STA 6167 – Exam 3 – Spring 2014 PRINT Name __________________________________________________________
Conduct all tests at  = 0.05 significance level.
Q.1. Three Poisson regression models relating the rate of fatalities for the British rail system versus year (1967-2003)
were fit. The rate was (fatalities/million miles of train service). The British railway system was privatized in 1994, and an
indicator variable for privatized was created. The following models were fit (where i / ti = mean rate of fatalities,
X1 = Year – 1967, X2 = 1 if privatized (post 1994), 0 if not):

1. ln  i
 ti

   0  1 X 1


2. ln  i
 ti
 2 ln L1  486.64
 
3. ln  i    0  1 X 1   2 X 2  3 X 1 X 2
 ti 

   0  1 X 1   2 X 2

 2 ln L2  443.86
 2 ln L3  443.36
Parameter Estimates and Standard Errors are given below for each model.
Parameter
Intercept
X1
X2
X1*X2
Model1
Estimate (SE)
-1.9069 (0.0431)
-0.0209 (0.0022)
#N/A
#N/A
Model2
Estimate (SE)
-2.0430 (0.0492)
-0.0058 (0.0032)
-0.5932 (0.0913)
#N/A
Model3
Estimate (SE)
-2.0469 (0.0496)
-0.0055 (0.0032)
-0.0462 (0.7803)
-0.0172 (0.0245)
p.1.a. Test H0: 2 = 3 = 0 versus HA: 2 and/or 3 ≠ 0
Test Statistic: _____________________________________
Rejection Region: _____________________________
p.1.b. Assuming the interaction is not significant, Use the Wald Test, and the Likelihood Ratio tests to test for a
privatization effect: H0: 2 = 0 versus HA:  2 ≠ 0
Wald Statistic: __________________ LR Statistic: ______________________ Rejection Region: _____________
p.1.c. Based on model 2, the fitted values for 1993 (X1=1993-1967=26, X2=0, t1993=425) and for 1995 (28, 1, 423) are:
Y-hat(1993)= _______________________________ Y-hat(1995) = _______________________________
Q.2. A nonlinear regression model was fit, relating beer foam height (Y, centimeters) to time since pouring (X, seconds)
for three brands of beer. We have 3 dummy variables, one for each brand due to the nonlinearity. The model fit is:
1 if Brand 1 for observation i
1 if Brand 2 for observation i
1 if Brand 3 for observation i
Z1i  
Z 2i  
Z 3i  
0 otherwise
0 otherwise
0 otherwise
Model 1: (Brand Specific Equations): Yi   01Z1i exp  11 XZ1i    02 Z 2i exp  12 XZ 2i    03 Z 3i exp  13 XZ 3i    i
Model 2: (Common Equations): Yi   0 exp  1 X    i   0  Z1i  Z 2i  Z 3i  exp  1 X  Z1i  Z 2i  Z 3i    i
Model 3: (Brand 1 vs Common 2&3): Yi   01Z1i exp  11 XZ1i    023  Z 2i  Z 3i  exp  123 X  Z 2i  Z 3i    i
Note: Each brand was observed at 15 time points between 0 and 360 seconds, thus the overall sample size is n=45.
Results for the 3 models are given below.
Model1 (Brand Specific Curves)
Model2 (Common Curves)
Parameter Estimate t
Parameter Estimate t
b01
16.5
79.3 b0
14.3
b11
0.0034
29.0 b1
0.0049
b02
13.23
53.6
b12
0.0068
26.7
b03
13.37
57.0
b13
0.0056
26.6
SSE1
6.03
SSE2
184.09
Model 3 (Brands 2&3 vs 1)
Parameter Estimate t
20.8 b01
16.5
9.2 b023
0.0034
b11
13.29
b123
0.0061
SSE3
62.2
22.7
61.3
29.5
10.31
p.2.a. Give the fitted values (predicted beer foam height) based on model 1 for each brand at X=120 seconds.
Brand 1 ________________________ Brand 2 _________________________ Brand 3 ________________________
p.2.b. Use Models 1 and 2 to test H0:  (Common Curves for All Brands)
Test Statistic: _______________________________ Rejection Region: ____________________________
p.2.b. Use Models 1 and 3 to test H0: (Common Curves for Brands 2 and 3)
Test Statistic: _______________________________ Rejection Region: ____________________________
Q.3. A study was conducted to relate probability of returning to a whale viewing boat tour (Y=1 if Yes, Y=0, if No) to
several predictors, based on a sample of n=410 tourists after the tour:





