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Applied Linear Statistical Models

Michael H. Kutner, Christopher J. Nachtsheim, John Neter

Chapter 29

Exploratory Experiments: Two-Level Factorial and Fractional Factorial Designs - all with Video Answers

Educators

Chapter Questions

12:18

Problem 1

A plant manager used a $2^{4}$ factorial design with two replicates for each treatment to study the effects of four process variables $\left(X_{1}, \ldots, X_{4}\right)$ on product quality ( $Y$ ). State the response model in the form of $(29.2 a) .$ How many two-factor interaction terms are there? How many three-factor interaction terms? How many four-factor interaction terms?

Shu Naito
Shu Naito
Numerade Educator
02:36

Problem 2

A scientist observed: "Two-level factorial designs are useful if the number of factors is small. But I am concerned when there are 10 or more factors; the number of trials required for a $2^{10}$ experiment is simply too large." Discuss.

Shu Naito
Shu Naito
Numerade Educator
04:34

Problem 3

Reaction yield. A chemical engineer decided to employ a single replicate of a $2^{6}$ factorial design to study the effects of the process variables on the yield of a chemical reaction.
a. How may factors are involved? How many levels are there for each factor? How many experimental trials will be required for the single replicate of the experiment?
b. Can a test for lack of fit be obtained here?

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
09:19

Problem 4

A biologist considered studying the effects of various environmental pollutants on the health of mice by using a $2^{7-4}$ fractional factorial design.
a. How many factors are involved? How many levels are there for each factor? How many trials will be required for a single replicate of the experiment? Can a test for lack of fit be obtained?
b The biologist decided to augment the design with six center-point replicates. Can a test for lack of fit now be obtained? If so, can the biologist determine which factors caused a curvature effect?

JW
Jack Wickhem
Numerade Educator
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Problem 5

State the $\mathbf{X}$ matrix (including all main effects and interaction columns) for a single replicate of a $2^{3}$ factorial design, with the rows listed in standard order. Show numerically that (29.3) holds for your $\mathbf{X}$ matrix.

Nick Johnson
Nick Johnson
Numerade Educator
02:33

Problem 6

Refer to Reaction yield Problem $29.3 .$ Past experience indicates that the standard deviation of reaction yield is $\sigma=5$
a Find the variance of the estimated main effect coefficient $b_{1}$, Is the variance of the interaction effect coefficient $b_{12}$ the same? Should it be?
b. How many replicates of the experiment are required in order to estimate factor effect coefficient $b_{1}$ within ±.5 with 95 percent confidence?

Sheryl Ezze
Sheryl Ezze
Numerade Educator
00:01

Problem 7

Pilot training An unreplicated $2^{5}$ full factorial design was used to investigate the effects of five factors on the learning rates of flight trainees when using flight simulators. The factors were display type $\left(X_{1}=-1: \text { symbolic } ; X_{1}=1: \text { pictorial }\right),$ display orientation $\left(X_{2}=-1:\right.$outside in; $\left.X_{2}=1: \text { inside out }\right),$ crosswind $(X_{3}=-1: \text { no wind present; } X_{3}=1: \text{crosswind . present})$, command guidance $(X_{4}=-1: \text { constant guidance; } X_{4}=1:$ guidance only when trainee strays far from best flight path$)$, and flight path prediction $(X_{5}=-1:$ no prediction; . $\left.X_{5}=1: \text { constant prediction }\right)$. The response $Y$ is the average squared distance from the optimal flight path for 12 landing attempts by the trainee. The smaller is $Y$, the better is the trainee's performance. Thirty-two subjects (trainees) were selected at random from a large group of trainees with no prior flying experience. The design matrix for the experiment and the observed trainee flight scores ( $Y$ ) follow.
a. State the regression model in the form (29.2a). Fit this model and obtain the estimated factor effect coefficients. Does it appear from the magnitudes of the estimated coefficients that some factors may be active here?
b. Prepare a dot plot of the estimated factor effect coefficients. Which effects appear to be active?
c. Obtain a normal probability plot of the estimated factor effect coefficients. Which effects appear to be active? Do the estimated factor cffccts appear to be normally distributed? How do your results compare with those in parts (a) and (b)?

