DS 303
Spring 2004
Exam # 3
Name: _____KEY______________
Show All your Work
1.
Mid-Valley Travel Agency (MVTA) has offices in 12 cities. The company
believes that its monthly airline bookings are related to the mean income in those
cities and has collected the following data:
Location Bookings
1
1098
2
1131
3
1120
4
1142
5
971
6
1403
7
855
8
1054
9
1081
10
982
11
1098
12
1387
Income
43299
45021
40290
41893
30620
48105
27482
33025
34687
28725
37892
46198
Simple linear regression model was used to analyze the data. The partial computer
output is given below:
SUMMARY OUTPUT
Regression Statistics
Multiple R
0.879189
R Square
0.772974
Adjusted R Square 0.750271
Standard Error
78.16735
Observations
12
ANOVA
df
Regression
Residual
Total
Intercept
Income
1
10
11
SS
MS
F
208036.3 208036.3 34.04775
61101.35 6110.135
269137.7
Coefficients Standard Error t Stat
P-value
371.6758
128.5571 2.891133 0.016076
0.019381
0.003322
a)
What is the estimated least square regression line?
ŷ = 371.68 + .019x
ŷ = (bookings)
b)
x = (income)
What is the value of the coefficient of determination (R2)? What does it
mean?
R2 = .77
77% of the variability in the number of bookings is due to the income.
c)
Forecast the number of bookings when the mean income is $51385.
ŷ = 371.68 + .019(51385) = 1347.99
≈ 1348
d)
Is there a significant relation between monthly airline bookings and the mean
income? Test this at 5% level (state the null and alternative hypothesis, the
value of your test statistic, the p-value or the decision rule, and your
conclusion).
Ho: β1 = 0
Ha: β1 = ≠ 0
t = b1 = .019381 = 5.834
S(b1) .003322
Since t = 5.384 > 2.228 Reject Ho. There is statistically significant relation
Between number of booking s and the average income.
e)
Give a 95% confidence interval estimate of the average increase in monthly
bookings. Explain what it means.
b1 ± t*s(b1)
t* = t(.025,10) = 2.228
.019381 ± 2.228(.003322)
.019381± .0074
(.012, .03)
For every $1000 increase in mean income there will be 12 to more bookings
A tanning parlor located in a major shopping center near a large New England city has the
following history of customers over the last four years (data are in hundreds of customers):
Year
1
2
3
4
Number of
Moving
Centered
CMA
Seasonal
Seasonal
Cycle
Quarters Customers
Average
Moving Average
Trend
Factor
Index
Factor
1
3.50
2
2.90
3
2.00
4
1
.73
1.028 .98
1.247 1.27
.986 1.01
2.9
2.975
2.90
.672
3.20
3.05
3.113
3.10
4.10
3.175
3.288
3.29
2
3.40
3.4
3.45
3.49
3
2.90
3.5
3.638
3.68
.797
4
3.60
3.775
3.913
3.88
.920
1
5.20
4.05
4.075
4.08
1.276
2
4.50
4.1
4.213
4.27
1.068
3
3.10
4.325
4.438
4.47
.699
4
4.50
4.55
4.613
4.66
.976
1
6.10
4.675
4.838
4.86
1.261
2
5.00
5
5.188
5.05
.964
3
4.40
5.375
4
6.00
1.026
1.004
.999
.989
.989
1.009
.999
.987
.993
.990
.995
1.027
5.25
5.45
a) Find a four period moving average for each quarter.
b) Find the centered moving average for the sample.
c) Find the seasonal factors and the seasonal indexes.
ASF
.723
.975
1.261
1.006
3.965
Quarters
Q3
Q4
Q1
Q2
SI
Quarter
(4/3.965)*.723 = .73
Q3
(4/3.965)*.975 =.98
Q4
(4/3.965)*1.26 = 1.271
Q1
(4/3.965)*1.006 = 1.01
Q2
d) Find the cycle factors.
e) Use the multiplicative decomposition method to forecast the number of customers
for each quarter of year 4.
FY1 = (4.86)(1.27)(.995) = 6.09
FY2 = (5.05)(1.01)(1.027) = 5.24
FY3 = (5.25)(.73)(.993) = 3.81
FY4 = (5.45)(.98)(.990) = 5.29
Multiple Choice Questions
Select the best answer
1.
In the linear model, the slope coefficient i measures the expected change in Y
per unit change in Xi given the other independent variables are fixed.
A) True
2.
t- distribution with 9 degrees of freedom.
t- distribution with 8 degrees of freedom.
t- distribution with 19 degrees of freedom.
t- distribution with 18 degrees of freedom.
None of the above.
Stepwise regression is an approach to choosing the independent variables to be
included in a multiple regression equation.
A) True
4.
B) False
C) Not enough information
A company has computed a seasonal index for its quarterly sales. Which of the following
statements is not correct?
A)
B)
C)
D)
E)
5.
C) Not enough information
In a test of the distribution of the anti-fungus activity of a chemical compound,
fungus is grown in petri dishes with different concentrations of the compound and
the diameter of the fungus colonies is measured after one day. There are 20
dishes, two at each of 10 concentrations. A plot of diameter against concentration
shows a straight-line pattern, with higher concentrations giving smaller diameters.
Least squares regression is used to analyze the data. What distribution is used in
the test of the hypothesis that concentration has no effect on diameter?
A)
B)
C)
D)
E)
3.
B) false
The sum of the four quarterly seasonal index numbers is 4.
An index of .75 for quarter-one sales indicates that sales were 25 percent lower
than average sales.
An index of 1.10 indicates sales 10% above the norm.
The index for any quarter must be between 0 and 1.
The average index is 1.
The long-term trend of a time series in the decomposition model is estimated using
A)
B)
C)
D)
E)
a nonlinear time trend.
the actual un smoothed data.
the centered moving average data.
the series of seasonal factors.
All of the above.
6.
The F-statistic reported in standard multiple regression computer packages tests which
hypothesis?
A)
B)
C)
D)
7.
The Y-intercept of the simple regression model
A)
B)
C)
D)
E)
8.
When Y increases by one, X increases by 3.5.
When X increases by one, Y increases by 3.5.
The regression line crosses the Y-axis at -14.
X and Y are positively related.
None of the above.
Income is used to predict savings. For the regression equation Y = 1,000 + .10X, which
of the following is true?
A)
B)
C)
D)
10.
rarely has a useful interpretation.
almost always has a useful interpretation.
is always a positive number.
is always positive when the correlation between the dependent and independent
variable is positive.
All the above.
The Y-intercept of a regression line is -14 and the slope is 3.5. Which of the following is
not correct?
A)
B)
C)
D)
E)
9.
H0: 1 ≠ 2 ≠ 3 ≠ .. ≠ K ≠ 0.
H0: 1 + 2 + 3 + .. + K = 0.
H0: 1 = 2 = 3 = .. = K = 0.
H0: The set of independent variables has a significant linear influence on the
dependent variable..
Y is income, X is savings, and income is the independent variable.
Y is income, X is savings, and savings is the independent variable.
Y is savings, X is income, and savings is the independent variable.
Y is savings, X is income, and income is the independent variable.
The least squares procedure minimizes the sum of
A)
B)
C)
D)
E)
the residuals.
squared maximum error.
absolute errors.
squared residuals.
None of the above.
11.
In simple linear regression model, testing the null hypothesis that the slope
coefficient is zero uses what sampling distribution?
A)
B)
C)
D)
E)
Normal.
Chi-square.
t distribution with n-1 degrees of freedom.
Standard Normal.
None of the above.