Operations and Production Management
MGMT 405
Answer set 4
MGMT 405 Operations and Production Management
Answer set 4
(Reference chapters 3– William J. Stevenson-2007, ninth edition)
Problems and Solutions
1. Using the following data set and
Sales
Advertising
1
1.3
1.6
2
2.5
3
0
2
4
6
8
10
(a) Plot the relationship between sales and advertising and also shows the data points
on a graph.
(b) Write theoretically an equation of the form that describes the relationship in the
graph. (i.e., y=….)
(c) Determine the regression equation of the firm’s sales revenue (Y) on its
advertising expenditure (X).
(d) Calculate the values of Y for X=$0, $5, $10 and $15 million respectively and
briefly explain why would you not to be confident in the reliability of the
estimated value of Y for X=15?
© 2010/11, Sami Fethi, EMU, All Right Reserved, McGraw-Hill, 2007, 9. Ed.
1
Operations and Production Management
MGMT 405
Answer set 4
Ans:
(a)
sale
Sales
3.5
3
2.5
2
1.5
1
0.5
0
0
2
4
6
8
10
ads
(b) y= a + bx where a is the value of y when x=0 (i.e., where the line intersects the yaxis) and b is the slope of the line (the amount by which y changes for a one-unit
change in x).
(c) y=1 + 0.2 x, a=1 and b=(3-1)/(10-0)=0.2, . Where =(3-1) is the change in y, and
(10-0) is the change in x.
(d) y=1 + 0.2 (5)=2, y=1 + 0.2 (10)=3, y=1 + 0.2 (15)=4
© 2010/11, Sami Fethi, EMU, All Right Reserved, McGraw-Hill, 2007, 9. Ed.
2
Operations and Production Management
MGMT 405
Answer set 4
We are not confident in the reliability of the estimated value of Y for X=15 because 4 is
out of the data series.
2. The owner of a small hardware store has noted a sales pattren for window
locks that seems to parallel the number of break-ins reported each week in the
newspaper. The data are:
Sales
46
18
20
22
27
34
14
37
30
Break-ins
9
3
3
5
4
7
2
6
4
(a) Plot the data to determine which type of equation, linear or nonlinear is
appropriate.
(b) Obtain a regression equation for the data.
(c) Estimate sales when the number of break-ins is five as well as ten.
Ans:
(a)
50
Sales
40
30
20
10
0
0
2
4
6
8
10
Break-ins
© 2010/11, Sami Fethi, EMU, All Right Reserved, McGraw-Hill, 2007, 9. Ed.
3
Operations and Production Management
MGMT 405
Answer set 4
Sales
50
Sales
40
30
20
10
0
2
3
4
6
9
break-ins
The graph supports a linear relationship.
(b)
Regression Statistics
Multiple R
0.927622591
R Square
0.860483672
Adjusted R Square
0.840552768
Standard Error
4.09223514
Observations
9
Intercept
Break-ins (X)
Observation
1
2
3
4
5
6
7
8
9
Coefficients
(a) 7.129213483
(b) 4.275280899
Standard Error
3.394834629
0.650664086
Forecast (Y)
45.60674157
19.95505618
19.95505618
28.50561798
24.23033708
37.05617978
15.67977528
32.78089888
24.23033708
Error
0.393258427
-1.95505618
0.04494382
-6.505617978
2.769662921
-3.056179775
-1.679775281
4.219101124
5.769662921
t Stat
2.100018
6.570642
© 2010/11, Sami Fethi, EMU, All Right Reserved, McGraw-Hill, 2007, 9. Ed.
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Operations and Production Management
MGMT 405
Answer set 4
(c) Y= 7.129+ 4.275 (5)= 28.5 is within the data series.
Y= 7.129+ 4.275 (10)= 49.9 is out of the data series.
3. The manager of a seafood restaurant was asked to establish a pricing policy on
lobster dinners. Experimenting with prices produced the following data:
DAY(t)
Average number
sold per day (y)
Price (x)
DAY(t)
Average number
sold per day (y)
Price (x)
1
200
$6
7
160
8
2
190
6.5
8
155
8.25
3
188
6.75
9
156
8.50
4
180
7
10
148
8.75
5
170
7.25
11
140
9
6
162
7.50
12
133
9.25
a)
Estimate the regression equation of the restaurant’s lobster (Y) lobster on
price (X)
b)
Plot the estimated regression line and also shows the data points on a
graph.
c)
Calculate the values of Y for X=$5 and $10 respectively and briefly
explain why would you not to be confident in the reliability of the
estimated value of Y for X=5 and X=10?
d)
Calculate the standard error and t-test of the slope parameter.
e)
Calculate the adjusted coefficients determination (R2 ) and overall
statistical significance (F-value).
f)
Determine the correlation coefficient and interpret it.
© 2010/11, Sami Fethi, EMU, All Right Reserved, McGraw-Hill, 2007, 9. Ed.
5
Operations and Production Management
MGMT 405
Answer set 4
Ans:
(a)
b=
t
Y
$X
1
2
3
4
5
6
7
8
9
10
11
12
200
190
188
180
170
162
160
155
156
148
140
133
6
6.5
6.75
7
7.25
7.5
8
8.25
8.5
8.75
9
9.25
1982
∑Y
92.75
∑X
165.1666667
7.729166667
n xy   x y
n x   x 
2
-19.56
2
X*Y
Y²
X²
X-Xbar
(X-Xbar)²
1200
40000
36
-1.72917
2.990017
1235
36100
42.25
-1.22917
1.510851
1269
35344
45.5625
-0.97917
0.958767
1260
32400
49
-0.72917
0.531684
1232.5
28900
52.5625
-0.47917
0.229601
1215
26244
56.25
-0.22917
0.052517
1280
25600
64
0.270833
0.073351
1278.75
24025
68.0625
0.520833
0.271267
1326
24336
72.25
0.770833
0.594184
1295
21904
76.5625
1.020833
1.042101
1260
19600
81
1.270833
1.615017
1230.25
17689
85.5625
1.520833
2.312934
15081.5
∑XY
332142
∑Y²
729.0625
∑X²
-3.6E-15
a
12.18229
Y  b  X
n
316.35
© 2010/11, Sami Fethi, EMU, All Right Reserved, McGraw-Hill, 2007, 9. Ed.
