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Statistics for Managers Study Set 1
Exam 17: Getting Ready to Analyze Data
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Question 341
Short Answer
SCENARIO 17-11 A logistic regression model was estimated in order to predict the probability that a randomly chosen university or college would be a private university using information on mean total Scholastic Aptitude Test score (SAT) at the university or college, the room and board expense measured in thousands of dollars (Room/Brd), and whether the TOEFL criterion is at least 550 (Toefl550 = 1 if yes, 0 otherwise.) The dependent variable, Y, is school type (Type = 1 if private and 0 otherwise). The Minitab output is given below:
-Referring to Scenario 17-11, which of the following is the correct expression for the estimated model? a)
Y
=
−
27.118
+
0.015
Y = - 27.118 + 0.015
Y
=
−
27.118
+
0.015
SAT
−
0.390
- 0.390
−
0.390
Toefl
550
+
2.078
550 + 2.078
550
+
2.078
Room / Brd b)
Y
^
=
−
27.118
+
0.015
S
A
T
−
0.390
\hat { Y } = - 27.118 + 0.015 \mathrm { SAT } - 0.390
Y
^
=
−
27.118
+
0.015
SAT
−
0.390
Toefl550
+
2.078
R
o
o
m
/
B
r
d
+ 2.078 \mathrm { Room } / \mathrm { Brd }
+
2.078
Room
/
Brd
c)
ln
(
\ln (
ln
(
odds ratio
)
=
−
27.118
+
0.015
) = - 27.118 + 0.015
)
=
−
27.118
+
0.015
SAT
−
0.390
- 0.390
−
0.390
Toefl
550
+
2.078
R
o
o
m
/
B
r
d
550 + 2.078 \mathrm { Room } / \mathrm { Brd }
550
+
2.078
Room
/
Brd
d)
ln
(
\ln (
ln
(
estimated odds ratio
)
=
−
27.118
+
0.015
) = - 27.118 + 0.015
)
=
−
27.118
+
0.015
SAT
−
0.390
- 0.390
−
0.390
Toefl
550
+
2.078
R
o
o
m
/
B
r
d
550 + 2.078 \mathrm { Room } / \mathrm { Brd }
550
+
2.078
Room
/
Brd
Question 342
Multiple Choice
SCENARIO 17-1 A real estate builder wishes to determine how house size (House) is influenced by family income (Income) , family size (Size) , and education of the head of household (School) . House size is measured in hundreds of square feet, income is measured in thousands of dollars, and education is in years. The builder randomly selected 50 families and ran the multiple regression. Microsoft Excel output is provided below: SUMMARY OUTPUT Regression Statistics
Multiple R
0.865
R Square
0.748
Adjusted R Square
0.726
Standard Error
5.195
Observations
50
\begin{array} { l l } \text { Multiple R } & 0.865 \\ \text { R Square } & 0.748 \\ \text { Adjusted R Square } & 0.726 \\ \text { Standard Error } & 5.195 \\ \text { Observations } & 50 \end{array}
Multiple R
R Square
Adjusted R Square
Standard Error
Observations
0.865
0.748
0.726
5.195
50
ANOVA
d
f
SS
MS
F
Signif
F
Regression
3605.7736
1201.9245
0.0000
Residual
1214.2264
26.3962
Total
49
4820.0000
\begin{array} { l c c c c c } & d f & \text { SS } & \text { MS } & F & \text { Signif } F \\ \text { Regression } & & 3605.7736 & 1201.9245 & & 0.0000 \\ \text { Residual } & & 1214.2264 & 26.3962 & & \\ \text { Total } & 49 & 4820.0000 & & & \end{array}
Regression
Residual
Total
df
49
SS
3605.7736
1214.2264
4820.0000
MS
1201.9245
26.3962
F
Signif
F
0.0000
Coeff
StdError
t
Stat
P
-value
Intercept
−
1.6335
5.8078
−
0.281
0.7798
Income
0.4485
0.1137
3.9545
0.0003
Size
4.2615
0.8062
5.286
0.0001
School
−
0.6517
0.4319
−
1.509
0.1383
\begin{array} { l c l c c } & \text { Coeff } & \text { StdError } & t \text { Stat } & P \text {-value } \\ \text { Intercept } & - 1.6335 & 5.8078 & - 0.281 & 0.7798 \\ \text { Income } & 0.4485 & 0.1137 & 3.9545 & 0.0003 \\ \text { Size } & 4.2615 & 0.8062 & 5.286 & 0.0001 \\ \text { School } & - 0.6517 & 0.4319 & - 1.509 & 0.1383 \end{array}
Intercept
Income
Size
School
Coeff
−
1.6335
0.4485
4.2615
−
0.6517
StdError
5.8078
0.1137
0.8062
0.4319
t
Stat
−
0.281
3.9545
5.286
−
1.509
P
-value
0.7798
0.0003
0.0001
0.1383
-Referring to Scenario 17-1, when the builder used a simple linear regression model with house size (House) as the dependent variable and education (School) as the independent variable, he Obtained an r2 value of 23.0%. What additional percentage of the total variation in house size has Been explained by including family size and income in the multiple regression?
