Download Exercise 1

Practice Problems
Problem 12.1:
We presented the data for the weight loss of a compound for different
amounts of time the compound was exposed to the air and the humidity of
the environment during exposure. The complete data is presented in Table
12.3.
Table 12.3
Weight
Loss (Y)
4.3
5.5
6.8
8.0
4.0
5.2
6.6
7.5
2.0
4.0
5.7
6.5
Exposure
Time
(X1)
Relative
Humidity
(X2)
4
5
6
7
4
5
6
7
4
5
6
7
0.2
0.2
0.2
0.2
0.3
0.3
0.3
0.3
0.4
0.4
0.4
0.4
(a) Set up the multiple regression equation for the model in Table 12.3.
Use the SAS Printout to answer the following problems:
SAS Printout for Problem 12.1
Model: MODEL1
Dependent Variable: Y
Analysis of Variance
Source
Model
Error
C Total
Root MSE
Dep Mean
C.V.
DF
2
9
11
Weight Loss (Y)
Sum of
Squares
31.12417
1.34500
32.46917
0.38658
5.50833
7.01810
Mean
Square
15.56208
0.14944
R-square
Adj R-sq
F Value
104.133
0.9586
0.9494
Prob>F
0.0001
2
Parameter Estimates
Variable
INTERCEP
X1
X2
DF
1
1
1
Variable
INTERCEP
X1
X2
DF
1
1
1
Parameter
Estimate
0.666667
1.316667
-8.000000
Standard
Error
0.69423219
0.09981464
1.36676829
T for H0:
Parameter=0
0.960
13.191
-5.853
Variable
Label
Intercept
Exposure Time (X1)
Relative Humidity (X2)
(b) What are the sample estimates for?
(c) What is the least squares prediction equation?
(d) Find SSE, MSE and s.
(e) Test H0: 1 = 0 against Ha: 1  0. Use  = 0.10.
(f) Test H0: 2 = 0 against Ha: 2 < 0. Use  = 0.01.
Prob > |T|
0.3620
0.0001
0.0002
3
(g) Find a 95% confidence interval for .
Problem 12.2:
A manufacturer of laundry detergent was interested in testing a new product
prior to market release. One area of concern was the relationship between
the height of the detergent suds in a washing machine as a function of the
amount of detergent added to the wash cycle and the degree of agitation in
the wash cycle (measured in minutes). The complete data are presented in
Table 12.4.
Table 12.4
Height
(Y)
Agitation
(X1)
Amount
(X2)
1
1
1
1
1
2
2
2
2
2
3
3
3
3
3
6
7
8
9
10
6
7
8
9
10
6
7
8
9
10
28.1
32.3
34.8
38.2
43.5
60.3
63.7
65.4
69.2
72.9
88.2
89.3
94.1
95.7
100.6
The SAS Printout for Table 12.4 is as follows:
SAS Printout for Problem 12.2
Model: MODEL1
Dependent Variable: Y
Analysis of Variance
Source
Model
Error
C Total
Root MSE
DF
2
12
14
Height (Y)
Sum of
Squares
8792.16533
20.53200
8812.69733
1.30805
Mean
Square
4396.08267
1.71100
R-square
F Value
2569.306
0.9977
Prob>F
0.0001
4
Dep Mean
C.V.
65.08667
2.00971
Adj R-sq
0.9973
Parameter Estimates
Parameter
Estimate
-19.406667
29.100000
3.286667
Variable
INTERCEP
X1
X2
Variable
INTERCEP
X1
X2
DF
1
1
1
Standard
Error
2.10917045
0.41364236
0.23881653
T for H0:
Parameter=0
-9.201
70.351
13.762
Prob > |T|
0.0001
0.0001
0.0001
Variable
Label
Intercept
Agitation (X1)
Amount (X2)
(a) Report the least squares prediction equation.
(b) Find the standard deviation of the regression model.
(c) Does the data provide sufficient evidence to conclude that the degree of
agitation in the wash cycle is important to the height of the detergent suds?
Use = 0.05.
(d) Test H0: 2 = 0 against Ha: 2 > 0 using  = 0.01. Why is it reasonable to
conduct a one-tailed test rather than a two-tailed test of this hypothesis?
What is the observed significance level for this test?