Pritesh Ranjan
Pritesh Ranjan
Numerade Educator
01:28

Problem 8

Refer to Pilot Training Problem 29.7 . The regression model was revised by dropping all three-factor and higher-order interactions.
a. State the revised regression model. Fit the revised regression model and prepare a plot of the residuals against the fitted values. Do the standard regression assumptions appear to be satisfied?
b. Obtain a normal probability plot of the residuals. Also conduct the correlation test for normality; use $\alpha=.05 .$ Docs the assumption of normality appear to be reasonable here?
c. Using the $P$ -values for the estimated factor effect coefficients, test for the significance of each factor effect. Control the family level of significance at $\alpha=.05$ using the Kimball inequality. Which effects appear to be active?
d. Summarize the results of the experiment with an appropriate set of plots of main effects and interactions. Interpret the results.

Dominador Tan
Dominador Tan
Numerade Educator
03:00

Problem 9

Computer monitors. A single replicate of a $2^{4}$ full factorial design, augmented by three replicates at the center point, was used to determine the most reliable design of a computer monitor base. Factors of interest were clearance under the base $\left(X_{1}\right),$ interface board beight $\left(X_{2}\right),$ side vent size $\left(X_{3}\right),$ and interface board angle $\left(X_{4}\right) .$ All factors are quantitative and are coded with $X_{i}=-1$ for the low level of the factor and $X_{l}=1$ for the high level. The response ( $Y$ ) is the failure rate of the interface board, with lower failure rates representing higher product quality. The design matrix for the experiment and the observed design failure rates $(Y)$ follow.
$$\begin{array}{crrrr}
Y & \boldsymbol{X}_{1} & \boldsymbol{X}_{2} & \boldsymbol{X}_{3} & \boldsymbol{X}_{4} \\
\hline 3.88 & -1 & -1 & -1 & -1 \\
3.17 & 1 & -1 & -1 & -1 \\
4.07 & -1 & 1 & -1 & -1 \\
\ldots & . . & . . & . . . & . . . \\
3.80 & 0 & 0 & 0 & 0 \\
3.99 & 0 & 0 & 0 & 0 \\
4.16 & 0 & 0 & 0 & 0
\end{array}$$
a. State the regression model in the form (29.2a). Fit this model and obtain the estimated factor effect coefficients. Does it appear from the magnitudes of the estimated coefficients that some factors may be active here?
b. Prepare a dot plot of the estimated factor effect coefficients, Which effects appear to be active?
c. Obtain a normal probability plot of the estimated factor effect coefficients. Which effects appear to be active? Do the estimated factor effects appear to be normally distributed? How do your results compare with those in parts (a) and (b)?
d. Obtain $M S P E$ using the three center-point replicates and $(29.17) .$ Use this estimate to determine the $P$ -value for each estimated factor effect coefficient. Determine which effects are active; use $\alpha=.05$ for each test.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
01:18

Problem 10

Refer to Computer monitors Problem 29.9 . The regression model was revised by including only the main effects of factors $1,3,$ and 4 and the 34 interaction.
a. Fit the revised model and prepare a plot of the residuals against the fitted values. Do the standard regression assumptions appear to be satisfied?
b. Obtain a normal probability plot of the residuals. Also conduct the correlation test for normality: use $\alpha=.05 .$ Does the assumption of normality appear to be a reasonable one here?
c. Using the $P$ -values for the estimated factor effect coefficients, test for the significance of each effect; use $\alpha=.01$ for each test. Which effects are active?
d. Conduct a test for lack of fit; use $\alpha=.05 .$ State the decision rule and conclusion.
e. Summarize the results of the experiment with an appropriate set of plots of main effects and interactions. Interpret the results, How should the monitor base be designed to achieve a minimum failure rate?

Tyler Moulton
Tyler Moulton
Numerade Educator
04:01

Problem 11

Refer to the $\mathrm{X}$ matrix for a $2^{4}$ full factorial design in Table 29.2
a. Identify the defining relation for the fractional design obtained by dropping treatments 3 to $6,9,10,15,$ and $16 .$ What is the resolution of the fractional design so obtained?
b. Give the complete confounding scheme for the fractional design obtained in part (a).

