Chapter 9, Three-Level and Mixed-Level Factorial and Fractional Factorial Designs Video Solutions, Design and Analysis of Experiments | Numerade (2024)

Douglas C. Montgomery

Chapter 9

Three-Level and Mixed-Level Factorial and Fractional Factorial Designs - all with Video Answers

Educators

Chapter Questions

02:32
Problem 1

The effects of developer strength $(A)$ and development time (B) on the density of photographic plate film are being studied. Three strengths and three times are used, and four replicates of a $3^2$ factorial experiment are run. The data from this experiment follow. Analyze the data using the standard methods for factorial experiments.
Table can't copy

Chapter 9, Three-Level and Mixed-Level Factorial and Fractional Factorial Designs Video Solutions, Design and Analysis of Experiments | Numerade (7)

Raymond Matshanda

Numerade Educator

Problem 2

Compute the $I$ and $J$ components of the two-factor interaction in Problem 9.1.

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02:19
Problem 3

An experiment was performed to study the effect of three different types of 32 -ounce botles $(A)$ and three different shelf types ( $B$ )-smooth permanent shelves, end-aisle displays with grilled shelves, and beverage coolers--on the time it takes to stock ten 12 -bottle cases on the shelves. Three workers (factor $C$ ) were employed in the experiment, and two replicates of a $3^3$ factorial design were run. The observed time data are shown in the following table. Analyze the data and draw conclusions.
Tables can't copy

Chapter 9, Three-Level and Mixed-Level Factorial and Fractional Factorial Designs Video Solutions, Design and Analysis of Experiments | Numerade (11)

Dominador Tan

Numerade Educator

04:59
Problem 4

A medical researcher is studying the effect of lidocaine on the enzyme level in the heart muscle of beagle dogs. Three different commercial brands of lidocaine ( $A$ ), three dosage levels ( $B$ ), and three dogs $(C)$ are used in the experiment, and two replicates of a $3^3$ factorial design are run. The observed enzyme levels follow. Analyze the data from this experiment.
Tables can't copy

Problem 5

Compute the $I$ and $J$ components of the two-factor interactions for Example 10.1.

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Problem 6

An experiment is run in a chemical process using a $3^2$ factorial design. The design factors are temperature and pressure, and the response variable is yield. The data that result from this experiment are as follows.
Pressure, psig
$$
\begin{aligned}
&\text { Temper- }\\
&\begin{array}{cccc}
\text { ature, }{ }^{\circ} \mathrm{C} & 100 & 120 & 140 \\
\hline 80 & 47.58,48.77 & 64.97,69.22 & 80.92 .72 .60 \\
90 & 51.86 .82 .43 & 88.47,84.23 & 93.95 .88 .54 \\
100 & 71.18 .92 .77 & 96.57,88.72 & 76.58,83.04
\end{array}
\end{aligned}
$$
(a) Analyze the data from this experiment by conducting an analysis of varnance. What conclusions can you draw?
(b) Graphically analyze the residuals. Are there any concerns about underlying assumptions or model adequacy?
(c) Verify that if we let the low, medium, and high levels of both factors in this design take on the levels $-1,0$, and +1 , then a least squares fit to a second-order model for yield is
$$
\begin{aligned}
y= & 86.81+10.4 x_1+8.42 x_2 \\
& -7.17 x_1^2-7.84 x_2^2-7.69 x_1 x_2
\end{aligned}
$$
(d) Confirm that the model in part (c) can be written in terms of the natural variables temperature $(T)$ and pressure $(P)$ as
$$
\begin{aligned}
\hat{y}= & -1335.63+18.56 T+8.59 P \\
& -0.072 T^2-0.0196 P^2-0.0384 T P
\end{aligned}
$$
(e) Construct a contour plot for yield as a fuaction of pressure and temperature. Based on examination of this plot, where would you recommend running this process?

Chapter 9, Three-Level and Mixed-Level Factorial and Fractional Factorial Designs Video Solutions, Design and Analysis of Experiments | Numerade (18)

Victor Salazar

Numerade Educator

Problem 7

(a) Confound a $3^3$ design in three blocks using the $A B C^2$ component of the three-factor interaction. Compare your results with the design in Figure 9.7.
(b) Confound a $3^3$ design in three blocks using the $A B^2 C$ component of the three-factor interaction. Compare. your results with the design in Figure 9.7.
(c) Confound a $3^3$ design in three blocks using the ABC component of the three-factor interaction. Compare your results with the design in Figure 9.7.
(d) After looking at the designs in parts (a), (b), and (c) and Figure 9.7, what conclusions can you draw?

