Question 1

Which of the following statements are not true?

Accepted Answer

A)  If the two three-factor interactions BCD and CDE are chosen for confounding, then their generalized interaction is BE. 
B)  If the two three-factor interactions ABC and CDE are chosen for confounding, then their generalized interaction is ABCDE. 
C)  When the number p of factors is large, a single replicate of a $2^3$
Experiment can be expensive and time consuming. 
D)  All of the above statements are true. 
E)  None of the above statements are true. 
A)  If the two three-factor interactions BCD and CDE are chosen for confounding, then their generalized interaction is BE. 
B)  If the two three-factor interactions ABC and CDE are chosen for confounding, then their generalized interaction is ABCDE. 
C)  When the number p of factors is large, a single replicate of a $2^3$
Experiment can be expensive and time consuming. 
D)  All of the above statements are true. 
E)  None of the above statements are true.

Question 2

In the fixed effects model with interaction, assume that there are 3 levels of factor A, 2 levels of factor B, and 3 observations for each of the six combinations of levels of the two factors. Then the critical value for testing the null hypothesis of no interaction between the levels of the two factors at the .05 significance level is

Accepted Answer

A)  3.49 
B)  3.89 
C)  3.00 
D)  3.55 
E)  3.11 
A)  3.49 
B)  3.89 
C)  3.00 
D)  3.55 
E)  3.11

Question 3

In a two-factor experiment where factor A consists of I levels, factor B consists of J levels, and there is only one observation on each of the IJ treatments, SSE has __________ degrees of freedom.

Accepted Answer

The answer of In a two-factor experiment where factor A...

Question 4

Which of the following statements are true?

Accepted Answer

A)  For experimental situations with more than three factors, there are often no replications, so sums of squares associated with nonconfounded higher-order interactions are usually pooled to obtain an error sum of squares that can be used in the denominators of the various F statistics. 
B)  One replicate of a $2 ^ { 6 }$
Factorial experiment involves an observation for each of the 64 different experimental conditions. 
C)  If an experimenter decides to include only  
Of the  
Possible conditions in the experiment; this is usually called a half-replicate. 
D)  The first step in selecting half-replicate is to select a defining effect as the nonestimable effect. 
E)  All of the above statements are true. 
A)  For experimental situations with more than three factors, there are often no replications, so sums of squares associated with nonconfounded higher-order interactions are usually pooled to obtain an error sum of squares that can be used in the denominators of the various F statistics. 
B)  One replicate of a $2 ^ { 6 }$
Factorial experiment involves an observation for each of the 64 different experimental conditions. 
C)  If an experimenter decides to include only  
Of the  
Possible conditions in the experiment; this is usually called a half-replicate. 
D)  The first step in selecting half-replicate is to select a defining effect as the nonestimable effect. 
E)  All of the above statements are true.

Question 5

Which of the following statements are true?

Accepted Answer

A)  Blocking is always effective in reducing variation associated with extraneous sources. 
B)  It is often not possible to carry out all  
Experimental conditions of a  
Factorial experiment in a homogeneous experimental environment. 
C)  When the  
Experimental conditions are placed in $2 ^r$
Homogeneous blocks (r<p), the price paid for this blocking is that $2 ^ { r } - 1$
Of the factor effects cannot be estimated. 
D)  All of the above statements are true. 
E)  None of the above statements are true. 
A)  Blocking is always effective in reducing variation associated with extraneous sources. 
B)  It is often not possible to carry out all  
Experimental conditions of a  
Factorial experiment in a homogeneous experimental environment. 
C)  When the  
Experimental conditions are placed in $2 ^r$
Homogeneous blocks (r<p), the price paid for this blocking is that $2 ^ { r } - 1$
Of the factor effects cannot be estimated. 
D)  All of the above statements are true. 
E)  None of the above statements are true.

Question 6

Assume the existence of I parameters $\alpha _ { 1 } , \alpha _ { 2 } , \ldots \ldots , \alpha _ { i }$ and J parameters $\beta _ { 1 } , \beta _ { 2 } , \ldots \ldots , \beta _ { i }$ such that $X _ { y } = \alpha _ { 2 } + \beta _ { j } + \varepsilon _ { y } ( i = 1 , \ldots \ldots , I \text { and } j = 1 , \ldots \ldots , J ) \text { and } \mu _ { y } = \alpha _ { 2 } + \beta _ { j } \text {. }$ The model specified by the above equations is called an __________ model because each mean response $\mu _ { y}$ is the __________ of an effect due to factor A at level $i \left( \alpha _ { 1 } \right)$ and an effect due to factor B at level $j \left( \beta _ { j } \right)$ .

Accepted Answer

The answer of Assume the existence of I parameters \(\alpha...

