Consider the experiment. Suppose that a center point with five replicates is added to the factorial runs and the responses are $2800,5600,4500,5400$, and 3600 . Compute an ANOVA with the sum of squares for curvature and conduct an $F$-test for curvature. Use $\alpha=0.05$.

The percentage of hardwood concentration in raw pulp, the freeness, and the cooking time of the pulp are being investigated for their effects on the strength of paper. The data from a three-factor factorial experiment are shown in the following table.

The residuals are found from $e_{i j k l}=y_{i j k l}-\bar{y}_{i j k}$. Graphically analyze the residuals from this experiment.

An article in Rubber Age describes an experiment on the manufacture of a product in which two factors were varied. The factors are reaction time (hr) and temperature $\left({ }^{\circ} \mathrm{C}\right)$. These factors are coded as $x_1=($ time -12$) / 8$ and $x_2=($ temperature -250$) / 30$.

Fit a second-order model to the data. Is the second-order model adequate?

The percentage of hardwood concentration in raw pulp, the freeness, and the cooking time of the pulp are being investigated for their effects on the strength of paper. The data from a three-factor factorial experiment are shown in the following table.

Compute approximate $P$-values for the $F$-ratios in part (a).

An article in Industrial Quality Control describes an experiment to investigate the effect of two factors (glass type and phosphor type) on the brightness of a television tube. The response variable measured is the current (in microamps) necessary to obtain a specified brightness level. The data are shown in the following table:

Test the hypotheses in part (a) and draw conclusions using the analysis of variance with $\alpha=0.05$.

An article in Industrial and Engineering Chemistry ["More on Planning Experiments to Increase Research Efficiency" (1970, pp. 60-65)] uses a $2^{5-2}$ design to investigate the effect on process yield of $A=$ condensation temperature, $B=$ amount of material 1,C= solvent volume, $D=$ condensation time, and $E$ $=$ amount of material 2. The results obtained are as follows:

Write down the complete defining relation and the aliases from the design.

An article in Industrial Quality Control describes an experiment to investigate the effect of two factors (glass type and phosphor type) on the brightness of a television tube. The response variable measured is the current (in microamps) necessary to obtain a specified brightness level. The data are shown in the following table:

State the hypotheses of interest in this experiment.

Consider a

$$2^2$$

factorial experiment with four center points. The data are (1) = 21, a = 125, b = 154, ab = 352, and the responses at the center point are 92, 130, 98, 152. Compute an ANOVA with the sum of squares for curvature and conduct an F-test for curvature. Use α = 0.05.

Consider the data . Suppose that the data from the second replicate were not available. Analyze the data from replicate I only and comment on your findings.

The percentage of hardwood concentration in raw pulp, the freeness, and the cooking time of the pulp are being investigated for their effects on the strength of paper. The data from a three-factor factorial experiment are shown in the following table.

|

$$\begin{array}{l}\text { Hardwood } \\\text { Concentration \% }\end{array}$$

|

$$\begin{array}{c}\text { Cooking } \\\text { Time } \\1.5 \text { hours } \\\text { Freeness }\end{array}$$

| | |

$$\begin{array}{c}\begin{array}{c}\text { Cooking } \\\text { Time }\end{array} \\\text { 2.0 hours } \\\text { Freeness }\end{array}$$

| | | | :---: | :---: | :---: | :---: | :---: | :---: | :---: | | | | | | | | | | | 350 | 500 | 650 | 350 | 500 | 650 | | 10 | 96.6 | 97.7 | 99.4 | 98.4 | 99.6 | 100.6 | | | 96.0 | 96.0 | 99.8 | 98.6 | 100.4 | 100.9 | | 15 | 98.5 | 96.0 | 98.4 | 97.5 | 98.7 | 99.6 | | | 97.2 | 96.9 | 97.6 | 98.1 | 96.0 | 99.0 | | 20 | 97.5 | 95.6 | 97.4 | 97.6 | 97.0 | 98.5 | | | 96.6 | 96.2 | 98.1 | 98.4 | 97.8 | 99.8 |

Analyze the data using the analysis of variance assuming that all factors are fixed. Use $\alpha=0.05$.

Four factors are thought to influence the taste of a soft-drink beverage: type of sweetener $(A)$, ratio of syrup to water $(B)$, carbonation level $(C)$, and temperature $(D)$. Each factor can be run at two levels, producing a $2^4$ design. At each run in the design, samples of the beverage are given to a test panel consisting of 20 people. Each tester assigns the beverage a point score from 1 to 10 . Total score is the response variable, and the objective is to find a formulation that maximizes total score. Two replicates of this design are run, and the results are shown in the table. Analyze the data and draw conclusions. Use $\alpha=0.05$ in the statistical tests.

$$\begin{array}{ccc} a b & 166 & 168 \\ c & 175 & 178 \\ a c & 179 & 183 \\ b c & 173 & 168 \\ a b c & 179 & 182 \\ d & 164 & 159 \\ a d & 187 & 189 \\ b d & 163 & 159 \\ a b d & 185 & 191 \\ c d & 168 & 174 \\ a c d & 197 & 199 \\ b c d & 170 & 174 \\ a b c d & 194 & 198 \end{array}$$

An article in Industrial and Engineering Chemistry ["More on Planning Experiments to Increase Research Efficiency" (1970, pp. 60-65)] uses a $2^{5-2}$ design to investigate the effect on process yield of $A=$ condensation temperature, $B=$ amount of material 1,C= solvent volume, $D=$ condensation time, and $E$ $=$ amount of material 2. The results obtained are as follows:

$$\begin{array}{rlrl} a e & =23.2 & c d & =23.8 \\ a b & =15.5 & a c e & =23.4 \\ a d & =16.9 & b d e & =16.8 \\ b c & =16.2 & a b c d e & =18.1 \end{array}$$

Verify that the design generators used were $I=A C E$ and $I=B D E$.

An article in Rubber Age describes an experiment on the manufacture of a product in which two factors were varied. The factors are reaction time (hr) and temperature $\left({ }^{\circ} \mathrm{C}\right)$. These factors are coded as $x_1=($ time -12$) / 8$ and $x_2=($ temperature -250$) / 30$.

The following data were observed where $y$ is the yield (in percent):

$$\begin{array}{cccc}\text { Run Number } & \boldsymbol{x}_{\mathbf{1}} & \boldsymbol{x}_{\mathbf{2}} & \boldsymbol{y} \\ 1 & -1 & 0 & 83.8 \\ 2 & 1 & 0 & 81.7 \\ 3 & 0 & 0 & 82.4 \\ 4 & 0 & 0 & 82.9 \\ 5 & 0 & -1 & 84.7 \\ 6 & 0 & 1 & 75.9 \\ 7 & 0 & 0 & 81.2 \\ 8 & -1.414 & -1.414 & 81.3 \\ 9 & -1.414 & 1.414 & 83.1 \\ 10 & 1.414 & -1.414 & 85.3 \\ 11 & 1.414 & 1.414 & 72.7 \\ 12 & 0 & 0 & 82.0\end{array}$$

Plot the points at which the experimental runs were made.

An article in Plant Disease ["Effect of Nitrogen and Potassium Fertilizer Rates on Severity of Xanthomonas Blight of Syngonium Podophyllum"] showed the effect of the variable nitrogen and potassium rates on the growth of "White Butterfly" and the mean percentage of disease. Data representative of that collected in this experiment is provided in the following table.

Estimate the appropriate variance components.