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02:36
Problem 8

Confound a $3^4$ design in three blocks using the $A B^2 C D$ component of the four-factor interaction.

Chapter 9, Three-Level and Mixed-Level Factorial and Fractional Factorial Designs Video Solutions, Design and Analysis of Experiments | Numerade (22)

Raymond Matshanda

Numerade Educator

Problem 9

Consider the data from the first replicate of Problem 9.3. Assuming that not all 27 observations could be run on the same day, set up a design for conducting the experiment over three days with $A B^2 C$ confounded with blocks. Analyze the data.

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02:15
Problem 10

Outline the analysis of variance table for the $3^4$ design in nine blocks. Is this a practical design?

Chapter 9, Three-Level and Mixed-Level Factorial and Fractional Factorial Designs Video Solutions, Design and Analysis of Experiments | Numerade (26)

Raymond Matshanda

Numerade Educator

03:33
Problem 11

Consider the data in Problem 9.3. If $A B C$ is confounded in replicate 1 and $A B C^2$ is confounded in replicate II, perform the analysis of variance.

Chapter 9, Three-Level and Mixed-Level Factorial and Fractional Factorial Designs Video Solutions, Design and Analysis of Experiments | Numerade (29)

Raymond Matshanda

Numerade Educator

Problem 12

Consider the data from replicate & of Problem 9.3. Suppose that only a one-third fraction of this design with $I=$ $A B C$ is run. Construct the design, determine the alias structure, and analyze the data.

Problem 13

From examining Figure 9.9, what type of design would remain if after completing the first nine suns, one of the three factors could be dropped?

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02:21
Problem 14

Construct a $3_{\mathrm{NV}}^{+-1}$ design with $I=A B C D$. Write out the alias structure for this design.

Chapter 9, Three-Level and Mixed-Level Factorial and Fractional Factorial Designs Video Solutions, Design and Analysis of Experiments | Numerade (34)

Khoobchandra Agrawal

Numerade Educator

00:51
Problem 15

Verify that the design in Problem 9.14 is a resolution IV design.

Chapter 9, Three-Level and Mixed-Level Factorial and Fractional Factorial Designs Video Solutions, Design and Analysis of Experiments | Numerade (37)

Khoobchandra Agrawal

Numerade Educator

02:21
Problem 16

Construct a $3^{5-2}$ design with $l=A B C$ and $l=C D E$. Write out the alias structure for this design. What is the resolution of this design?

Chapter 9, Three-Level and Mixed-Level Factorial and Fractional Factorial Designs Video Solutions, Design and Analysis of Experiments | Numerade (40)

Khoobchandra Agrawal

Numerade Educator

Problem 17

Construct a $3^{9-6}$ design and verify that it is a resolution III design.

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Problem 18

Construct a $4 \times 2^3$ design confounded in two blocks of 16 observations each. Outline the analysis of variance for this design.

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02:15
Problem 19

Outline the analysis of variance table for a $2^2 3^2$ factorial design. Discuss how this design may be confounded in blocks.

Chapter 9, Three-Level and Mixed-Level Factorial and Fractional Factorial Designs Video Solutions, Design and Analysis of Experiments | Numerade (45)

Raymond Matshanda

Numerade Educator

01:39
Problem 20

Starting with a $16-\operatorname{run} 2^4$ design. show how two threelevel factors can be incorporated in this experiment. How many two-level factors can be included if we want some information on two-factor interactions?

Chapter 9, Three-Level and Mixed-Level Factorial and Fractional Factorial Designs Video Solutions, Design and Analysis of Experiments | Numerade (48)

Raymond Matshanda

Numerade Educator

01:39
Problem 21

Starting with a $16-\mathrm{run} 2^4$ design, show how one threelevel factor and three two-level factors can be accommodated and still allow the estimation of two-factor interactions.

Chapter 9, Three-Level and Mixed-Level Factorial and Fractional Factorial Designs Video Solutions, Design and Analysis of Experiments | Numerade (51)

Raymond Matshanda

Numerade Educator

Problem 22

In Problem 8.26, you met Harry and Judy PetersonNedry, two friends of the author who have a winery and vineyard in Newberg, Oregon. That problem described the application of two-level fractional factorial designs to their 1985 Pinot Noir product. In [987, they wanted to conduct another Pinot Noir experiment. The variables for this experiment were
Variable
Clone of Pinot Noir
Beny size
Fermentation temperature
Whole berty
Maceration time
Yeast type
Oak type

Levels
Wadenswil, Pommard
Small, large
$80^{\circ}, 85^{\circ}, 90 / 80^{\circ}$, and $90^{\circ} \mathrm{F}$
None, $10 \%$
10 and 21 days
Assmanhau, Champagne
Tronçais, Allier

