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Stats for Strategy Spring 2015
Final Exam Information and Study Tips
A. Final Exam
• Time: 3:00–5:00 pm Thursday, May 14
• You MUST bring and show your UI ID in order to take the exam.
• Location:
◦ The ﬁnal exam location is
AUD MH (Macbride Hall)
◦ Use triple seating where possible (2 seats between students)
• 50 multiple-choice questions. (There is a base score of 25% built-in partial credit, the
same as for midterm exams.)
• Roughly 60% of ﬁnal exam questions review Topics 1–9 (Exams 1–3) and the remaining
40% cover Topics 10 and 11 (Logistic Regression and Time Series.)
B. Study Tips
• Since no exam covers all topics, a review of Exams 1–3 alone is not a suﬃcient review
of Topics 1–9, but is an excellent place to start!
Print a clean copy of the exam from the website and take the exam again.
(Time yourself for 60 minutes.) How many questions did you answer correctly?
Did you do better than you did the ﬁrst time?
Then focus on questions which you missed — study the related Topic Notes
and get help during oﬃce hours and Friday Stats Lab.
• Consider reviewing/reworking Notebook Examples and Homework Questions from earlier in the course, especially for Topics which you found most challenging.
• Another idea: Review or Re-take Discussion Quizzes. (Recall that solutions for your
version of the quiz are posted at ICON −→ CONTENT.)
• This handout contains multiple-choice Practice Questions for Topics 10 and 11.
(Additional Practice Questions for Topic 11 are available on the course website.)
• Final Exam Formulas are found at the end of the Notebook and on the last page of this
handout.
(See Review Week Schedule and Course Outline next page)
C. Review Week Schedule
• Monday, May 4
In-Class Practice Questions for Topics 10/11
• Wednesday, May 6
• Friday, May 8
Q/A session with Prof. Whitten over Topics 10/11
Q/A session with TAs and Prof. Whitten:
* TAs return this week’s quizzes during lecture.
* Individual Q/A with TAs in the BACK of W10.
* Group Q/A with Whitten at the FRONT of W10 Auditorium.
* Ask questions over:
◦ Topics 10/11 practice questions
◦ Exams 1–3. Practice questions for Exams 1–3.
◦ Topics 1–11 Notes and Homework
• Friday Stats Lab, May 8
2:30–4:30 in C207
D. Exam Week Schedule
Also TA Huan Zhang has
volunteered office hours Tuesday
May 12 9:30-11:00 AM in S321
• Bring UI ID and attend correct location for Friday exam.
Regular oﬃce hours don’t meet during Finals Week.
•
But Prof. Whitten will hold his usual (Monday and Wednesday) oﬃce
hours during Finals Week to assist with review questions.
E. Stats for Strategy Course Outline
(Exam 1)
• Topics 1-3 (Review/Transition from ﬁrst Business Stats course)
POPULATIONS, SAMPLES, CONFIDENCE INTERVALS,
HYPOTHESIS TESTS, COMPARING TWO PROPORTIONS
(Exam 2)
• Topic 4: CHI-SQUARE TESTS (Multiple Proportions in categorical data)
• Topic 5: COMPARING TWO MEANS
• Topic 6: ANOVA (Multiple Means from numerical data)
(Exam 3)
• Topic 7: CORRELATION
• Topic 8: SIMPLE REGRESSION
• Topic 9: MULTIPLE REGRESSION
(After Exam 3)
• Topic 10: LOGISTIC REGRESSION
• Topic 11: TIME SERIES
Statistics for Strategy
Topics 10 and 11 In-Class Practice Questions
Monday Week 16
Directions: There are 24 multiple choice questions. Choose the single best answer for each
question. The answers will be posted (on the last page) by 5:00 p.m. Monday on the Exams
page of the main Stats website. (Additional practice questions for Topic 11 are also available
there.) Carry all calculations to at least four decimal places.
Disclaimer: These practice questions are intended to familiarize you with the style of the
ﬁnal exam. The content of actual exam questions will diﬀer.
Questions 1–2.
1. The odds in favor of an event can be any number within which range?
(a) all positive and negative numbers
(b) all nonnegative numbers
(c) all numbers between 1 and inﬁnity
(d) all numbers between 0 and 1, inclusive
(e) None of the above
2. In multiple logistic regression,
(a) the response variable has more than two possible values and the predictor variable
has two possible values.
(b) the response variable has two possible values and there are several predictor variables.
(c) the predictor variable has two possible values and there are several response variables.
(d) the response variable has more than two possible values and there are several
predictor variables.
