Exercise
1a
1. A chemist used a 24 factorial design to study the effects of temperature, pH, concentration, and agitation rate on yield (measured in grams).If the standard deviation of an individual observation is 6 grams, what is the variance of the temperature effect?
2. A metallurgical engineer is about to begin a comprehensive study to determine the effects of six variables on the strength of a certain type of alloy.
If a 26 factorial design were used, how many runs would be made?
i. If (Sigma)2 is the experimental error variance of an individual observation, what is the variance of a main effect?
ii. What is the usual formula for a 99% confidence interval for the main effect of variable 1?
iii. On the basis of some previous work it is believed that Sigma = 8000 pounds. If the experimenter wants 99% confidence intervals for the main effects and interactions whose lengths are equal to 4000 pounds (i.e.,the upper limit minus the lower limit is equal to 4000 pounds), how many replications of the 26 factorial design will be required?
3. Write a design matrix for a 32 factorial design and one for a 23 factorial design.
3. Using a 23 factorial design, a chemical engineer studied three variables (temperature, pH, and agitation rate) on the yield of a chemical reaction and obtained the following data. Using Yate`s algorithm, estimate the main effects and interactions. Plot the data. Say what the results might mean, stating the assumptions you make
|
X1 |
X2 |
X3 |
Y |
|
-1 |
-1 |
-1 |
60 |
|
+1 |
-1 |
-1 |
61 |
|
-1 |
+1 |
-1 |
54 |
|
+1 |
+1 |
-1 |
75 |
|
-1 |
-1 |
+1 |
58 |
|
+1 |
-1 |
+1 |
61 |
|
-1 |
+1 |
+1 |
55 |
|
+1 |
+1 |
+1 |
75 |
where
X1 = ( temperature – 150 degC ) / 10 degC
X2 = ( pH – 8.0 ) / 0.5
X3 = ( agitation rate – 30 rpm ) / 5 rpm
Y = yield ( % theoretical)