Raymond Matshanda
Raymond Matshanda
Numerade Educator
03:15

Problem 12

a. Construct a design for four two-level factors with cight cxperimental trials that has the highest possible resolution. What is the resolution of this design?
b. Verify the projection property for the design constructed in part (a) that any subset of three (or fewer) factors yields a full factorial design in those factors.

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
00:42

Problem 13

Is it possible to construct a resolution III design for four two-level factors with four experimental trials? If so, construct such a design. If not, indicate why this is not possible.

Mayukh Banik
Mayukh Banik
Numerade Educator
00:36

Problem 14

Construct a $2_{\mathrm{111}}^{4-1}$ design using the defining relation $0=123 .$ Is there an alternative eight-run design of higher resolution?

AG
Ankit Gupta
Numerade Educator
01:12

Problem 15

Obtain the complete defining relation and the confounding scheme for the eight-run, five-factor design that is fractionated on the basis of the relation $0=123=245 .$ What is the resolution of this design? Is there an alternative design with higher resolution?

Nick Johnson
Nick Johnson
Numerade Educator
00:25

Problem 16

The following design matrix was used in an eight-run, five-factor experiment:
$$\begin{array}{rrrrr}
x_{1} & x_{2} & x_{3} & x_{4} & x_{5} \\
\hline-1 & -1 & -1 & -1 & -1 \\
1 & -1 & -1 & -1 & 1 \\
-1 & 1 & -1 & 1 & 1 \\
1 & 1 & -1 & 1 & -1 \\
-1 & -1 & 1 & 1 & 1 \\
1 & -1 & 1 & 1 & -1 \\
-1 & 1 & 1 & -1 & -1 \\
1 & 1 & 1 & -1 & 1
\end{array}$$
Obtain the defining relation and the complete confounding scheme for this design. What is the resolution of this design? Can an alternative five-factor, eight-run design with higher resolution be constructed?

Heather Zimmers
Heather Zimmers
Numerade Educator
03:15

Problem 17

Construct a $2^{6-3}$ fractional factorial design of highest resolution using Table $29.6 .$ What is the defining relation for this design? What is its resolution?

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
03:56

Problem 18

Peanut solids. A food scientist conducted a single replicate of a $2^{7-3}$ fractional factorial design in an cffort to identify factors that affect the extraction of food solids from peanuts using water. Factors of interest were the p $\mathrm{H}$ level of the water $\left(X_{1}=-1: 6.95 ; X_{1}=1: 8.00\right)$ water termperature $\left(X_{2}=-1: 20^{\circ} \mathrm{C} ; X_{2}=1: 60^{\circ} \mathrm{C}\right),$ extraction time $\left(X_{3}=-1: 15 \mathrm{minutes}\right.$
$\left.X_{3}=1: 40 \text { minutes }\right),$ water-to-peanuts ratio $\left(X_{4}=-1: 5 ; X_{4}=1: 9\right),$ agitation speed $\left(X_{5}=-1: 5,000 \mathrm{rpm} ; X_{5}=1: 10,000 \mathrm{rpm}\right), \text { hydrolysis }\left(X_{6}=-1: \text { unhydrolyzed; } X_{6}=1:\right.$ hydrolyzed), and presoaking level $(X_{7}=-1: \text { dry; } X_{7}=1:$ soaked$)$. The experimental units \right. were 16 randomly selected batches of peanuts. The response $(Y)$ is the percentage of the total solids removed from each batch. The defining relation used to construct the $2^{7-3}$ fractional design (excluding generalized interactions) is $0=1235=2346=1247 .$ The design matrix for the experiment and the observed percentage extractions $(Y)$ follow.
$$\begin{array}{crrrrrrr}
\boldsymbol{r} & x_{1} & x_{2} & x_{3} & x_{4} & x_{5} & x_{6} & x_{7} \\
\hline 10.82 & -1 & -1 & -1 & -1 & -1 & -1 & -1 \\
10.59 & 1 & -1 & -1 & -1 & 1 & -1 & 1 \\
8.19 & -1 & 1 & -1 & -1 & 1 & 1 & 1 \\
\ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots \\
5.12 & 1 & -1 & 1 & 1 & -1 & -1 & -1 \\
5.60 & -1 & 1 & 1 & 1 & -1 & 1 & -1 \\
5.73 & 1 & 1 & 1 & 1 & 1 & 1 & 1
\end{array}$$
a Obtain the generalized interactions and the complete defining relation. What is the resolution of the design? Could a design of higher resolution have been used here?
b. Using the defining relation in part (a), determine the confounding pattern for all main effects and two-factor interactions.
c. State the regression model in the form ( 29.2 a). Remember that confounded effects must not be included in your model. Fit this model and obtain the estimated factor effect coefficients. Prepare a dot plot of the estimated factor effect coefficients. Which effects appear to be active?
d. Obtain a normal probability plot of the estimated factor effect coefficients, Which effects appear to be active? Do the estimated effects appear to be normally distributed? How do your results compare with those in part (c)?
e. Test whether all two-factor interaction effects can be dropped from the model; use $\alpha=.01$ State the alternatives, decision rule, and conclusion.