1
Questions 3–9. The following table and graph show natural gas sales by quarter measured
in British thermal units (Btu) for a Midwestern U.S. energy company.
(Use at least 4 decimal places precision in all calculations.)
Quarter
1
2
3
4
5
6
7
8
9
10
11
12
13
Natural
Date
Gas Sales
4th Quarter 2003
170
1st Quarter 2004
148
2nd Quarter 2004
141
3rd Quarter 2004
150
4th Quarter 2004
161
1st Quarter 2005
137
2nd Quarter 2005
132
3rd Quarter 2005
158
4th Quarter 2005
160
1st Quarter 2006
145
2nd Quarter 2006
128
3rd Quarter 2006
134
4th Quarter 2006
160
Time Series Plot of Natural Gas Sales
)u
tB
no
il
ib
(
se
la
S
sa
Gl
ar
ut
aN
170
160
150
140
130
6
5
6
6
6
4
5
5
5
3
4
4
4
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
2
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2
r
r
r
r
r
r
r
r
r
r
r
r
r
e
e
e
e
e
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e
e
e
e
e
e
e
rt
rt
rt
rt
rt
rt
rt
rt
rt
rt
rt
rt
rt
a
a
a
a
a
a
a
a
a
a
a
a
a
u
u
u
u
u
u
u
u
u
u
u
u
u
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
Q
d
d
d
st
st
st
th
th
th
th
rd
rd
rd
n
n
n
1
1
1
4
4
4
4
3
3
3
2
2
2
Date
2
Below are a MINITAB worksheet, commands applied to that worksheet, and resulting output.
C1
Quarter
1
2
3
4
5
6
7
8
9
10
11
12
13
C2
Date
4th Quarter 2003
1st Quarter 2004
2nd Quarter 2004
3rd Quarter 2004
4th Quarter 2004
1st Quarter 2005
2nd Quarter 2005
3rd Quarter 2005
4th Quarter 2005
1st Quarter 2006
2nd Quarter 2006
3rd Quarter 2006
4th Quarter 2006
C3
Natural Gas Sales
170
148
141
150
161
137
132
158
160
145
128
134
160
C4
S1
0
1
0
0
0
1
0
0
0
1
0
0
0
C5
S2
0
0
1
0
0
0
1
0
0
0
1
0
0
C6
S3
0
0
0
1
0
0
0
1
0
0
0
1
0
Stat > Regression > Regression > Fit Regression Model > (Response: Natural Gas Sales)
> (Predictors: Quarter S1 S2 S3 ) > OK
Regression Analysis: Natural Gas Sales versus Quarter, S1, S2, S3
The regression equation is
Natural Gas Sales = 170 - 1.08 Quarter - 20.5 S1 - 29.1 S2 - 14.3 S3
Predictor
Constant
Quarter
S1
S2
S3
Coef
170.307
-1.0795
-20.496
-29.083
-14.337
S = 6.30115
SE Coef
4.580
0.4750
4.836
4.813
4.836
R-Sq = 84.7%
T
37.18
-2.27
-4.24
-6.04
-2.96
P
0.000
0.053
0.003
0.000
0.018
R-Sq(adj) = 77.0%
Analysis of Variance
Source
Regression
Residual Error
Total
DF
4
8
12
SS
1758.36
317.64
2076.00
MS
439.59
39.70
F
11.07
3
P
0.002
3. Calculate the 3rd-quarter seasonal factor for natural gas sales.
(a) 0.9984
(b) 1.0999
(c) 0.9028
(d) 0.9875
(e) 1.0016
4. Use seasonal factors to calculate the company’s seasonally-adjusted natural gas sales
in the 3rd of 2004, in billions of Btu.
(a) 149.76
(b) 150.24
(c) 138.89
(d) 162.00
(e) 166.15
5. Is there an increase or a decrease in the core (nonseasonal) rate of natural gas sales
from the 3rd quarter of 2005 to the 4th quarter of 2005, and why?
(a) Increase since 158.00 < 160.00
(b) Increase since 158.25 > 158.00
(c) Decrease since 160.00 > 157.75
(d) Decrease since 157.75 > 145.47
(e) Increase since 158.25 < 175.84
6. Is there an increase or a decrease in the core (nonseasonal) rate of natural gas sales
from the 3rd quarter of 2005 to the 3rd quarter of 2006, and why?
(a) Decrease since 158.00 > 134.00
(b) Decrease since 157.75 > 134.00
(c) Increase since 157.75 < 164.32
(d) Decrease since 157.75 > 145.47
(e) Increase since 158.00 > 134.00
7. Make the best possible forecast for natural gas sales in the 4th quarter of 2008 (in
billions Btu), when seasonality is modeled by addition.