Question 343
Short Answer
SCENARIO 17-12 The marketing manager for a nationally franchised lawn service company would like to study the characteristics that differentiate home owners who do and do not have a lawn service. A random sample of 30 home owners located in a suburban area near a large city was selected; 15 did not have a lawn service (code 0) and 15 had a lawn service (code 1). Additional information available concerning these 30 home owners includes family income (Income, in thousands of dollars), lawn size (Lawn Size, in thousands of square feet), attitude toward outdoor recreational activities (Atitude 0 = unfavorable, 1 = favorable), number of teenagers in the household (Teenager), and age of the head of the household (Age). The Minitab output is given below:
-Referring to Scenario 17-12, what should be the decision ('reject' or 'do not reject') on the null hypothesis when testing whether Teenager makes a significant contribution to the model in the presence of the other independent variables at a 0.05 level of significance?
Question 344
True/False
SCENARIO 17-5 You worked as an intern at We Always Win Car Insurance Company last summer. You notice that individual car insurance premiums depend very much on the age of the individual, the number of traffic tickets received by the individual, and the population density of the city in which the individual lives. You performed a regression analysis in EXCEL and obtained the following information:
Regression Statistics
Multiple R
0.63
R Square
0.40
Adjusted R Square
0.23
Standard Error
50.00
Observations
15.00
ANOVA
d
f
SS
MS
F
Significance
F
Regression
3
5994.24
2.40
0.12
Residual
11
27496.82
Total
45479.54
Coefficients
Standard
Error
t Stat
P-value
Lower 99.0%
Upper 99.0%
Intercept
123.80
48.71
2.54
0.03
−
27.47
275.07
AGE
−
0.82
0.87
−
0.95
0.36
−
3.51
1.87
TICKETS
21.25
10.66
1.99
0.07
−
11.86
54.37
DENSITY
−
3.14
6.46
−
0.49
0.64
−
23.19
16.91
\begin{array}{l}\begin{array}{lr}\hline {\text { Regression Statistics }} \\\hline \text { Multiple R } & 0.63 \\\text { R Square } & 0.40 \\\text { Adjusted R Square } & 0.23 \\\text { Standard Error } & 50.00 \\\text { Observations } & 15.00 \\\hline\end{array}\\\\\text { ANOVA }\\\begin{array}{lrcccr}\hline & d f &{\text { SS }} & \text { MS } & \text { F }&{\text { Significance } F} \\\hline \text { Regression } & 3 & & 5994.24 & 2.40 & 0.12 \\\text { Residual } & 11 & 27496.82 & & & \\\text { Total } & & 45479.54 & & & \\\hline\end{array}\\\\\begin{array}{lrrrrrr}\hline & \text { Coefficients } & \begin{array}{c}\text { Standard } \\\text { Error }\end{array} & \text { t Stat } & \text { P-value } & \text { Lower 99.0\% } & \text { Upper 99.0\% } \\\hline \text { Intercept } & 123.80 & 48.71 & 2.54 & 0.03 & -27.47 & 275.07 \\\text { AGE } & -0.82 & 0.87 & -0.95 & 0.36 & -3.51 & 1.87 \\\text { TICKETS } & 21.25 & 10.66 & 1.99 & 0.07 & -11.86 & 54.37 \\\text { DENSITY } & -3.14 & 6.46 & -0.49 & 0.64 & -23.19 & 16.91 \\\hline\end{array}\end{array}
Regression Statistics
Multiple R
R Square
Adjusted R Square
Standard Error
Observations
0.63
0.40
0.23
50.00
15.00
ANOVA
Regression
Residual
Total
df
3
11
SS
27496.82
45479.54
MS
5994.24
F
2.40
Significance
F
0.12
Intercept
AGE
TICKETS
DENSITY
Coefficients
123.80
−
0.82
21.25
−
3.14
Standard
Error
48.71
0.87
10.66
6.46
t Stat
2.54
−
0.95
1.99
−
0.49
P-value
0.03
0.36
0.07
0.64
Lower 99.0%
−
27.47
−
3.51
−
11.86
−
23.19
Upper 99.0%
275.07
1.87
54.37
16.91
AGE -0.82 0.87 -0.95 0.36 -3.51 1.87 TICKETS 21.25 10.66 1.99 0.07 -11.86 54.37 DENSITY -3.14 6.46 -0.49 0.64 -23.19 16.91 -Referring to Scenario 17-5, the multiple regression model is significant at a 10% level of significance.