Question 7

Which of the following statements are not true?

Accepted Answer

A)  An experiment in which there are p factors, each at two levels, is referred to as a $p ^ { 2 }$
Factorial experiment. 
B)  A  
Factorial experiment provides a simple setting for introducing the important concepts of confounding and fractional replications. 
C)  A  
Experiment, with four factors A, B, C, and D, has 16 different experimental conditions. 
D)  All of the above statements are true. 
E)  None of the above statements are true. 
A)  An experiment in which there are p factors, each at two levels, is referred to as a $p ^ { 2 }$
Factorial experiment. 
B)  A  
Factorial experiment provides a simple setting for introducing the important concepts of confounding and fractional replications. 
C)  A  
Experiment, with four factors A, B, C, and D, has 16 different experimental conditions. 
D)  All of the above statements are true. 
E)  None of the above statements are true.

Question 8

In the three-factor fixed effects model, assume that there are 4 levels of factor A, 2 levels of factor B, 4 levels of factor C, and 3 observations for each combination of levels of the three factors. Then, the number of degrees of freedom for the three-factor interaction sum of squares (SSABC) is

Accepted Answer

A)  32 
B)  10 
C)  9 
D)  12 
E)  13 
A)  32 
B)  10 
C)  9 
D)  12 
E)  13

Question 9

In the fixed effects model with interaction, assume that there are I levels of factor A, J levels of factor B, and K observations (replications) for each of the IJ combinations of levels of the two factors. Then SST (the total sum of squares) has df = __________.

Accepted Answer

The answer of In the fixed effects model with interaction,...

Question 10

The accompanying data was obtained in an experiment to investigate whether compressive strength of concrete cylinders depends of the type of capping material used or variability in different batches. Each number is a cell total

Accepted Answer

Summary quantities include @#IMG-DLM& resulting in

Question 11

Which of the following statements are not true?

Accepted Answer

A)  Tukey's multiple comparison procedure can be used in two-factor ANOVA but not in three-factor (or more) ANOVA. 
B)  When several factors are to be studied simultaneously, an experiment in which there is at least one observation for every possible combination of levels is referred to as complete layout. 
C)  A three-factor experiment, with I levels of factor A, J levels of factor B, and K levels of factor C, in which fewer than IJK observations are made is called an incomplete layout. 
D)  There are some incomplete layouts in which the pattern of combinations of factors is such that the analysis is straightforward. One such three-factor design is called a Latin square. 
E)  All of the above statements are true. 
A)  Tukey's multiple comparison procedure can be used in two-factor ANOVA but not in three-factor (or more) ANOVA. 
B)  When several factors are to be studied simultaneously, an experiment in which there is at least one observation for every possible combination of levels is referred to as complete layout. 
C)  A three-factor experiment, with I levels of factor A, J levels of factor B, and K levels of factor C, in which fewer than IJK observations are made is called an incomplete layout. 
D)  There are some incomplete layouts in which the pattern of combinations of factors is such that the analysis is straightforward. One such three-factor design is called a Latin square. 
E)  All of the above statements are true.

Question 12

The following equation SST = SSA + SSB +SSC + SSAB + SSAC + SSBC + SSABC + SSE applies to which ANOVA model?

Accepted Answer

A)  One-factor ANOVA 
B)  Two-factor ANOVA with interaction 
C)  Three-factor ANOVA with interactions 
D)  Randomized block design 
E)  Latin square design 
A)  One-factor ANOVA 
B)  Two-factor ANOVA with interaction 
C)  Three-factor ANOVA with interactions 
D)  Randomized block design 
E)  Latin square design

Question 13

A data from an experiment to assess the effects of vibration (A), temperature cycling (B), altitude cycling (C), and temperature for altitude cycling and firing (D) on thrust duration are shown below. Use the Yates method to obtain sums of squares and the ANOVA table. Then assume that three- and four-factor interactions are absent, pool the corresponding sums of squares to obtain an estimate of $\sigma ^ { 2 }$ and test all appropriated hypotheses at level .05.

Accepted Answer

@#IMG-DLM& SSE = .051 + .055 + 1.020 + .

Question 14

Which of the following statements are not true regarding the model $X _ { y } = \mu + \alpha _ { i} + \beta _ { j} + \varepsilon _ { y }$ where $\sum _ { l = 1 } ^ { \prime } \alpha _ { 1 } = 0 \text { and } \sum _ { j = 1 } ^ { j } \beta _ { j } = 0 ?$

Accepted Answer

A)  $\mu$
Is the true grand mean; that is, the mean response averaged over all levels of both factors A and B. 
B)  $\alpha _ { i }$
Is the effect of Factor A at level i. 
C)  $\beta _ { j }$
Is the effect of Factor B at level j. 
D)  $\varepsilon _ { y }$
Are assumed independent and normally distributed with mean 0 and variance 1. 
E)  All of the above statements are true. 
A)  $\mu$
Is the true grand mean; that is, the mean response averaged over all levels of both factors A and B. 
B)  $\alpha _ { i }$
Is the effect of Factor A at level i. 
C)  $\beta _ { j }$
Is the effect of Factor B at level j. 
D)  $\varepsilon _ { y }$
Are assumed independent and normally distributed with mean 0 and variance 1. 
E)  All of the above statements are true.