(a) 141.75
(b) 147.73
(c) 162.75
(d) 138.96
(e) 147.64
8. Make the best possible forecast for natural gas sales in the 4th quarter of 2008 (in
billions Btu), when seasonality is modeled by multiplication.
(a) 141.75
(b) 147.73
(c) 162.75
(d) 138.96
(e) 147.64
9. What is the interpretation of the coeﬃcient for S2 in the regression output?
(a) Second-quarter natural gas sales average 29.083 billion Btu less than natural gas
sales in an average quarter.
(b) Second-quarter natural gas sales average 29.083 billion Btu less than fourthquarter natural gas sales.
(c) Second-quarter natural gas sales are divided by a factor of 29.083 to obtain a
seasonal adjustment.
(d) Second-quarter natural gas sales are multiplied by a factor of 29.083 to obtain a
seasonal adjustment.
(e) Second-quarter natural gas sales are multiplied by a factor of e−29.083 to obtain a
seasonal adjustment.
4
(more space for Questions 3–9)
5
Questions 10–17.
The table below shows annual oil production in the United States, in billions of barrels.
Also shown are the predictions from exponential smoothing, where 80% of the weight for a
prediction in the next year is placed on the current year. These predictions are missing for
some years, as indicated by the symbol ∗ in the table.
(Carry all calculations to at least 3 decimal places.)
Time
Period
1
2
3
4
5
6
7
8
9
10
11
12
13
14
..
.
Year
1949
1950
1951
1952
1953
1954
1955
1956
1957
1958
1959
1960
1961
1962
..
.
Oil Production
1.84
1.97
2.25
2.29
2.36
2.31
2.48
2.62
2.62
2.45
2.57
2.57
2.62
2.68
..
.
Exponentially-Smoothed
Predictions
∗
∗
∗
2.189
2.270
2.342
2.316
2.447
2.585
2.613
2.483
2.553
2.567
2.609
..
.
47
48
49
50
51
1995
1996
1997
1998
1999
2.39
2.37
2.35
2.28
2.15
2.450
2.402
2.376
2.355
∗
10. Find the exponentially-smoothed prediction for the year 1949.
(a) 1.840 (b) 1.944 (c) 1.970 (d) 2.020
(e) The answer cannot be calculated.
11. Find the exponentially-smoothed prediction for the year 1950.
(a) 1.840 (b) 1.944 (c) 1.970 (d) 2.020
(e) The answer cannot be calculated.
12. Find the exponentially-smoothed prediction for the year 1951.
(a) 1.840 (b) 1.944 (c) 1.970 (d) 2.020
(e) The answer cannot be calculated.
(continued)
6
13. Find the exponentially-smoothed prediction for the year 1999.
(a) 1.944 (b) 2.288 (c) 2.355 (d) 2.295
(e) None of the answers is correct to the third decimal place.
14. Find the exponentially-smoothed forecast for the year 2000.
(a) 1.944 (b) 2.288 (c) 2.355 (d) 2.295
(e) None of the answers is correct to the third decimal place.
15. Find the moving-average prediction for the year 1950, using k = 4.
(a) 1.840 (b) 1.944 (c) 1.970 (d) 2.020
(e) The answer cannot be calculated.
16. Find the moving-average forecast for the year 2000, using k = 4.
(a) 1.944 (b) 2.288 (c) 2.355 (d) 3.130
(e) None of the answers is correct to the third decimal place.
17. Oil production in the year 2000 was 2.14 billion barrels. Which of the two time series
provides a better forecast, the exponentially-smoothed series or the moving-average
series?
(a) Exponential smoothing
(b) Moving average
(c) Neither
7
Questions 18–21.
A franchise is a license granted to a local ﬁrm by a national corporation which allows the
ﬁrm to operate under the corporate brand name. (For instance, Burger King restaurants
and Shell service stations are franchise ﬁrms.)
A study of franchise ﬁrms considered two issues: whether or not the ﬁrm is proﬁtable, and
whether or not the ﬁrm has an exclusive territory. (An exclusive territory means that no
other franchise ﬁrm operates in the local area.) The data are shown below. Use α = 0.05.
Observed Numbers of Firms
Exclusive Territory
Profitable Yes
No Total
Yes
108
15
123
No
34
13
47
Total
142
28
170
18. Which combination of MINITAB worksheet and MINITAB commands shown on page 9
correctly analyze this problem?
(a) A
(b) B
(c) C
(d) D
19. Refer to the MINITAB output. Which of the following conclusions is correct?
(a) Exclusivity is a signiﬁcant predictor of proﬁtability.