Sana Riaz
Sana Riaz
Numerade Educator
01:28

Problem 19

Refer to Peanut Solids Problem 29.18 . The regression model was revised by dropping all interaction effects.
a. Fit the revised model and prepare a plot of the residuals against the fitted values, Do the standard regression assumptions appear to be satisfied?
b. Cases 3 and 14 have fairly large absolute residuals. Conduct the Bonferroni outlier test for each of these cases; use $\alpha=.05$ for each test. What do you conclude?
c. Obtain a normal probability plot of the residuals. Also conduct the correlation test for normality; use $\alpha=.025 .$ Does the assumption of normality appear to be reasonable here?
d. Using the $P$ -values of the estimated factor effect coefficients, test for the significance of each effect; use $\alpha=.02$ for each test. Which effects are active?
e. Summarize the results of the experiment with an appropriate set of plots of main effects. Interpret the results. How should maximum food solids extraction be achieved?

Dominador Tan
Dominador Tan
Numerade Educator
03:00

Problem 20

Fiber optics. A chemist conducted a screening experiment to identify factors that affect the viscosity of a gel used in the manufacture of fiber optic cabling. To minimize the loss of telephone signal, the inner glass fibers must be allowed to move freely within the cabling for a range of temperatures, A lubricant (gel) is used to promote this movement. The viscosity of the gel must be sufficiently low to allow such movement; yet it must not be so low as to lead to dripping (leakage) from the ends. A single replicate of a $2^{9-5}$ fractional factorial design was conducted. The factors of interest were silica particle size $\left(X_{1}=-1: 200 ; X_{1}=1: 380\right)$ silica weight $\left(X_{2}=-1: \text { low; } X_{2}=1: \text { high }\right),$ oil ratio $\left(X_{3}=-1: \text { low; } X_{3}=1: \text { high }\right),$ oil temperature $\left(X_{4}=-1: \operatorname{low} ; X_{4}=1: \text { high }\right),$ stabilizer level $\left(X_{5}=-1: \operatorname{low}, X_{5}=1: \text { high }\right)$ premix time $\left(X_{6}=-1: \text { short; } X_{6}=1: \text { Iong }\right),$ postmix time $\left(X_{7}=-1: \text { short; } X_{7}=1\right.$ long, postmix vacuum $\left(X_{B}=-1: \text { no; } X_{8}=1: \text { yes }\right),$ and filter mesh size $\left(X_{9}=-1:\right.$ small; $X_{9}=1:$ large). The response of interest is gel viscosity $(Y) ;$ management feels that an optimal (target) gel viscosity is $74.5 .$ The design matrix for the experiment and the chserved viscosities $(Y)$ follow.
a. State the regression model containing only factor main effects in the form (29.2a). Fit this model and obtain the estimated factor effect coefficients. Does it appear from the magnitudes of the estimated coefficients that some factors may be active here?
b. Prepare a Pareto plot of the estimated factor effect coefficients. Which effects appear to be active?
c. Obtain a normal probability plot of the estimated factor effect coefficients. Which effects appear to be active? Do the estimated factor effects appear to be normally distributed? How do your results compare with those in part (b)?
d. Using the $P$ -values of the estimated factor effect coefficients, test for the significance of each effect term; use $\alpha=.10$ for each test. Which effects are active?