Question 15

In an experiment to assess the effects of curing time (factor A) and type of mix (factor B) on the compressive strength of hardened cement cubes, three different curing times were used in combination with four different mixes, with three observations obtained for each of the 12 curing time-mix combinations. The resulting sums of squares were computed to be SSA = 30,763.0, SSB = 34,185.6, SSE = 97,436.8, and SST = 205,966.6.
a. Construct an ANOVA table.
b. Test at level .05 the null hypothesis $H _ { o A B } : \text { all } \gamma _ { \mathrm { ij } } { } ^ { \prime } s = 0$ 
(no interaction of factors) against $H _ { o A B } \text { : at least one } \gamma _ { \mathrm { ij } } \neq 0$ 
c. Test at level .05 the null hypothesis $H _ { o A } : \alpha _ { 1 } = \alpha _ { 2 } = \alpha _ { 3 } = 0$ 
(factor A main effects are absent) against $H _ { o A } \text { : at least one } \alpha _ { i } \neq 0$ 
d. Test $H _ { o B } : \beta _ { 1 } = \beta _ { 2 } = \beta _ { 3 } = \beta _ { 4 } = 0 \text { ver sus } H _ { o B } :$ 
at least one $\beta _ { j } \neq 0$ 
using a level .05 test. 
e. The values of the $\bar { x } _ { 1 00 } \text { 's were } \bar { x } _ { 100 } = 4010.88 , \bar { x } _ { 2 00 } = 4029.10 \text {, and } \bar { x } _ { 300 } = 3960.02$ 
Use Tukey's procedure to investigate significant differences among the three curing times.

Accepted Answer

a. @#IMG-DLM& b. @#IMG-DLM& cannot be rejected, and we concl

Question 16

A two-factor experiment where factor A consists of I levels, factor B consists of J levels, and there is only one observation on each of the IJ treatments, can be represented by the model $X _ { 4 } = \mu + \alpha _ { i } + \beta _ { j } + \varepsilon _ { y } \text { where } \sum _ { i = 1 } ^ { j} \alpha _ { i } = 0 \text { and } \sum _ { j = 1 } ^ { j } \beta _ { j } = 0$ . Which of the following is the correct form in testing the null hypothesis that the different levels of factor A have no effect on true average response?

Accepted Answer

A)    : $\alpha _ { 1 } = \alpha _ { 2 } = \ldots \ldots = \alpha _ { j } = 0$ 
B)   $ : \beta _ { 1 } = \beta _ { 1 } = \ldots \ldots = \beta _ { 3 } = 0$ 
C)   $ \text { : at least one } \alpha _ { i } \neq 0$ 
D)   $ \text { : at least one } \beta _ { \mathrm { j } } \neq 0$ 
E)  None of the above answers are correct. 
A)    : $\alpha _ { 1 } = \alpha _ { 2 } = \ldots \ldots = \alpha _ { j } = 0$ 
B)   $ : \beta _ { 1 } = \beta _ { 1 } = \ldots \ldots = \beta _ { 3 } = 0$ 
C)   $ \text { : at least one } \alpha _ { i } \neq 0$ 
D)   $ \text { : at least one } \beta _ { \mathrm { j } } \neq 0$ 
E)  None of the above answers are correct.

Question 17

In two-factor ANOVA, additivity means that the difference in true average responses for any two levels of one of the factors is the same for each level of the other factor. When additivity does not hold, we say that there is __________ between the different levels of the factors.

Accepted Answer

The answer of In two-factor ANOVA, additivity means that the...

Question 18

In the three-factor fixed effects model, assume that there are 3 levels for each of the three factors A, B, and C, and 2 observations for each combination of levels of the three factors. Then the number of degrees of freedom for the error sum of squares (SSE) is

Accepted Answer

A)  54 
B)  27 
C)  11 
D)  18 
E)  16 
A)  54 
B)  27 
C)  11 
D)  18 
E)  16

Question 19

The current (in $\mu A$ ) necessary to produce a certain level of brightness of a television tube was measured for two different types of glass and three different types of phosphor, resulting in the accompanying data: $$\begin{array} { l | c c c } 
\hline & { \text { Phosphor Type } } \
& \mathbf { 1 } & \mathbf { 2 } & \mathbf { 3 } \
\hline \text { Glass 1 } & 280,290,285 & 300,310,295 & 270,285,290 \
\text { Type 2 } & 230,235,240 & 260,240,235 & 220,225,230 \
\hline
\end{array}$$ Assuming that both factors are fixed, test $H _ { o A B } \text { versus } H _ { oA B }$ at level .01. Then if $H _ { o A B }$ cannot be rejected, test the two sets of main effect hypotheses.

