Exersise 9
1.The following results were obtained by a chemist ( where all data are given in coded units):
|
Temperature X1 |
pH X2 |
Yield ot chemical reaction Y |
|
-1 |
-1 |
6 |
|
1 |
-1 |
14 |
|
-1 |
1 |
13 |
|
1 |
1 |
7 |
|
-1 |
-1 |
4 |
|
1 |
-1 |
14 |
|
-1 |
1 |
10 |
|
1 |
1 |
8 |
(a)Fit the model Mu = Beta0 + Beta1 * X1 + Beta2 * X2 by the method of least squares ( Mu is the mean value of the yield).
(b)Obtain an estimate of the experimental error variance of an individual yield reading , assuming this model is adequate.
(c)Fit the model Mu = Beta0 + Beta1 * X1 + Beta2 * X2 + Beta11 * X12+ Beta22 * X22+ Beta12 * X1 * X2. What difficulties arise? Explain these . What parameters and linear combinations of parameters can be estimated with this design?
(d)Consider what experimental design is being used here , and make an analysis in terms of factorial effects . Relate the two analyses.
2.Suppose that a chemical engineer uses the method of least squares with the data given below to estimate the parameters Beta0, Beta1, and Beta 2 in the model
Mu = Beta0 + Beta1 * X1 + Beta2 * X2
where Mu is the mean response ( peak gas temperature , deg.R ). He obtains the following fitted equation:
Yhat = 1425.8 + 123.3X1 + 96.7X2
where Yhat is the predicted value of the response, and X1 and X2 are given by the equations
X1 = 0.5( compression ratio – 12.0)
X2 = ( cooling water temperature – 550 deg.R) / 10 deg.R
|
Test |
Compression ratio |
Cooling water Temperature (deg.R) |
Peak gas Temperature (deg.R) Y |
Residuals Y - Yhat |
|
1 |
10 |
540 |
1220 |
14 |
|
2 |
14 |
540 |
1500 |
48 |
|
3 |
10 |
560 |
1430 |
41 |
|
4 |
14 |
560 |
1650 |
4 |
|
5 |
8 |
550 |
1210 |
31 |
|
6 |
16 |
550 |
1700 |
29 |
|
7 |
12 |
530 |
1200 |
-32 |
|
8 |
12 |
570 |
1600 |
-29 |
|
9 |
12 |
550 |
1440 |
14 |
|
10 |
12 |
550 |
1450 |
24 |
|
11 |
12 |
550 |
1350 |
-76 |
|
12 |
12 |
550 |
1360 |
-66 |
The tests were run in random order . Check the engineer’s calculations. Do you think the model form is adequate?
3.The following data were obtained from a study on chemical reaction system:
|
Trial |
Temperature (deg.R) |
Concentration (%) |
pH |
Yield |
|
|
|
1 |
150 |
40 |
6 |
73 |
70 |
|
|
2 |
160 |
40 |
6 |
75 |
74 |
|
|
3 |
150 |
50 |
6 |
78 |
80 |
|
|
4 |
160 |
50 |
6 |
82 |
82 |
|
|
5 |
150 |
40 |
7 |
75 |
||
|
6 |
160 |
40 |
7 |
76 |
79 |
|
|
7 |
150 |
50 |
7 |
87 |
85 |
82 |
|
8 |
160 |
50 |
7 |
89 |
88 |
The 16 runs were performed in random order , trial 7 being run three times , trial 5 once and all the rest twice . ( the intention was to run each test twice , but a mistake was made in setting the concentration level on one of t he tests.)
(a)Analyze these data . One thing you might wish to consider is fitting the following model to them : Mu = Beta0 + Beta1 * X1 + Beta2 * X2 + Beta3 * X3, where X1, X2 , and X3 (preferably, though not necessarily, in coded units ) refer to the variables t emperature , concentration , and pH, respectively , and Mu is the mean value of the yield. Alternatively , a good indication of how elaborate a model is needed will be obtained by averaging the available data and making a preliminary factorial anal ysis on the results.
(b)Compare these two approaches.