Question 1

A research project studied the physical properties of wood materials constructed by bonding together small flakes of wood.The two factors considered were the size of the flakes and species of tree.The sizes of the flakes were S1: 0.15 inches by 2 inches and S2: 0.25 inches by 2 inches,and the species of tree used were aspen,birch,and maple.For each combination of flake size and tree species,three samples of wood material were constructed.For each sample,the physical property measured was the tension modulus of elasticity in the direction perpendicular to the alignment of the flakes,in pounds per square inch (psi). The table below gives the means for the different size-species groups:  For these data,a two-way ANOVA was run and the partial ANOVA table is given below.  What is the value of the F statistic for testing interaction?
A)0.59
B)1.70
C)3.40
D)202.33

Accepted Answer

The F statistic for testing interaction is found in the last row of the ANOVA table, under the heading "Interaction." The value is 1.70.

Question 2

A researcher wished to compare the effect of the different rates of stepping on volunteers' heart rates in a step-aerobics workout.A collection of thirty adult volunteers,fifteen women and fifteen men,were selected from a local gym.The men were randomly divided into three groups of five subjects each.Each group did a standard step-aerobics workout,with Group 1 at a low rate of stepping,Group 2 at a medium rate of stepping,and Group 3 at a rapid rate.The women were also randomly divided into three groups of five subjects each.As with the men,each group did a standard step-aerobics workout,with Group 1 at a low rate of stepping,Group 2 at a medium rate of stepping,and Group 3 at a rapid rate.The mean heart rate at the end of the workout for all the subjects was determined in beats per minute. A partial ANOVA table for these data is given below.  If the data were analyzed as a one-way ANOVA with six "treatments" corresponding to the six combinations of stepping rate and gender,what would have been the sum of squares for error?
A)2623.9
B)2855.6
C)5762.0
D)8617.6

Accepted Answer

The answer of A researcher wished to compare the effect...

Question 3

An experiment is done as a 3 × 5 ANOVA and the experiment is replicated twice. What is the total number of observations in this experiment?
A)30
B)15
C)It cannot be determined from the given information.
D)None of the above

Accepted Answer

In a 3 × 5 ANOVA, there are 3 levels for one factor and 5 levels for another, resulting in 3*5 = 15 unique conditions. Since the experiment is replicated twice, each condition is observed twice, leading to a total of 15*2 = 30 observations.

Question 4

A researcher wished to compare the effect of the different rates of stepping on volunteers' heart rates in a step-aerobics workout.A collection of thirty adult volunteers,fifteen women and fifteen men,were selected from a local gym.The men were randomly divided into three groups of five subjects each.Each group did a standard step-aerobics workout,with Group 1 at a low rate of stepping,Group 2 at a medium rate of stepping,and Group 3 at a rapid rate.The women were also randomly divided into three groups of five subjects each.As with the men,each group did a standard step-aerobics workout,with Group 1 at a low rate of stepping,Group 2 at a medium rate of stepping,and Group 3 at a rapid rate.The mean heart rate at the end of the workout for all the subjects was determined in beats per minute. A partial ANOVA table for these data is given below.  What is the value of the degrees of freedom for interaction?
A)1
B)2
C)3
D)6

Accepted Answer

The degrees of freedom for interaction can be calculated by multiplying the number of levels in one factor by the number of levels in another factor. In this case, there are three levels in the factor of stepping rate for both men and women, so the degrees of freedom for the interaction is 3 x 3 = 9. However, since the total sample size is 30, the degrees of freedom for error is 30 - (3 x 2) - 1 = 24. Therefore, the degrees of freedom for interaction is 9 - 24 = 2. So, the best choice is B.

Question 5

A researcher wished to compare the effect of the different rates of stepping on volunteers' heart rates in a step-aerobics workout.A collection of thirty adult volunteers,fifteen women and fifteen men,were selected from a local gym.The men were randomly divided into three groups of five subjects each.Each group did a standard step-aerobics workout,with Group 1 at a low rate of stepping,Group 2 at a medium rate of stepping,and Group 3 at a rapid rate.The women were also randomly divided into three groups of five subjects each.As with the men,each group did a standard step-aerobics workout,with Group 1 at a low rate of stepping,Group 2 at a medium rate of stepping,and Group 3 at a rapid rate.The mean heart rate at the end of the workout for all the subjects was determined in beats per minute. A partial ANOVA table for these data is given below.  What are the factors in this experiment?
A)Rate of stepping and heart rate
B)Rate of stepping and gender
C)Heart rate and gender
D)Gym membership and gender

Accepted Answer

The factors in this experiment are the rate of stepping and gender. The study was designed to compare the effect of different rates of stepping on heart rates, with men and women each randomly assigned to one of three groups based on the rate of stepping. The gender factor is included in the design to control for potential gender differences in heart rate response to exercise. Gym membership and heart rate are not factors in this experiment.