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
01:28

Problem 21

Refer to Fiber optics Problem $29.20 .$ The regression model was revised to include only the main effects for factors $1,5,$ and 7
a. Fit the revised regression model and prepare a plot of the residuals against the fitted values. Do the standard regression assumptions appear to be satisfied?
b. Obtain a normal probability plot of the residuals. Also conduct the correlation test for normality; use $\alpha=.05 .$ Does the assumption of normality appear to be reasonable here?
c. Conduct a lack of fir test for the revised regression model; use $\alpha=.05 .$ State the alternatives, decision rule, and conclusion. What does your conclusion suggest about the possible presence of interactions?

Dominador Tan
Dominador Tan
Numerade Educator
07:32

Problem 22

Refer to Fiber optics Problems 29.20 and $29.21 .$ Since the experimental design consists of two complete replicates of a $2^{3}$ factorial in the three active factors $1,5,$ and $7,$ consider now a revised model containing the main effects of factors $1,5,$ and 7 and all interactions among these three factors.
a State the revised regression model and fit it. Using the $P$ -values of the estimated factor effect coefficients, test for the significance of each factor effect; use $\alpha=.01$ for each test. Which effects are active?
b. Obtain a normal probability plot of the residuals. Compare this plot to that obtained in Problem $29.21 \mathrm{b}$. What do you conclude?
c. Summarize the experimental results with an appropriate set of plots of the main effects and interactions. Interpret the results.
How might you proceed to determine the levels of factors $1,5,$ and 7 so that the expected viscosity of the resulting gel would be on target at $74.5 ?$

OC
Omer Ceyhan
Numerade Educator
09:38

Problem 23

Windshield molding manufacture. An experimental study was undertaken in an effort to reduce the occurrence of dents in a windshield molding manufacturing process. The dents are caused by picces of metal or plastic that are carried into the dies during stamping and forming operations. Four factors were identified for use in an eight-run experiment: poly-film thickness- used to protect the metal strip during manufacturing to reduce surface blemishes $\left(X_{1}=-1: .00175 ; X_{1}=1: .0025\right),$ oil mixture ratio for surface lubrication $\left(X_{2}=-1\right.$ $\left.05 ; X_{2}=1: .10\right),$ operator glove type $(X_{3}=-1: \text { cotton; } X_{3}=1: \text {nylon})$, underside oil coating $\left(X_{4}:=-1: \text { no coating } ; X_{4}=1: \text { coating }\right) .$ During each run of the experiment, 1,000 moldings were fabricated; the response ( $Y$ ) is the number of defect-free moldings produced. The design matrix for the experiment and the observed numbers of defect-free moldings produced ( $Y$ ) follow.
$$\begin{array}{rrrrr}
Y & X_{1} & X_{2} & X_{3} & X_{4} \\
\hline 338 & 1 & -1 & -1 & -1 \\
826 & 1 & -1 & 1 & 1 \\
350 & 1 & 1 & -1 & -1 \\
647 & 1 & 1 & 1 & 1 \\
917 & -1 & -1 & -1 & 1 \\
977 & -1 & -1 & 1 & -1 \\
953 & -1 & 1 & -1 & 1 \\
972 & -1 & 1 & 1 & -1
\end{array}$$
a. Determine the defining relation and the complete confounding scheme used in the experiment. Could a design of higher resolution have been used?
b. State the regression model in the form (29.2a). Remember that confounded factor effects must not be included in your model. Fit this model and obtain the estimated factor effect coefficients.
c. Prepare a dot plot of the estimated factor effect coefficients, Which effects appear to be active?
d. Obtain a normal probability plot of the estimated factor effect coefficients. Which effects appear to be active? How do your results compare with those in part (c)? Do the estimated factor effects appear to be normally distributed?

Carson Merrill
Carson Merrill
Numerade Educator
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Problem 24

Refer to Windshield molding manufacture Problem $29.23 .$ The regression model was revised to include only the main effects for the four factors.
a. Fit the revised regression model. Using the $P$ -values for the estimated factor effect coefficients, test for the significance of each effect; use $\alpha=.05$ in each case. Which effects are active?
b. Summarize the results of the experiment with an appropriate set of plots of main effects. Interpret the results. Identify the settings of the experimental factors within the operating range that lead to the maximum number of defect-free moldings.

