Stat 301 A -- Fall 2015 -- Midterm exam 2
3 November 2015
Information and JMP output
Problem 1: Intelligence and future salary
These data were analyzed in an influential (and contentious) book on intelligence. The observations in
this data set are a subset of the individuals in a nationwide random sample of youth. The subset is those
individuals who were 16 or 17 year olds in 1978, took the Armed Forces Qualifying Test, and were still
alive in 2005. The study goal is to evaluate associations between the subject’s characteristics (including
socioeconomic status and intelligence) and their income in 2005 (INCOME) and then to develop a model
to predict 2005 income. The X variables used in this problem are:
MotherEd: number of years of schooling for the mother (e.g. 12 means finished high school)
FatherEd: number of years of schooling for the mother
FamilyIncome78: family income in 1978, in dollars
AFQT: subject’s score on the Armed Forces Qualifying Test, no units
The AFQT score was treated as a measure of intelligence; higher values are considered to be more
intelligent. The other three variables are measures of socioeconomic status; higher values are
considered to be higher status.
The response variable, INCOME, is the subject’s income in 2005, in dollars.
The output includes:
1. Correlation coefficients between each pair of variables
2. Summary of Fit, ANOVA table, and Parameter Estimates from Fit Model with
MotherEd, FatherEd, FamilyIncome78, and AFQT.
3. Summary of Fit, ANOVA table, and Parameter Estimates from Fit Model with 6 variables:
MotherEd, FatherEd, FamilyIncome78, and AFQT, one squared term and FamilyIncome78*AFQT.
1. Correlation coefficients for each pair of variables
MotherEd
FatherEd
FamilyIncome78
AFQT
Income
MotherEd
1.0000
0.6148
0.3229
0.4409
0.1648
FatherEd FamilyIncome78
0.6148
0.3229
1.0000
0.3526
0.3526
1.0000
0.4495
0.3061
0.1904
0.1743
AFQT
0.4409
0.4495
0.3061
1.0000
0.2981
Income
0.1648
0.1904
0.1743
0.2981
1.0000
2. Response Income, Model: MotherEd, FatherEd, FamilyIncome78, and AFQT
Summary of Fit
Root Mean Square Error
Mean of Response
Observations (or Sum Wgts)
9254.748
104379.8
2584
Analysis of Variance
Source
Model
Error
C. Total
DF Sum of Squares
4
2.4071e+10
2579
2.2089e+11
2583
2.4496e+11
Mean Square
6.0178e+9
85650352
F Ratio
70.2601
Prob > F
<.0001*
Parameter Estimates
Term
Intercept
MotherEd
FatherEd
FamilyIncome78
AFQT
Estimate
96878.474
-13.83038
139.91682
0.05642
88.45966
Std Error
844.7635
91.1411
68.0730
0.0143
7.6366
t Ratio
114.68
-0.15
2.06
3.93
11.58
Prob>|t|
<.0001
0.8794
0.0399
<.0001
<.0001
Std Beta
0
-0.00372
0.051069
0.080278
0.252173
VIF
.
1.7201567
1.7655996
1.192387
1.3554371
3. Response Income, Model: MotherEd, FatherEd, FamilyIncome78, AFQT, 1 squared term,
and AFQT*FamilyIncome78
Response Income
Summary of Fit
Root Mean Square Error
Mean of Response
Observations (or Sum Wgts)
9262.602
104382.4
2584
Analysis of Variance
Source
Model
Error
C. Total
DF Sum of Squares
6
2.4714e+10
2577
2.211e+11
2583
2.4581e+11
Mean Square
4.1191e+9
85795801
F Ratio
48.0102
Prob > F
<.0001*
Parameter Estimates
Term
Intercept
MotherEd
FatherEd
FamilyIncome78
AFQT
FamilyIncome78*FamilyIncome78
AFQT*FamilyIncome78
Estimate
95834.972
-26.6895
142.1196
0.1115
112.6681
0.00000038
-0.0012
Std Error
995.0404
91.241
68.076
0.043
12.768
0.00000061
0.0005
t Ratio
96.31
-0.29
2.09
2.59
8.82
0.62
-2.38
Prob>|t|
<.0001
0.77
0.037
0.0096
<.0001
0.54
0.018
Problem 2: longevity of mammal species
Investigators compiled information about the average body mass (in kg), average metabolic rate (units
unknown) and typical longevity (in years) for 95 species of mammals. These can be assumed to be a
simple random sample of all mammal species. There is one row of data for each species. The
investigators want to model longevity as a function of body mass and metabolic rate. Preliminary
inspection of the data indicates that is necessary to log transform longevity. All analyses use log
longevity as the Y variable.
The output includes:
4. Scatterplot matrix of mass, metabolic rate, log mass, log metabolic rate, and log longevity
5. Fit Model output for log longevity as response and model effects of mass and metabolic rate
6. Fit Model output for log longevity as response and model effects of log mass and log metabolic rate
7. Information about predictions at selected combinations of mass and metabolic rate using output 6.
4. Scatterplot Matrix of mass, metabolic rate, log mass, log metab, and log longevity
5. Response: log Longevity, model effects: mass, metabolic rate
Summary of Fit
Root Mean Square Error
Mean of Response
Observations (or Sum Wgts)
0.87127
2.022222
95
Analysis of Variance
Source
Model
Error
C. Total
DF Sum of Squares
2
33.88288
92
69.83819
94
103.72107
Mean Square
16.9414
0.7591
F Ratio
22.3175
Prob > F
<.0001*
Parameter Estimates
Term
Intercept
Mass
Metab
Estimate
1.7364868
-0.008884
0.0001748
Std Error
0.099417
0.001875
0.000032
t Ratio
17.47
-4.74
5.37
Prob>|t|
<.0001*
<.0001*
<.0001*
6. Response: log Longevity, model effects: log mass, log metabolic rate
Summary of Fit
Root Mean Square Error
Mean of Response
Observations (or Sum Wgts)
0.377345
2.022222
95
Analysis of Variance
Source
Model
Error
C. Total
DF Sum of Squares
2
90.62126
92
13.09981
94
103.72107
Mean Square
45.3106
0.1424
F Ratio
318.2166
Prob > F
<.0001*
Parameter Estimates
Term
Intercept
log Mass
log Metab
Estimate
3.7193428
0.5346157
-0.316104
Residual by Predicted Plot
Std Error
0.484075
0.064361
0.085577
t Ratio
7.68
8.31
-3.69
Prob>|t|
<.0001*
<.0001*
0.0004*
Std Beta
0
1.641772
-0.73007
VIF
.
28.455897
28.455897
7. Predictions at selected combinations of mass and metabolic rate using output 6.
Note: The row labelled mean is the prediction at the average mass and metabolic rate.
Common
Name
Sloth
Camel
Cat
Beluga
whale
Bat
Mean
Mass
Metab
log
Mass
log
Metab
Predicted
StdErr
log
Predicted
longevity
Y
3.79
407
3
331
23600
546
1.33
6.01
1.10
5.80
10.07
6.30
2.60
3.75
2.31
0.0812
0.0792
0.0419
170
0.022
65
23000
15.6
7560
5.14
-3.82
4.17
10.04
2.75
8.93
3.29
0.81
0.0875
0.0631
0.0634
1.32
346
0.28
5.85
2.02
0.0387