X1 = Perceived Crowding (1 if the # of boats affected their enjoyment, 0 if Not)
X2 = Reported Crowding (# of boats subject saw near whales)
X3 = Subjective Norm (1 if Societal Pressure to do act, such as conservation, 0 if Not)
X4 = Income ($1000s/Month)
X5 = Prices of Substitute activities (Scuba-Diving, Rafting, and Snorkeling)
p.3.a. Complete the following Table.
Variable
Perceived Crowding
Reported Crowding
Subjective Norm
Income
Prices of Substitute
Estimate Std Err Lower Bound CI
-0.563
0.248
-0.125
0.059
0.421
0.217
0.540
0.329
0.172
0.106
Upper Bound CI Odds Ratio
Odds Ratio LB
Odds Ratio UB
p.3.b. Give the predicted probabilities of return for the boat tour for the following levels of the independent variables.
i) X1 = 1, X2 = 6, X3 = 0, X4 = 1, X5=1
ii) X1 = 0, X2 = 2, X3 = 1, X4 = 3, X5=3
Q.4. An Analysis of Covariance is conducted to compare three exercise regimens with respect to conditioning. Each
participant is given a test to measure their baseline strength prior to training (X). Out of the 30 participants, 10 are
assigned to method 1 (Z1=1, Z2=0), 10 are assigned to method 2 (Z1=0, Z2=1), and the remaining 10 receive method 3
(Z1=0, Z2=0). After training, each participant is given a test of their strength (Y) .
Model 1: E(Y) =  X
Model 2: E(Y) =  X + 1Z1 + 2Z2
Model 3: E(Y) =  X + 1Z1 + 2Z2+ 1XZ1 + 2XZ2



R12 = .204
R22 = .438
R32 = .534
p.4.a. Test whether the slopes relating Y to X differ for the three exercise regimen. H0: __________________
Test Statistic ________________________________ Rejection Region: _____________________ Reject H0 ? ______
p.4.b. Assuming no interaction, test whether the three regimens differ, after controlling for X H0: __________________
Test Statistic ________________________________ Rejection Region: _____________________ Reject H0 ? ______
p.4.c. Based on Model 2, give the adjusted means for the 3 regimens. (X-bar=29)
Coefficientsa
Model
1
2
3
(Constant)
X
(Constant)
X
Z1
Z2
(Constant)
X
Z1
Z2
XZ1
XZ2
Unstandardized
Coefficients
B
Std. Error
18.088
7.382
.676
.252
17.258
7.639
.721
.238
3.224
2.590
-4.650
2.503
9.456
11.772
.970
.373
28.333
15.588
-12.390
17.222
-.885
.526
.299
.577
Standardized
Coefficients
Beta
.452
.482
.228
-.328
.649
2.002
-.875
-1.735
.606
t
2.450
2.683
2.259
3.030
1.245
-1.857
.803
2.604
1.818
-.719
-1.684
.519
Sig.
.021
.012
.032
.005
.224
.075
.430
.016
.082
.479
.105
.609
a. Dependent Variable: Y
Regimen 1 _________________________ Regimen 2 _______________________ Regimen 3 ___________________
Q.5. A study compared 3 foods on serum glucose levels. A sample of 12 subjects were selected, and randomized such
that 4 people received Food A, 4 received Food B, and 4 received Food C. Each subject’s glucose levels were observed at
3 time points after eating the meal (15, 30, and 45 minutes).
p.5.a. Complete the following ANOVA table.
Source
Food
Subject(Food)
Time
Food*Time
Error2
Total
df
SS
MS
F
F(.05)
Significant?
1020.67
413.33
170.17
869.67
128.17
2602.00
p.5.b. The means for each food/time combination are given below. Use Bonferroni’s method to compare all
pairs of food, separately for each time (treat each time as a separate “family”, when making adjustment).
Food\Time
1
2
3
Time 1:
Food1
1
19
22
26
2
35
20
27.25
Food2
Food3
3
31.5
11.5
35.75
Time 2:
Food2
Food3
Food1
Time 3:
Food2
Food1
Food3
Q.6. A study was conducted to compare 5 Nitrogen levels (0, 45, 90, 135, 180 kg/hectare) and Rice Straw
(Absent/Present) on Rice Grain Yield (100s of kg/hectare). The experiment was conducted as a split-plot design, with
whole plot factor being Nitrogen level, and sub-plot factor being Rice Straw. The experiment was conducted in 3 blocks
(Years).
p.6.a. Complete the following Analysis of Variance Table.
ANOVA
Source
WP
Block
WP*Block
SP
WP*SP
Error
Total
df
SS
MS
F
F(.05)
5693.63
216.82
60.18
110.82
0.96
6098.94
p.6.b. Assuming no significant Nitrogen/Rice Straw Interaction, compute Bonferroni’s Minimum significant Difference
for comparing pairs of Nitrogen level effects. Show which levels are significantly different.
Nitrogen
0
45
90
135
180
Mean
48.65
75.19
79.07
85.85
85.96
Rice Straw
0
1
Mean
73.02
76.87
Bonferoni MSD = ________________________
Nit0
Nit45
Nit90
Nit135
Nit180
p.6.c. Compute a 95% Confidence Interval for the effect of Using Rice Straw, versus not using rice straw.
Have an Excellent Summer!