14-10 RESPONSE SURFACE METHODS AND DESIGNS (CD ONLY) 569MIND-EXPANDING EXERCISESx 1 x 2 yx 1 x 2 y
1 1 211 0 1.5 168
1 1 92 0 1.5 179
1 1 216 0 0 122
1 1 99 0 0 175
1.5 0 222 0 0 157
1.5 0 48 0 0 14614-68. Consider the experiment in Exercise 14-19.
Suppose that a center point had been run (replicated five
times) and the responses were 45, 40, 41, 47, and 43.
(a) Estimate the experimental error using the center
points. Compare this to the estimate obtained origi-
nally in Exercise 14-19 by pooling apparently non-
significant effects.
(b) Test for look of fit, using 0.05.
14-69. Consider the data from Exercise 14-13, repli-
cate 1 only. Suppose that a center point with four
replicates is added to these eight factorial runs and the
responses are 425, 400, 437, and 418.
(a) Estimate the facter effects.
(b) Test for lack of fit using 0.05
(c) Test for main effects and interactions using 0.05.
(d) Analyze residuals and draw conclusions.
To work problem 14-70 through 14-74 you will need to
read Section 14-10 on the CD.
14-70. An article in Rubber Age(Vol. 89, 1961, pp.
453 – 458) describes an experiment on the manufacture of a
product in which two factors were varied. The factors are
reaction time (hr) and temperature (C). These factors are
coded asx 1 (time12)8 and x 2 (temperature
250)30. The following data were observed where yis the
yield (in percent):Run
Number x 1 x 2 y
1 1 0 83.8
2 1 0 81.7
3 0 0 82.4
4 0 0 82.9
50 1 84.7
6 0 1 75.9
7 0 0 81.2
8 1.414 0.414 81.3
9 1.414 1.414 83.1
10 1.414 1.414 85.3
11 1.414 1.414 72.7
12 0 0 82.0(a) Plot the points at which the experimental runs were
made.(b) Fit a second-order model to the data. Is the second-
order model adequate?
(c) Plot the yield response surface. What recommenda-
tions would you make about the operating condi-
tions for this process?
14-71. Consider the first-order modelwhere 1 xi1. Find the direction of steepest ascent.
14-72. A manufacturer of cutting tools has devel-
oped two empirical equations for tool life (y 1 ) and tool cost
(y 2 ). Both models are functions of tool hardness (x 1 ) and
manufacturing time (x 2 ). The equations areand both equations are valid over the range1.5
xi1.5. Suppose that tool life must exceed 12 hours
and cost must be below $27.50.
(a) Is there a feasible set of operating conditions?
(b) Where would you run this process?
14-73. An article in Tappi(Vol. 43, 1960, pp. 38–44)
describes an experiment that investigated the ash value of
paper pulp (a measure of inorganic impurities). Two vari-
ables, temperature Tin degrees Celsius and time tin hours,
were studied, and some of the results are shown in the fol-
lowing table. The coded predictor variables shown areand the response yis (dry ash value in %) 103.x 1 1 T 7752
115
, x 2 1 t 32
1.5yˆ 2 23 3 x 1 4 x 2yˆ 1 10 5 x 1 2 x 2yˆ 50 1.5x 1 0.8x 2(a) What type of design has been used in this study? Is
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