(b) Exclusivity is not a signiﬁcant predictor of proﬁtability.
(c) Proﬁtability is a signiﬁcant predictor of exclusivity.
(d) Proﬁtability is not a signiﬁcant predictor of exclusivity.
(e) None of the conclusions is correct.
20. Refer to the MINITAB output. Which of the following interpretations is correct?
(a) The odds that a proﬁtable ﬁrm has an exclusive territory are approximately 1.013
times the odds that an unproﬁtable ﬁrm has an exclusive territory.
(b) The chances that a proﬁtable ﬁrm has an exclusive territory are approximately
1.013 times the chances that an unproﬁtable ﬁrm has an exclusive territory.
(c) The chances that a ﬁrm with an exclusive territory is proﬁtable are approximately
1.013 times the chances that a ﬁrm without exclusive territory is proﬁtable.
(d) The odds that a ﬁrm with an exclusive territory is proﬁtable are approximately
1.013 times the odds that a ﬁrm without exclusive territory is proﬁtable.
(e) None of the interpretations is correct.
8
A.
C1
Exclusive
1
0
C2
Proﬁtable
108
34
C3
Total
123
47
Stat > Regression > Binary Logistic Regression > Fit Binary Logistic Model
> (Choose Response in event/trial format)
> (Number of events: Exclusive , Number of trials: Total)
> (Categorical predictors: Profitable) > Results > (Display of results: Expanded tables)
> OK > OK
B.
C1
Exclusive
1
0
C2
Proﬁtable
108
15
C3
Total
142
28
Stat > Regression > Binary Logistic Regression > Fit Binary Logistic Model
> (Choose Response in event/trial format)
> (Number of events: Exclusive , Number of trials: Total)
> (Categorical predictors: Profitable) > Results > (Display of results: Expanded tables)
> OK > OK
C.
C1
Exclusive
1
0
C2
Proﬁtable
108
15
C3
Total
142
28
Stat > Regression > Binary Logistic Regression > Fit Binary Logistic Model
> (Choose Response in event/trial format)
> (Number of events: Profitable , Number of trials: Total)
> (Categorical predictors: Exclusive) > Results > (Display of results: Expanded tables)
> OK > OK
D.
C1
Exclusive
1
0
C2
Proﬁtable
108
34
C3
Total
123
47
Stat > Regression > Binary Logistic Regression > Fit Binary Logistic Model
> (Choose Response in event/trial format)
> (Number of events: Profitable , Number of trials: Total)
> (Categorical predictors: Exclusive) > Results > (Display of results: Expanded tables)
> OK > OK
9
MINITAB output:
Binary Logistic Regression: Profitable versus Exclusive
Response Information
Variable
Value
Profitable Event
Non-event
Total
Total
Coefficients
Term
Coef
Constant
0.143
Exclusive
1
1.013
Count
123
47
170
Name
Event
SE Coef
0.379
95% CI
(-0.600, 0.886)
Z-Value
0.38
P-Value
0.706
VIF
0.427
( 0.176, 1.849)
2.37
0.018
1.00
Odds Ratios for Categorical Predictors
Level A
Level B Odds Ratio
95% CI
Exclusive
1
0
2.7529 (1.1923, 6.3561)
21. Suppose that Firm A does not have an exclusive territory, but Firm B does. Which of
the following conclusions is correct?
The odds that Firm A is successful are approximately
(a) 0.987 times the odds that Firm B is successful.
(b) 0.364 times the odds that Firm B is successful.
(c) 0.268 times the odds that Firm B is successful.
(d) 1.013 times the odds that Firm B is successful.
(e) 2.750 times the odds that Firm B is successful.
10
Questions 22–24.
Undergraduate students at Miami University in Oxford, Ohio were surveyed in order to
evaluate the eﬀects of price on the purchase of pizza from Pizza Hut. 220 students were
asked to suppose that they were going to have a large two-topping pizza delivered to their
residence. They were asked to select from either Pizza Hut or from another pizza shop of
their choice.
The price they would have to pay to get a Pizza Hut pizza varied from survey to survey. For
example, some surveys used the price currently charged by the Oxford Pizza Hut, $11.49.
Other prices investigated were $8.49, $9.49, $10.49, $12.49, $13.49, and $14.49.
The variables are deﬁned as
{
1 if Pizza Hut is choice of pizza shop
y=
0 otherwise
x = price of Pizza Hut pizza (dollars)
and MINITAB output is shown below. Use α = 0.05.