Shu Naito
Shu Naito
Numerade Educator
01:39

Problem 25

Construct a $2_{111}^{5-2}$ design in two blocks of size four such that main effects are not confounded with the block effect.

Raymond Matshanda
Raymond Matshanda
Numerade Educator
03:35

Problem 26

Team effectiveness A researcher employed a single replicate of a $2^{6}$ full factorial design, with eight blocks containing eight treatments each, to study the effects of team member's ability level and motivation level on the performance of three-person military teams consisting of an operator, a loader, and a mover. The factors studied were operator's ability $\left(X_{1}\right),$ operator's motivation $\left(X_{2}\right),$ loader's ability $\left(X_{3}\right),$ loader's motivation $\left(X_{4}\right),$ mover's ability $\left(X_{5}\right),$ and mover's motivation $\left(X_{6}\right)$. All factors are quantitative and are coded with $X_{l}=-1$ referring to the low level of the factor and $X_{l}=1$ referring to its high level. The 64 teams were formed by assigning persons to teams in accordance with the $2^{6}$ full factorial design.
The team ratings ( $Y$ ) were assigned by unit commanders following two months of military activity. Because unit commanders could observe at most 10 teams, and because it was expected that some scoring biases might result, the teams were assigned to commanders in blocks of size eight. Levels of the interaction terms $X_{135}, X_{146},$ and $X_{245}$ were used to determine the blocks. The observed team ratings, the design matrix, and the blocking arrangement follow.
a. Obtain a scatter plot of team ratings against block number. Does it appear that blocking was effective here?
b. Identify the complete confounding scheme for blocks, Are any main effects confounded with blocks? Any two-factor interactions?
c. State the regression model in the form (29.2a). Fit this model and obtain the estimated factor effect coefficients. Prepare a dot plot of the estimated factor effect coefficients. Which effects appear to be active?
d. Obtain a normal probability plot of the estimated factor effect coefficients. Which effects appear to be active? How do your findings compare with those in part(c)? Do the estimated factor effects appear to be normally distributed?

Raymond Matshanda
Raymond Matshanda
Numerade Educator
05:28

Problem 27

Refer to Team effectiveness Problem $29.26 .$ The regression model was revised to include only the factor main effects, two-factor interactions, and block main effects.
a. Fit the revised model and prepare a plot of the residuals against the fitted values. Do the standard regression assumptions appear to be satisfied?
b. Obtain a normal probability plot of the residuals. Also conduct the correlation test for normality; use $\alpha=.05 .$ Does the assumption of normality appear to be reasonable here?
c. Using the $P$ -values for the estimated factor effect coefficients, test for the significance of each factor effect; use $\alpha=.01$ for each test. Which effects are active?

Neel Faucher
Neel Faucher
Numerade Educator
05:31

Problem 28

Refer to Team effectiveness Problems 29.26 and 29.27 . The finally revised regression model consists of all block main effects and all factor main effects only.
a. Fit the finally revised regression model.
b. Summarize the results of the experiment with an appropriate set of plots of the factor main effects, Interpret the results. How is maximum team effectiveness achieved?
c. Obtain a 95 percent prediction interval for the team performance for a single new team formed as described in part (b); assume that the rater (block) effect is zero in making your prediction.