Question 345
True/False
SCENARIO 17-10 Given below are results from the regression analysis where the dependent variable is the number of weeks a worker is unemployed due to a layoff (Unemploy) and the independent variables are the age of the worker (Age), the number of years of education received (Edu), the number of years at the previous job (Job Yr), a dummy variable for marital status (Married:
1
=
1 =
1
=
married,
0
=
0 =
0
=
otherwise), a dummy variable for head of household (Head:
1
=
1 =
1
=
yes,
0
=
0 =
0
=
no) and a dummy variable for management position (Manager:
1
=
1 =
1
=
yes,
0
=
0 =
0
=
no). We shall call this Model 1. The coefficient of partial determination (
R
y
2
R _ { \mathrm { y } } ^ { 2 }
R
y
2
(All raiables excopt
j
j
j
) ) of each of the 6 predictors are, respectively,
0.2807
0.2807
0.2807
,
0.0386
,
0.0317
,
0.0141
,
0.0958
0.0386,0.0317,0.0141,0.0958
0.0386
,
0.0317
,
0.0141
,
0.0958
, and
0.1201
0.1201
0.1201
.
Regression Statistics
Multiple R
0.7035
R Square
0.4949
Adjusted R
0.4030
Square
Standard
18.4861
Error
Observations
40
\begin{array}{lr}\hline{\text { Regression Statistics }} \\\hline \text { Multiple R } & 0.7035 \\\text { R Square } & 0.4949 \\\text { Adjusted R } & 0.4030 \\\text { Square } & \\\text { Standard } & 18.4861 \\\text { Error } & \\\text { Observations } & 40 \\\hline\end{array}
Regression Statistics
Multiple R
R Square
Adjusted R
Square
Standard
Error
Observations
0.7035
0.4949
0.4030
18.4861
40
ANOVA
\text { ANOVA }
ANOVA
Coefficients
Standard Error
t
Stat
P-value
Lower 95%
Upper 95%
Intercept
32.6595
23.18302
1.4088
0.1683
−
14.5067
79.8257
Age
1.2915
0.3599
3.5883
0.0011
0.5592
2.0238
Edu
−
1.3537
1.1766
−
1.1504
0.2582
−
3.7476
1.0402
Job Yr
0.6171
0.5940
1.0389
0.3064
−
0.5914
1.8257
Married
−
5.2189
7.6068
−
0.6861
0.4974
−
20.6950
10.2571
Head
−
14.2978
7.6479
−
1.8695
0.0704
−
29.8575
1.2618
Manager
−
24.8203
11.6932
−
2.1226
0.0414
−
48.6102
−
1.0303
\begin{array}{l}\begin{array} { l r r r r r r } \hline & \text { Coefficients } & \text { Standard Error } & { t \text { Stat } } & \text { P-value } & \text { Lower 95\% } & \text { Upper 95\% } \\\hline \text { Intercept } & 32.6595 & 23.18302 & 1.4088 & 0.1683 & - 14.5067 & 79.8257 \\\text { Age } & 1.2915 & 0.3599 & 3.5883 & 0.0011 & 0.5592 & 2.0238 \\\text { Edu } & - 1.3537 & 1.1766 & - 1.1504 & 0.2582 & - 3.7476 & 1.0402 \\\text { Job Yr } & 0.6171 & 0.5940 & 1.0389 & 0.3064 & - 0.5914 & 1.8257 \\\text { Married } & - 5.2189 & 7.6068 & - 0.6861 & 0.4974 & - 20.6950 & 10.2571 \\\text { Head } & - 14.2978 & 7.6479 & - 1.8695 & 0.0704 & - 29.8575 & 1.2618 \\\text { Manager } & - 24.8203 & 11.6932 & - 2.1226 & 0.0414 & - 48.6102 & - 1.0303 \\\hline\end{array}\end{array}
Intercept
Age
Edu
Job Yr
Married
Head
Manager
Coefficients
32.6595
1.2915
−
1.3537
0.6171
−
5.2189
−
14.2978
−
24.8203
Standard Error
23.18302
0.3599
1.1766
0.5940
7.6068
7.6479
11.6932
t
Stat
1.4088
3.5883
−
1.1504
1.0389
−
0.6861
−
1.8695
−
2.1226
P-value
0.1683
0.0011
0.2582
0.3064
0.4974
0.0704
0.0414
Lower 95%
−
14.5067
0.5592
−
3.7476
−
0.5914
−
20.6950
−
29.8575
−
48.6102
Upper 95%
79.8257
2.0238
1.0402
1.8257
10.2571
1.2618
−
1.0303
-Referring to Scenario 17-10 and using both Model 1 and Model 2, there is sufficient evidence to conclude that the independent variables that are not significant individually are also not significant as a group in explaining the variation in the dependent variable at a 5% level of significance?
Question 346
Multiple Choice
The Tampa International Airport (TIA) has been criticized for the waiting times associated with departing flights. While the critics acknowledge that many flights have little or no waiting times, Their complaints deal more specifically with the longer waits attributed to some flights. The critics Are interested in showing, mathematically, exactly what the problems are. Which type of Distribution would best model the waiting times of the departing flights at TIA?
Question 347
Multiple Choice
Suppose the light bulbs in a factory burn out at a rate of 50 bulbs per month. Which of the following distributions would you use to determine the probability that the next two light bulbs Will burn out 2 days apart?