums of squares include:
SSR(X1) = 400 SSR(X2) = 600 SSR(X3) = 800 SSR(X1,X2) = 700 SSR(X1,X3) = 1000 SSR(X2,X3)=900
SSR(X1,X2,X3)= 1200
SSR(X3|X1,X2) = _________________ SSR(X2|X1,X3) = _______________ SSR(X1,X2|X3) = ___________________
S.6. A researcher reports that for a linear regression model, the regression sum of squares is three times larger than the
error sum of squares. Compute R2 for this model
R2 = ____________________________________________________
S.7. In multiple regression, when predictor variables are highly correlated, the model is said to display multicollinearity.
Effects of multicollinearity include (select all that are appropriate):
i) Decreased t-statistics for some of the tests of H 0: k = 0 (k=1,…,p-1)
ii) Wider confidence intervals for some of the k (k=1,…,p-1)
iii) Inflated standard errors for the least squares estimates of some of the b k (k=1,…,p-1).
Q.1. A simple linear regression model is to be fit: Y i = 0 + 1Xi + I . The data are as follows: Complete the following parts
in matrix form (Note: SSTO=82):
X
0
0
3
3
6
6
Y
14
10
9
7
6
2
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
Coefficients
Standard Errort Stat
Intercept
X
Q.2. A regression model is fit, relating height (Y, in cm) to hand length (X 1, in cm) and foot length (X2, in cm) for a sample
of n=75 adult females. The following results are obtained from a regression analysis of:
Y = 0 + 1X1 + 2X2 +  ~ NID(0,2)
Regression Statistics
R Square
ANOVA
df
Regression
Residual
Total
Intercept
X1
X2
SS
1105.52
F*
F(0.95)
#N/A
#N/A
#N/A
#N/A
StdErr t*
7.97 #N/A
0.49
0.37
t(.975)
#N/A
1793.85
Coeff
74.41
2.38
1.73
MS
#N/A
#N/A
#N/A
p.2.a. Complete the tables.
p.2.b. The first woman in the sample had a hand length of 19.56cm, a foot length of 25.70cm, and a height of 160.60cm.
Obtain her fitted value and residual.
Fitted value = _____________________________________ Residual = ______________________________________
p.2.c. Obtain simultaneous 95% Confidence Intervals for 0, 1, and 2 (Hint: z(.9917) ≈2.395)
Q.3. Regression models are fit, relating bursting strength of knit fabric (Y) to yarn count (X 1) and stitch length (X2).
The following 5 models were fit on centered yarn counts and stitch lengths to reduce collinearity.
Model 1: E {Y } = b 0 + b1 x1 Model 2: E {Y } = b 0 + b 2 x2
Model 3: E {Y } = b0 + b1 x1 + b 2 x2
Model 4: E {Y } = b 0 + b1 x1 + b 2 x2 + b3 x12 + b 4 x22 + b 5 x1 x2
Model 5: E {Y } = b 0 + b 0 + b1 x1 + b 2 x2 + b 3 x12
where: x1 = X 1 - X 1 x2 = X 2 - X 2
ANOVA
Model1
df
Regression
1
Residual
54
Total
55
Model4
df
Regression
5
Residual
50
Total
55
SS
20.2509
18.7953
39.0462
MS
20.2509
0.3481
SS
35.7314
3.3148
39.0462
MS
7.1463
0.0663
ANOVA
Model2
df
1
54
55
Model5
df
3
52
55
SS
17.4997
21.5466
39.0462
MS
17.4997
0.3990
SS
35.5942
3.4520
39.0462
MS
11.8647
0.0664
Model3
df
2
53
55
SS
34.4914
4.5548
39.0462
MS
17.2457
0.0859
Complete the following parts (all parts are based on the centered values).
p.3.a. Based on model 3, test whether either centered yarn count (x 1) and/or centered stitch length (x2) are associated
with bursting strength.
H0: _______________ HA: ________________ Test Statistic: ___________________ Rejection Region: _____________
p.3.b. Compute
SSR ( x1 ) = ____________ SSR ( x2 | x1 ) = ______________________
RY2 2|1 = ________________________
SSR ( x2 ) = ____________ SSR ( x1 | x2 ) = ______________________
RY21|2 = ________________________
p.3.c. Based on models 4 and 5, test whether after controlling for yarn count, stitch length, and squared yarn count, that
neither squared stitch length or the cross-product between yarn count and stitch length are associated with bursting
strength. That is H0: 4 = 
Test Statistic: ____________________________________ Rejection Region: ______________________________
^
Q.4. You obtain the following spreadsheet from a regression model. The fitted equation is Y = -2.67 + 3.75 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
12
16
18
20
Ybar(Grp)
Y-hat
Pure Error
Lack of Fit
df
SS
MS
F
F(0.05)
Q.5. A firm that produces technical manuscripts is interested in the relationship between cost of correcting
typographical errors (Y, in dollars) and the total number of galleys (pages, X). They wish to determine whether a
regression-through-the-origin model is appropriate. You are given the following results for the model Y = X + :
Y
1 28
2 13
1 91
1 78
2 50
4 46
5 40
4 57
3 24
1 77
X
7
12
10
10
14
25
30
25
18
10
Y-hat
126.19
216.32
180.27
180.27
252.38
450.67
540.81
450.67
324.48
180.27
e
1.81
-3.32
10.73
-2.27
-2.38
-4.67
-0.81
6.33
-0.48
-3.27
å X = 161 å Y = 2904 å X = 3163 å Y = 1028088 å XY = 57019
å ( X - X ) = 571 å (Y - Y ) = 2904 å ( X - X )(Y - Y ) = 10265 å e = 214
n = 10
2
2
2
2
2
b1 = ____________ MSE = ________ s {b1} = ________ 95% CI for b1 : _____________________
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

|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
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
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
 .295
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.95,1
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
F.95,2
F.95,3
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
F.95,4
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
F.95,5
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
F.95,6
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
F.95,7
F.95,8
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
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|

Download Report