Question

The yield of a chemical process is being studied. The two most important variables are thought to be the pressure and the temperature. Three levels of each factor are selected, and a factorial experiment with two replicates is performed. The yield data are as follows:

	Pressure (psig)
Temperature (ºC)	200	215	230
150	90.4	90.7	90.2
	90.2	90.6	90.4
160	90.1	90.5	89.9
	90.3	90.6	90.1
170	90.5	90.8	90.4
	90.7	90.9	90.1

1. Use the LSD test to determine which levels of the pressure factor are significantly different.
2. Use the LSD test to determine which levels of the temperature factor are significantly different.
3. Suppose that we wish to reject the null hypothesis with a high probability if the difference in the true mean yield at any two pressures is as great as 0.5. If a reasonable prior estimate of the standard deviation of yield is 0.1, how many replicates should be run?

Answer 1

Use the LSD test to determine which levels of the pressure factor are significantly different.

The output is:

Post hoc analysis
p-values for pairwise t-tests for Factor 2
		230 psi	200 psi	215 psi
		90.18	90.37	90.68
230 psi	90.18
200 psi	90.37	.0411
215 psi	90.68	.0001	.0026
Tukey simultaneous comparison t-values (d.f. = 9)
		230 psi	200 psi	215 psi
		90.18	90.37	90.68
230 psi	90.18
200 psi	90.37	2.38
215 psi	90.68	6.50	4.11
critical values for experimentwise error rate:
		0.05	2.79
		0.01	3.84

215 psi and 230 psi pressure are statistically significant.

215 psi and 200 psi pressure are statistically significant.

All other pairs are significantly different.

Use the LSD test to determine which levels of the temperature factor are significantly different.

The output is:

Post hoc analysis
p-values for pairwise t-tests for Factor 1
		160C	150C	170C
		90.25	90.42	90.57
160C	90.25
150C	90.42	.0586
170C	90.57	.0026	.0832
Tukey simultaneous comparison t-values (d.f. = 9)
		160C	150C	170C
		90.25	90.42	90.57
160C	90.25
150C	90.42	2.17
170C	90.57	4.11	1.95
critical values for experimentwise error rate:
		0.05	2.79
		0.01	3.84

170C and 160C temperature are statistically significant.

All other pairs are significantly different.

Suppose that we wish to reject the null hypothesis with a high probability if the difference in the true mean yield at any two pressures is as great as 0.5. If a reasonable prior estimate of the standard deviation of yield is 0.1, how many replicates should be run?

The number of replications should be 3.

The output would look like as follows:

			Factor 2
	Means:
		200 psi	215 psi	230 psi
	150C	90.23	90.60	90.17	90.33
Factor 1	170C	90.50	90.77	90.20	90.49
		90.37	90.68	90.18	90.41
	3	replications per cell
ANOVA table
Source	SS	df	MS	F	p-value
Factor 1	0.109	1	0.1089	3.44	.0884
Factor 2	0.768	2	0.3839	12.12	.0013
Interaction	0.041	2	0.0206	0.65	.5399
Error	0.380	12	0.0317
Total	1.298	17
Post hoc analysis
p-values for pairwise t-tests for Factor 2
		230 psi	200 psi	215 psi
		90.18	90.37	90.68
230 psi	90.18
200 psi	90.37	.0996
215 psi	90.68	.0004	.0095
Tukey simultaneous comparison t-values (d.f. = 12)
		230 psi	200 psi	215 psi
		90.18	90.37	90.68
230 psi	90.18
200 psi	90.37	1.78
215 psi	90.68	4.87	3.08
critical values for experimentwise error rate:
		0.05	2.67
		0.01	3.56