1. A experiment using a single replicate of a 2⁴ factorial design. After analyzing the data, the following results are available:

Model Term	Effect Estimate	Sum of Squares	% Contribution
Intercept	X	X	X
A-A	-9.75	380.25	40.4952
B-B	-0.25		0.0266241
C-C	-0.25	0.25	0.0266241
D-D		90.25	9.61129
AB	4.5	81
AC	0.5	1	0.106496
AD	-3.5	49	5.21832
BC	-1	4	0.425985
BD	-2	16	1.70394
CD	1	4	0.425985
ABC	-6.25	156.25	16.64
ABD	4.75	90.25	9.61129
ACD	-0.25	0.25	0.0266241
BCD	-2.75	30.25	3.22151
ABCD	3		3.83387

The grand average of all 16 observations is 83.75 and the total sum of squares is 939.00.

Answer the following questions.

1. Fill in the 4 blanks in the table above,

1. Is this an I-optimal design (circle the correct answer)? Yes

1. A experimenter also ran 4 center points (the response data is not included in the calculations above) and observed the following results: 88 89, 92, and 87. The mean square for pure error is

1. 2.00
2. 3.85
3. 4.67
4. 5.25
5. None of the above

1. The value of the F-statistic for pure quadratic curvature is
   1. 10.51
   2. 9.98
   3. 8.75
   4. 9.50
   5. None of the above

1. The table below shows the coded and natural (engineering) units for the factors. Find the regression model(s) for predicting the response in terms of the natural units assuming that the model contains only the main effects of A and D and the AD interaction.

Factor	Natural levels	Coded levels (x’s)
A – time	80, 120 (minutes)	-1, 1
B - temperature	150, 200 (deg C)	-1, 1
C - pressure	(100, 150) psi	-1, 1
D - catalyst type	Organic, inorganic	-1, 1

1. Consider the 2² factorial design shown
2. 
   1. Write the design matrix in coded units and the appropriate response totals in the table below.

Run			Response
1
2
3
4

1. What is the estimate of the AB interaction effect?

1. What is the model regression coefficient for the AB interaction effect? ­­­­­­­­­­­­­­­­__________________

1. In the full model ANOVA how many degrees of freedom are available for pure error?

6.8. A bacteriologist is interested in the effects of two dif- ferent culture media and two different times on the growth of a particular virus. He or she performs six replicates of a 2 design, making the runs in random order. Analyze the bacte- rial growth data that follow and draw appropriate conclu- sions. Analyze the residuals and comment on the model's adequacy. Time (h) 12 18 21 23 20 37 38 35 1 Culture Medium 22 28 26 39 38 36 25 24 29 31 29 30 2 26 25 27 34 33 35