What blocking scheme would you recommend if it were necessary to run a

$$2^4$$

design in four blocks of four runs each?

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.

Construct a

$$2^5$$

design in two blocks. Select the ABCDE interaction to be confounded with blocks.

Construct a $2_{\mathrm{III}}^{6-3}$ fractional factorial design. Write down the aliases, assuming that only main effects and two factor interactions are of interest.

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.

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.

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

Consider the $2^{6-2}$ design in table below. Suppose that after analyzing the original data, we find that factors C and E can be dropped. What type of $2^{k}$ design is left in the remaining variables?

$$\scriptstyle\begin{array}{rrrrrrr} \text { Run } & A & B & C & D & E=A B C & F=B C D & (\times 10) \\ \hline 1 & - & - & - & - & - & - & 6 \\ 2 & + & - & - & - & + & - & 10 \\ 3 & - & + & - & - & + & + & 32 \\ 4 & + & + & - & - & - & + & 60 \\ 5 & - & - & + & - & + & + & 4 \\ 6 & + & - & + & - & - & + & 15 \\ 7 & - & + & + & - & - & - & 26 \\ 8 & + & + & + & - & + & - & 60 \\ 9 & - & - & - & + & - & + & 8 \\ 10 & + & - & - & + & + & + & 12 \\ 11 & - & + & - & + & + & - & 34 \\ 12 & + & + & - & + & - & - & 60 \\ 13 & - & - & + & + & + & - & 16 \\ 14 & + & - & + & + & - & - & 5 \\ 15 & - & + & + & + & - & + & 37 \\ 16 & + & + & + & + & + & + & 52 \\ \hline \end{array}$$

Montgomery (2001) describes a $2^{4-1}$ fractional factorial design used to study four factors in a chemical process. The factors are A = temperature, B = pressure, C = concentration, and D = stirring rate, and the response is filtration rate. The design and the data are as follows:

$$\scriptstyle\begin{array}{cccccc} & & & & &\text{Treatment} & \text{Filtration} \\ \text { Run } & A & B & C & D=A B C & \text { Combination } & \text { Rate } \\ \hline 1 & - & - & - & - & (1) & 45 \\ 2 & + & - & - & + & a d & 100 \\ 3 & - & + & - & + &b d & 45 \\ 4 & + & + & - & - & a b & 65 \\ 5 & - & - & + &+ & c d & 75 \\ 6 & + & - & + & - & a c & 60 \\ 7 & - & + & + & - &b c & 80 \\ 8 & + & + & + & - &a b c d & 96 \\ \hline \end{array}$$

(a) Write down the alias relationships. (b) Estimate the factor effects. Which factor effects appear large? (c) Project this design into a full factorial in the three apparently important factors and provide a practical interpretation of the results.

Consider the $2^{6}$ factorial design. Set up a design to be run in four blocks of 16 runs each. Show that a design that confounds three of the four-factor interactions with blocks is the best possible blocking arrangement.

An article in Industrial and Engineering Chemistry ("Factorial Experiments in Pilot Plant Studies) reports on an experiment to investigate the effect of temperature (A), gas throughput (B), and concentration (C) on the strength of product solution in a recirculation unit. Two blocks were used with ABC confounded, and the experiment was replicated twice. The data are as follows:

$$\begin{array}{ccc} {\text { Replicate 1 }} \\ \hline \text { Block 1 } & \text { Block 2 } \\ \hline(1)=99 & a=18 \\ a b=52 & b=51 \\ a c=42 & c=108 \\ b c=95 & a b c=35 \\ \hline \end{array}$$

$$\begin{array}{ccc} {\text { Replicate 2 }} \\ \hline \text { Block 3 } & \text { Block 4 } \\ \hline(1)=46 & a=18 \\ a b=47 & b=62 \\ a c=22 & c=104 \\ b c=67 & a b c=36 \\ \hline \end{array}$$

(a) Analyze the data from this experiment. (b) Analyze the residuals and comment on model adequacy. (c) Comment on the efficiency of this design. Note that we have replicated the experiment twice, yet we have no information on the ABC interaction. (d) Suggest a better design, specifically, one that would provide some information on all interactions.

A certain outpatient medical procedure has five steps that must be performed in sequence.

(a) Assuming that the time (in minutes) required for each step is an independent random variable, find the mean and standard deviation for the total time.

(b) Why might the assumption of independence be doubtful?

Which of the following terms come under the physiology heading?

Right atrium

Myocardium

Valve to aorta opens

Left ventricle

Valve between left atrium and left ventricle closes

Pressure in left atrium

Electrocardiogram

Endocardium

Superior vena cava

Heartbeat

A long-throated flume is installed in a trapezoidal channel using design B from the given table. Calculate the discharge for a head of $0.65 \mathrm{ft}$.

Figure bellow shows the graph of the function from previous Example giving the tip, T, as a function of the bill, B. Which of the points P, Q, and R in Figure bellow tells us that f(20)=4 ?