Harry and Judy decided to use a 16-run two-level fractional factonial design, treating the four levels of fermentation temperature as two two-level variables. As in Problem 8.27 , they used the rankings from a taste-test panel as the response variable. The design and the resulting average ranks are as follows.
$$
\begin{array}{|c|c|c|c|c|c|c|c|c|}
\hline \text { Run } & \text { Clone } & \begin{array}{l}
\text { Berry } \\
\text { Size }
\end{array} & \begin{array}{l}
\text { Ferm. } \\
\text { Teinp. }
\end{array} & \begin{array}{l}
\text { Whole } \\
\text { Berry }
\end{array} & \begin{array}{l}
\text { Macer, } \\
\text { Time }
\end{array} & \begin{array}{l}
\text { Yeast } \\
\text { Type }
\end{array} & \begin{array}{l}
\text { Oak } \\
\text { Type }
\end{array} & \begin{array}{l}
\text { Average } \\
\text { Rank }
\end{array} \\
\hline \text { । } & \text { - } & \text { - } & \cdots & - & - & \text { - } & \text { - } & 4 \\
\hline 2 & + & - & \text { - } & \text { - } & + & + & + & 10 \\
\hline 3 & \text { - } & \text { t } & - & + & \text { — } & + & + & 6 \\
\hline 4 & \text { + } & \text { + } & \cdots & + & + & \text { - } & \text { - } & 9 \\
\hline 5 & \text { - } & \text { - } & +- & + & + & + & = & \text { II } \\
\hline 6 & 4 & - & +- & + & \text { - } & \text { - } & t & 1 \\
\hline 7 & - & + & +- & - & + & \text { - } & + & 15 \\
\hline 8 & + & t & +- & - & \text { - } & + & \text { - } & 5 \\
\hline 9 & - & - & -+ & + & + & \text { - } & + & 12 \\
\hline 10 & + & - & -\div & + & \text { - } & + & - & 2 \\
\hline \text { (1 } & - & + & -+ & \text { - } & + & + & \text { - } & 16 \\
\hline 12 & + & + & -+ & \text { - } & \text { - } & - & + & 1 \\
\hline 13 & - & - & ++ & \text { - } & \text { - } & + & + & 8 \\
\hline 14 & + & - & ++ & \text { - } & + & - & \text { - } & 14 \\
\hline 15 & - & \text { † } & ++ & + & \text { - } & = & \text { - } & 7 \\
\hline 16 & + & + & ++ & + & \text { † } & + & + & 13 \\
\hline
\end{array}
$$
(a) Describe the aliasing in this design.
(b) Analyze the data and draw conclusions.
(c) What comparisons can you make between this experiment and the 1985 Pinot Noir experiment from Problem 8.27 ?

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Problem 23

An article by W. D. Baten in the 1956 volume of Industrial Qualing Control described an experiment to study the effect of three factors on the lengths of steel bars. Each bar was subjected to one of two heat treatment processes and was cut on one of four machines at one of three times during the day ( 8 A.M. II A.M. or 3 p.M.). The coded length data are as follows:
Table can't copy
(a) Analyze the data from this experiment, assuming that the four observations in each cell are replicates.
(b) Analyze the residuals from this experiment. Is there any indication that there is an outlier in one cell? If you find an outlier. remove it and repeat the analysis from part (a). What are your conclusions?
(c) Suppose that the observations in the cells are the lengths (coded) of bars processed together in heat treatment and then cut sequentially (that is, in order) on the four machines. Analyze the data to determine the effects of the three factors on mean length.
(d) Calculate the log variance of the observations in each cell. Analyze this response. What conclusions can you draw?
(e) Suppose the time at which a bar is cut really cannot be controlled during routine production. Analyze the average length and the log variance of the length for each of the 12 bars cut at each machine/heat treatment process combination. What conclusions can you draw?

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Problem 24

Reconsider the experiment in Problem 9.23 . Suppose that it was necessary to estimate all main effects and twofactor interactions, but the full factorial with 24 runs (nol counting replication) was too expensive. Recommend an altemative design.

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01:39
Problem 25

Suppose there are four three-level categorical factor and a style two-level continuous factor. What is the minimum number of nuns required to estimate all main effects and twofactor interactions? Construct this design.

Chapter 9, Three-Level and Mixed-Level Factorial and Fractional Factorial Designs Video Solutions, Design and Analysis of Experiments | Numerade (57)

Raymond Matshanda

Numerade Educator

Problem 26

Reconsider the experiment in Problem 9.25. Construct a design with $N=48$ runs and compare it to the design you constructed in Problem 9.25.