Neel Faucher
Neel Faucher
Numerade Educator
03:00

Problem 29

Whipped topping. Food scientists had developed a prototype soybean-based whipped topping, but the product suffered in that the volume of the whipped product did not meet expectations. In an effort to maximize the topping volume, a $2^{5-1}$ fractional factorial design of highest resolution was used in an experiment in two blocks of size eight each, with three center-point replicates in each block. The design confounded the block effect with the 45 interaction. The factors studied were soybean solids level $\left(X_{1}\right),$ fat level $\left(X_{2}\right),$ emulsifier level $\left(X_{3}\right),$ and the levels of two stabilisers: methocel $\left(X_{4}\right),$ and avicel $\left(X_{5}\right) .$ All factors are quantitative and are coded with $X_{1}=-1$ referring to the low level of the factor and $X_{f}=1$ referring to its high level. The response $(Y)$ is the percent increase in volume of the product due to whipping: large increases are desirable. The observed responses, the design matrix, and the blocking arrangement follow.
a. What is the defining relation for this design? What is the resolution, ignoring blocks?
b. State the regression model in the form (29.2a). Remember that confounded factor effects must not be included in your model. Fit this regression model and obtain the estimated factor effect coefficients. Prepare a dot plot of the estimated factor effect coefficients. Which effects appear to be active?
c. Obtain a normal probability plot of the estimated factor effect coefficients. Which effects appear to be active? How do your results compare with those in part (b)? Do the estimated factor effects appear to be normally distributed?
d. Test for the presence of block effects; use $\alpha=.05 .$ State the alternatives, decision rule. and conclusion.
Fit a revised regression model, omitting the block effect term. Obtain a pure error estimate of the error variance using the six center-point replicates and (29.17) and conduct a test for lack of fit; use $\alpha=.05 .$ State the decision rule and conclusion. Does your test indicate the presence of curvature?
f. Using the $P$ -values for the estimated factor effect coefficients obtained in part (c) based on the pure error estimate $M S P E$, test for the significance of the factor effects; use $\alpha=.025$ for each test. Which factors are active?

Khoobchandra Agrawal
Khoobchandra Agrawal
Numerade Educator
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Problem 30

Refer to Whipped topping Problem $29.29 .$ The model has been finally revised to include only the main effects for factors $1,2,$ and 5 and the 12 interaction term.
a. Fit the revised model and prepare a plot of the residuals against the fitted values. Do the standard regression assumptions appear to be satisfied?
b. Obtain a normal probability plot of the residuals. Also conduct the correlation test for normality; use $\alpha=.05 .$ Does the assumption of normality appear to be reasonable here?
c. Summarize the results of the experiment with an appropriate set of plots of the main effects and interactions. Interpret the results. How is maximum whippability achieved?
d. Obtain a 95 percent confidence interval for the expected percent volume increase for the whipped topping product when formulated as recommended in part (c).

Shu Naito
Shu Naito
Numerade Educator
01:52

Problem 31

Refer to Computer monitors Problem 29.9 . Suppose two more replicates were conducted for the $2^{4}$ full factorial design. Ignoring the center points, the design matrix for the new experiment with three replicate responses $Y_{11}, Y_{12},$ and $Y_{13}$ follows. Assume that the target failure rate is $T=0$
a. Obtain the sample variances and the logarithms of the sample variances for each of the control-factor-level combinations. Does the variance appear to be constant?
b. Fit the dispersion model (29.41) using the logarithm of the sample variances obtained in part (a). Prepare a Pareto plot of the estimated factor effect coefficients. Which dispersion effects appear to be active?
c. Using the subset dispersion model based on the estimates of the active dispersion effccts, provide estimates of the variance of the response for each control-factor-level combination. Are your estimates consistent with the sample variances obtained in part (a)?
Fit the location model (29.39) using weighted least squares. Obtain a normal probability plot of the estimated control-factor-effect coefficients. Which effects appear to be active? Use $\alpha=.05$
e. Using the subset dispersion and location models based on the active dispersion and location effects identified in parts (b) and (d), determine the control factor settings that minimize failure rate with minimum variance.
f. Give 95 percent confidence limits for the predicted variance for the optimal settings identified in part (c). How would these limits be used in a confirmation run?
g. Estimate the mean squared error in (29.38) for the optimal control-factor-level settings determined in part (c).