Accepted Answer

@#IMG-DLM& @#IMG-DLM& @#IMG-DLM& which yields the accompanying ANOVA t

Question 20

An experiment in which there are p factors, each at two levels, is referred to as a

Accepted Answer

@#IMG-DLM& factorial

Which of the following statements are not true?

In a two-factor experiment where factor A consists of I levels, factor B consists of J levels, and there is only one observation on each of the IJ treatments, SSE has __________ degrees of freedom.

Which of the following statements are true?

Which of the following statements are true?

Which of the following statements are not true?

In the three-factor fixed effects model, assume that there are 4 levels of factor A, 2 levels of factor B, 4 levels of factor C, and 3 observations for each combination of levels of the three factors. Then, the number of degrees of freedom for the three-factor interaction sum of squares (SSABC) is

In the fixed effects model with interaction, assume that there are I levels of factor A, J levels of factor B, and K observations (replications) for each of the IJ combinations of levels of the two factors. Then SST (the total sum of squares) has df = __________.

The accompanying data was obtained in an experiment to investigate whether compressive strength of concrete cylinders depends of the type of capping material used or variability in different batches. Each number is a cell total

Which of the following statements are not true?

The following equation SST = SSA + SSB +SSC + SSAB + SSAC + SSBC + SSABC + SSE applies to which ANOVA model?

Which of the following statements are not true regarding the model $X _ { y } = \mu + \alpha _ { i} + \beta _ { j} + \varepsilon _ { y }$ where $\sum _ { l = 1 } ^ { \prime } \alpha _ { 1 } = 0 \text { and } \sum _ { j = 1 } ^ { j } \beta _ { j } = 0 ?$

In two-factor ANOVA, additivity means that the difference in true average responses for any two levels of one of the factors is the same for each level of the other factor. When additivity does not hold, we say that there is __________ between the different levels of the factors.

In the three-factor fixed effects model, assume that there are 3 levels for each of the three factors A, B, and C, and 2 observations for each combination of levels of the three factors. Then the number of degrees of freedom for the error sum of squares (SSE) is

An experiment in which there are p factors, each at two levels, is referred to as a

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Exam 11: Multifactor Analysis of Variance

Which of the following statements are not true?

In a two-factor experiment where factor A consists of I levels, factor B consists of J levels, and there is only one observation on each of the IJ treatments, SSE has __________ degrees of freedom.

Which of the following statements are true?

Which of the following statements are true?

Which of the following statements are not true?

In the three-factor fixed effects model, assume that there are 4 levels of factor A, 2 levels of factor B, 4 levels of factor C, and 3 observations for each combination of levels of the three factors. Then, the number of degrees of freedom for the three-factor interaction sum of squares (SSABC) is

In the fixed effects model with interaction, assume that there are I levels of factor A, J levels of factor B, and K observations (replications) for each of the IJ combinations of levels of the two factors. Then SST (the total sum of squares) has df = __________.

The accompanying data was obtained in an experiment to investigate whether compressive strength of concrete cylinders depends of the type of capping material used or variability in different batches. Each number is a cell total

Which of the following statements are not true?

The following equation SST = SSA + SSB +SSC + SSAB + SSAC + SSBC + SSABC + SSE applies to which ANOVA model?

In two-factor ANOVA, additivity means that the difference in true average responses for any two levels of one of the factors is the same for each level of the other factor. When additivity does not hold, we say that there is __________ between the different levels of the factors.

In the three-factor fixed effects model, assume that there are 3 levels for each of the three factors A, B, and C, and 2 observations for each combination of levels of the three factors. Then the number of degrees of freedom for the error sum of squares (SSE) is

An experiment in which there are p factors, each at two levels, is referred to as a

Overview and Descriptive Statistics

Probability

Discrete Random Variables and Probability Distributions

Continuous Random Variables and Probability Distributions

Joint Probability Distributions and Random Samples

Point Estimation

Statistical Intervals Based on a Single Sample

Tests of Hypotheses Based on a Single Sample

Inferences Based on Two Samples

The Analysis of Variance

Simple Linear Regression and Correlation

Nonlinear and Multiple Regression

Goodness-Of-Fit Tests and Categorical Data Analysis

Distribution-Free Procedures

Quality Control Methods

Filters