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Operations and Production Management
Y^
e(Y-Y^)
e²
Y-Ybar
MGMT 405
(Y-Ybar)²
Answer set 4
Y^-Ybar
(Y^-Ybar)²
198.99
1.01
1.0201
34.83333
1213.361
33.82333
1144.0179
189.21
0.79
0.6241
24.83333
616.6944
24.04333
578.08188
184.32
3.68
13.5424
22.83333
521.3611
19.15333
366.85018
179.43
0.57
0.3249
14.83333
220.0278
14.26333
203.44268
174.54
-4.54
20.6116
4.833333
23.36111
9.373333
87.859378
169.65
-7.65
58.5225
-3.16667
10.02778
4.483333
20.100278
159.87
0.13
0.0169
-5.16667
26.69444
-5.29667
28.054678
154.98
0.02
0.0004
-10.1667
103.3611
-10.1867
103.76818
150.09
5.91
34.9281
-9.16667
84.02778
-15.0767
227.30588
145.2
2.8
7.84
-17.1667
294.6944
-19.9667
398.66778
140.31
-0.31
0.0961
-25.1667
633.3611
-24.8567
617.85388
135.42
-2.42
5.8564
-32.1667
1034.694
-29.7467
884.86418
143.3835
1.14E-13
1982.01
-0.01
Sb = 0.986
4781.667
143.38/(12-10)(12.18)
0.01
4660.8668
 (Y  Yˆ )
(n  k )  ( X  X )
t
t
 (Yˆ  Y )
 (Y  Y )
2
R 
2
2
t
t=
19.56/0.986
4660.86
0.987
4781.66
19.83
Y=a+bX
Y= 316.35-19.56X
(b)
© 2010/11, Sami Fethi, EMU, All Right Reserved, McGraw-Hill, 2007, 9. Ed.
7
2
Operations and Production Management
MGMT 405
Answer set 4
9
9.
25
8
8.
25
8.
5
8.
75
7
7.
25
7.
5
200
190
180
170
160
150
140
130
6
6.
5
6.
75
Sale
Lobster
price
sale
Lobster
200
190
180
170
160
150
140
130
0
2
4
6
8
10
price
(c) Y= 316.35-19.56(5)=218.55 is out of the data series.
Y= 316.35-19.56(10)=120.75 is out of the data series.
So we are not confident in the reliability of the estimated value of Y for Xs.
(d)
Sb =
 (Y  Yˆ )
(n  k )  ( X  X )
t
2
=
t
© 2010/11, Sami Fethi, EMU, All Right Reserved, McGraw-Hill, 2007, 9. Ed.
8
Operations and Production Management
Sb = 0.986
and
MGMT 405
Answer set 4
143.38/(12-10)(12.18)
t=b/ Sb=
t=
19.56/0.986
19.83
(e)
 (Yˆ  Y )
 (Y  Y )
2
R2 
2
=4660.86/4781.66=0.987
t
It appears that approximately 98% of the variation in sales can be accounted for by the price of
our product. This indicates that price is a good predictor of sales.
R2 /(k  1)
F
(1  R2 ) /(n  k )
F= (0.98/ (2-1)) / ((1-0.98)/ (12-2)) =490
(f)
r  R 2 withthe sign of bˆ
r=0.99
0r
r
n( xy)  ( x)(  y
n(  x 2 )  (  x ) 2 n(  y 2 )  (  y ) 2
r= (12)(15081)-(92.75)(1982)/√(12)(729.06)- (92.75)2 √(12)(332142)-(1982)=-0.99
© 2010/11, Sami Fethi, EMU, All Right Reserved, McGraw-Hill, 2007, 9. Ed.
9
Operations and Production Management
MGMT 405
Answer set 4
When we use the formula above, we can not only find its value but also its sign. –
means a high and negative relationship between price and sale (quantity demanded).
4. Long life Insurance has developed a linear model that it uses to determine the
amount of term life insurance a family of four should have based on the current age
of the household. The equation is:
Y = 150 – 0.1 X
Where
Y= insurance needed ($000).
X= current age of head of household.
(a) Plot the relationship on a graph.
(b) Use the equation to determine the amount of term life insurance to recommend for
a family of four if the head of the household is 30, 40, and 50 years old. Briefly
explain.
Ans:
(a) Give some values for X and find the corresponding values for Y.
Y
150
149
148
147
146
X
0
10
20
30
40
© 2010/11, Sami Fethi, EMU, All Right Reserved, McGraw-Hill, 2007, 9. Ed.
10
Operations and Production Management
MGMT 405
Answer set 4
insurance
an insurance company
151
150
149
148
147
146
145
144
0
10
20
30
40
household
(b)
Y = 150 – 0.1 (30)=147
Y = 150 – 0.1 (40)=146
Y = 150 – 0.1 (50)=145
As long as the age increases, the amount of insurance decreases. There is a
inverse relationship between the two variable under consideration.
© 2010/11, Sami Fethi, EMU, All Right Reserved, McGraw-Hill, 2007, 9. Ed.
11