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01:39
Problem 27

Reconsider the experiment in Problem 9.25. Suppose that you are only interested in main effects. Construct a design with $N=12$ runs for this experiment.

Chapter 9, Three-Level and Mixed-Level Factorial and Fractional Factorial Designs Video Solutions, Design and Analysis of Experiments | Numerade (61)

Raymond Matshanda

Numerade Educator

Chapter 9, Three-Level and Mixed-Level Factorial and Fractional Factorial Designs Video Solutions, Design and Analysis of Experiments | Numerade (2024)

FAQs

What is a 3 level 3 factor factorial design? ›

The model and treatment runs for a 3 factor, 3-level design. This is a design that consists of three factors, each at three levels. It can be expressed as a 3 x 3 x 3 = 33 design. The model for such an experiment is. Y i j k = μ + A i + B j + A B i j + C k + A C i k + B C j k + A B C i j k + ϵ i j k.

What is a 3x3 study design? ›

(b) A study with three levels of each variable is referred to as having a 3 X 3 design. This design would have 9 groups.

What are the levels of a factorial experiment? ›

A common experimental design is one, where all input factors are set at two levels each. These levels are termed high and low or + 1 and − 1, respectively. A design with all possible high/low groupings of all the input factors is termed as a full factorial design in two levels.

What is two level and three level factorial design? ›

A 2×3 factorial design is a type of experimental design that allows researchers to understand the effects of two independent variables on a single dependent variable. In this type of design, one independent variable has two levels and the other independent variable has three levels.

How do you solve a 3 factorial? ›

In short, a factorial is a function that multiplies a number by every number below it till 1. For example, the factorial of 3 represents the multiplication of numbers 3, 2, 1, i.e. 3! = 3 × 2 × 1 and is equal to 6.

What is an example of a mixed factorial design? ›

This is called a mixed factorial design. For example, a researcher might choose to treat cell phone use as a within-subjects factor by testing the same participants both while using a cell phone and while not using a cell phone (while counterbalancing the order of these two conditions).

What is an example of a factorial design experiment? ›

In principle, factorial designs can include any number of independent variables with any number of levels. For example, an experiment could include the type of psychotherapy (cognitive vs. behavioral), the length of the psychotherapy (2 weeks vs. 2 months), and the sex of the psychotherapist (female vs.

What is a factorial design for dummies? ›

Factorial design involves having more than one independent variable, or factor, in a study. Factorial designs allow researchers to look at how multiple factors affect a dependent variable, both independently and together. Factorial design studies are named for the number of levels of the factors.

What is the formula for a factorial experiment? ›

The formula is as follows: A main effect for a factor with s levels has s−1 degrees of freedom. The interaction of two factors with s1 and s2 levels, respectively, has (s1−1)(s2−1) degrees of freedom.

What are the basic types of factorial designs? ›

Types of factorial design

There are three main types of factorial designs, namely “Within Subject Factorial Design”, “Between Subject Factorial Design”, and “Mixed Factorial Design”. 1. Within Subject Factorial Design: In this factorial design, all of the independent variables are manipulated within subjects.

What are the advantages of factorial design? ›

A factorial design can help you determine how each factor affects the response variable, as well as how the factors interact with each other. For example, you can use a factorial design to test how different combinations of temperature and pressure affect the yield of a chemical reaction.

What is 2 factor design? ›

• A two-factor factorial design is an experimental design in which data is collected for all possible combinations of the levels of the two factors of interest. • If equal sample sizes are taken for each of the possible factor combinations then the design is a balanced two-factor factorial design.

What is a 3 factor 3 level ANOVA? ›

Three-way ANOVA, also called three-factor ANOVA, determines how a response is affected by three factors. For example, you might compare a response to drug vs. placebo in both men and women at two time points. Drug treatment is one factor, gender is the other, and time is the third.

What is the factorial of 3 factorial? ›

Topics
  • \frac{ 5 ! }{ 3 ! } The factorial of 5 is 120.
  • \frac{120}{3!} The factorial of 3 is 6.
  • \frac{120}{6} Divide 120 by 6 to get 20.

What is a factorial 2 factors 3 levels? ›

For example, if 2 factors at 3 levels each are to be used, 9 (3x3=9) different treatments are required for a full factorial experiment. If a third factor with 3 levels is added, 27 (3x3x3=27) treatments are required, and 81 (3x3x3x3=81) treatments are required if a fourth factor with three levels is added.

How many main effects does a 3x3 factorial design have? ›

O There are two main effects and one interaction. A study uses a 3x3 factorial design.

References

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