Binary Logistic Regression: Purchase versus Price
Response Information
Variable Value Count
Purchase 1
39
0
181
Total
220
Coefficients
Term
Coef
Constant
1.24327
Price
-0.250343
(Event)
SE Coef
1.02299
0.0933882
95% CI
( -0.76,
3.25)
(-0.4334, -0.0673)
Z-Value
1.22
-2.68
P-Value
0.224
0.007
VIF
1.00
Odds Ratios for Continuous Predictors
Odds Ratio
95% CI
Price
0.7785 (0.6483, 0.9349)
22. Which of the following conclusions is correct?
(a) The probability that pizza is purchased from Pizza Hut increases as the price of
Pizza Hut pizza increases.
(b) The odds that the price of Pizza Hut pizza increases go up as demand for Pizza
Hut pizza increases.
(c) The odds that pizza is purchased from Pizza Hut increase as the price of Pizza
Hut pizza increases.
(d) The probability that pizza is purchased from Pizza Hut is not related to the price
of Pizza Hut pizza.
(e) None of the conclusions is correct.
(continued)
11
23. Which of the following interpretations is correct?
We are 95% conﬁdent that
(a) the odds of ordering from Pizza Hut change by a factor of between 0.65 and 0.93
for every one-dollar increase in the price of Pizza Hut pizza.
(b) the odds of ordering from Pizza Hut are divided by between 0.65 and 0.93 for
every one-dollar increase in the price of Pizza Hut pizza.
(c) the odds of ordering from Pizza Hut are subtracted by between 0.65 and 0.93 for
every one-dollar increase in the price of Pizza Hut pizza.
(d) the odds of ordering from Pizza Hut are reduced by 0.250343 for every one-dollar
increase in the price of Pizza Hut pizza.
(e) None of the interpretations is correct.
24. Estimate the probability that a student orders from Pizza Hut if the price of pizza
is $8.99.
(a) 0.1954
(b) 0.0946
(c) 0.2675
(d) 0.3653
(e) 0.1635
12
Final Exam Formulas
√
s
x¯ ± t∗ √
n
pb ± z ∗
pb (1 − pb )
n
t=
√
(b
p1 − pb2 ) ± z ∗ σpb1 −bp2
Z = √(
1
n1
σpb1 −bp2 ≈
(b
p1 − pb2 )
)
1
+ n2 pb (1 − pb )
pb =
x¯ − µ0
√
s/ n
p0 (1−p0 )
n
pb1 (1 − pb1 ) pb2 (1 − pb2 )
+
n1
n2
x1 + x2
n1 + n2
χ2 =
s2p = (average of s21 , s22 , . . . , s2k ) = MSE
DFG = k − 1, where k = # of groups
pb − p0
Z=√
∑ (O − E)2
, df = (r−1)(c−1)
E
all cells
F =
MSG
MSE
DFE = N − k, where N = total # of measurements
1 ∑( xi − x¯ )( yi − y¯ )
r=
n − 1 i=1
sx
sy
n
F =
MSR
MSE
with p and n − p − 1 degrees of freedom
∑n
SSRegression
R =
SSTotal
2
2
s = MSE =
− ybi )2
n−p−1
i=1 (yi
Error df = (n − 2) for simple regression
Error df = (n − p − 1) for multiple regression (where p = # of predictors)
sy
bi
b1 = r
b0 = y¯−b1 x¯
bi ±t∗ SEbi
yb±t∗ SEµb
t=
sx
SEbi
F =
(R12 − R22 )/q
(1 − R12 )/(n − p − 1)
where
• R12 is from full model,
• numerator df = q
R22 is from reduced model
denominator df = n − p − 1
• p = # variables in full model,
odds =
Ft = ybt + ebt
p
1−p
ybt+1 =
q = # variables being tested as a group
p=
odds
1 + odds
log odds = β0 + β1 x
yt + yt−1 + · · · + yt−k−1
k
13
ybt+1 = wyt + (1 − w)b
yt
Solution
1. b
The number 0 is a possible value for the odds: If probability = 0 (i.e., an impossible
event), the odds in favor are also 0.
2. b
3. e
4. a
5. d
6. a
7. e
8. b
9. b
10. a
The ﬁrst prediction for exponential smoothing is yb1 = y1 (page 778 text reading)
11. a
12. b
13. d
14. e
2.179
15. e
16. b
17. a
18. c
19. a
20. e
The odds that a ﬁrm with an exclusive territory is proﬁtable are 2.75 times the
odds that a ﬁrm without exclusive territory is proﬁtable.
21. b
22. e
The probability that pizza is purchased from Pizza Hut decreases as the price of
Pizza Hut pizza increases.
23. a
24. c
14