Tyler Moulton
Tyler Moulton
Numerade Educator
02:30

Problem 32

Leaf springs. An enginecr conducted an experiment to identify factors that affect the height of an unloaded spring to improve a heat treatment process on truck leaf springs. The target value of the height $(Y)$ is $T=8$ inches. The heat treatment forms the camber (curvature) in leaf springs, and was conducted by heating in a high temperature furnace, processing by a forming machine, and quenching in an oil bath. The factors of interest were furnace temperature $\left(X_{1}=-1: 1840^{\circ} \mathrm{F} ; X_{1}=1: 1880^{\circ} \mathrm{F}\right)$. heating time $\left(X_{2}=-1: 23 \text { minutes; } X_{2}=1$ : \right. $25 \text { minutes }),$ transfer time $\left(X_{3}=-1: \text { short; } X_{3}=1: \text { long }\right),$ and hold-down time $\left(X_{4}=-1:\right.$ short; $\left.X_{4}=1: \text { long }\right) .$ The defining relation used to construct the $2^{4-1}$ design is $0=1234$ The design matrix for the experiment and the observed heights with 6 replicates $(Y)$ follow.
a. Obtain the sample variances and the logarithms of the sample variances for each of the control-factor-level combinations. Does the variance appear to be constant?
b. Fit the dispersion model (29.41) using the logarithms of the sample variances obtained in part (a). Prepare a Pareto plot of the estimated factor effect coefficients. Which dispersion effects appear to be active?
c. Using the subset dispersion model based on the estimates of the active dispersion effects, provide estimates of the variance of the response for each control-factor-level combination. Are your estimates consistent with the sample variances obtained in part (a)?
d. Fit the location model (29.39) using weighted least squares. Obtain a normal probability plot of the estimated control-factor-effect coefficients. Which effects appear to be active? Use $\alpha=.05$
e. Using the subset dispersion and location models based on the active dispersion and location effects identified in parts (b) and (d), determine the control factor settings that lead to a predicted mean height near $T=8$ with minimal variance.
f. Give simultaneous 95 percent confidence limits for the predicted variance for the optimal settings identified in part (c). How would these limits be used in a confirmation run?
g. Estimate the mean squared error in (29.38) for the optimal control-factor-level settings determined in part (e).

M Hassan Anwar
M Hassan Anwar
Numerade Educator
12:33

Problem 33

Show that (29.14) holds for balanced two-level experiments; use (2.51) and the additivity of the extra sums of squares in this situation.

Sandip Ranjan
Sandip Ranjan
Numerade Educator
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Problem 34

Suppose that the true (full) regression model in matrix form is:
\[
\mathbf{Y}=\mathbf{X}_{1} \boldsymbol{\beta}_{1}+\mathbf{X}_{2} \boldsymbol{\beta}_{2}+\boldsymbol{\varepsilon}
\]
However, the analyst assumes that the (reduced) model:
\[
\mathbf{Y}=\mathbf{X}_{1} \boldsymbol{\beta}_{1}+\boldsymbol{\varepsilon}
\]
is correct and uses it for purposes of estimation. For example, the $X$ matrix for the reduced model $\left(\mathbf{X}_{1}\right)$ might include only an intercept column and columns for first-order terms, while the true model involves first-order terms $\left(\mathbf{X}_{1}\right)$ and some two-factor interaction terms $\left(\mathbf{X}_{2}\right)$
a. Show that:
\[
\mathbf{E}\left\{\hat{\boldsymbol{\beta}}_{1}\right\}=\boldsymbol{\beta}_{1}+\mathbf{A} \boldsymbol{\beta}_{2}
\]
where $\mathbf{A}=\left(\mathbf{X}_{1}^{\prime} \mathbf{X}_{1}\right)^{-1} \mathbf{X}_{1}^{\prime} \mathbf{X}_{2}$ is called the alias matrix.
b. Let $\mathrm{X}_{1}$, be the $\mathrm{X}$ matrix (based on the intercept and first-order terms only) for the $2_{\mathrm{if}}^{3-1}$ design constructed from the defining relation $0=123 .$ Let $\mathbf{X}_{2}$ consist of the columns $X_{12}$ $X_{13},$ and $X_{23},$ corresponding to the omitred two-factor interaction effects $\beta_{12}, \beta_{13},$ and $\beta_{22}$ Use the result in part
(a) and $\left.\mathbf{b}=\left(\mathbf{X}_{\mathbf{i}}^{\prime} \mathbf{X}_{\mathbf{1}}\right)^{-\mathbf{t}} \mathbf{X}_{\mathbf{i}}^{\prime} \mathbf{Y}=\mathbf{X}_{\mathbf{1}}^{\prime} \mathbf{Y} / 8 \text { to show that } E | b_{1}\right\}=\beta_{1}+\beta_{23}$$E\left\{b_{2}\right\}=\beta_{2}+\beta_{13},$ and $E\left\{b_{3}\right\}=\beta_{3}+\beta_{12} .$ Thus, for this design we have: $1=23,2=13$ and $3=12$

Shu Naito
Shu Naito
Numerade Educator