Question 6

A study was done to determine how gender and stress level affect one's self-rated level of happiness on a scale of 1 to 10.The data are shown below.  What is the P-value for the interaction effect? (Note: Assume all assumptions have been reasonably met. )
A)Less than or equal to .01
B)Between .02 and .05
C)Between .06 and .08
D)Greater than .09

Accepted Answer

The answer of A study was done to determine how...

Question 7

An experiment is done as a 3 × 5 ANOVA and the experiment is replicated twice. How large should the F statistic be to reject the null hypothesis for the interaction at the .05 level?
A)1
B)3.22
C)2.64
D)None of the above

Accepted Answer

The F statistic for interaction in a 3 × 5 ANOVA with replication twice would have degrees of freedom for interaction as (3-1)*(5-1)=8 and for error as (3*5-1)*(2-1)=14. Using an F-distribution table, the critical value at the .05 level for F(8,14) is approximately 2.64.

Question 8

A research project studied the physical properties of wood materials constructed by bonding together small flakes of wood.The two factors considered were the size of the flakes and species of tree.The sizes of the flakes were S1: 0.15 inches by 2 inches and S2: 0.25 inches by 2 inches,and the species of tree used were aspen,birch,and maple.For each combination of flake size and tree species,three samples of wood material were constructed.For each sample,the physical property measured was the tension modulus of elasticity in the direction perpendicular to the alignment of the flakes,in pounds per square inch (psi). The table below gives the means for the different size-species groups:  For these data,a two-way ANOVA was run and the partial ANOVA table is given below.  What is the value of the sum of squares for error,SSE?
A)1021.8
B)14,512.0
C)20,853.5
D)147,144.0

Accepted Answer

The answer of A research project studied the physical properties...

Question 9

A researcher wished to compare the effect of the different rates of stepping on volunteers' heart rates in a step-aerobics workout.A collection of thirty adult volunteers,fifteen women and fifteen men,were selected from a local gym.The men were randomly divided into three groups of five subjects each.Each group did a standard step-aerobics workout,with Group 1 at a low rate of stepping,Group 2 at a medium rate of stepping,and Group 3 at a rapid rate.The women were also randomly divided into three groups of five subjects each.As with the men,each group did a standard step-aerobics workout,with Group 1 at a low rate of stepping,Group 2 at a medium rate of stepping,and Group 3 at a rapid rate.The mean heart rate at the end of the workout for all the subjects was determined in beats per minute. A partial ANOVA table for these data is given below.  What is the value of the F statistic for testing the main effect of gender?
A)0.01
B)0.29
C)5.32
D)81.49

Accepted Answer

The F statistic for testing the main effect of gender is found in the row labeled "gender" and has a value of 0.29. This means that the difference in mean heart rates between men and women is not statistically significant at the alpha=0.05 level.

Question 10

An experiment is done as a 3 × 5 ANOVA and the experiment is replicated twice. What are the degrees of freedom of the F statistic associated with the second main effect?
A)4 and 15
B)3 and 5
C)4 and 29
D)None of the above

Accepted Answer

The degrees of freedom for the second main effect in a 3 × 5 ANOVA are calculated as the number of levels minus one, which is 5 - 1 = 4. The total degrees of freedom are calculated as the total number of observations minus the number of groups, which is 30 - 15 = 15.

Question 11

An experiment is done as a 3 × 5 ANOVA and the experiment is replicated twice. What are the degrees of freedom of the F statistic associated with the interaction effect?
A)8 and 15
B)15 and 30
C)6 and 30
D)None of the above

Accepted Answer

The interaction effect has 2 degrees of freedom for the rows (3-1) and 4 degrees of freedom for the columns (5-1). Since the experiment is replicated twice, the degrees of freedom for the interaction effect is 2 x 4 = 8. The degrees of freedom for the error term is 15, which is the product of the degrees of freedom for the rows (2) and the degrees of freedom for the columns (4) multiplied by the number of replications (2).

Question 12

A study was done to determine how gender and stress level affect one's self-rated level of happiness on a scale of 1 to 10.The data are shown below.  What is the P-value for the main effect for stress? (Note: Include the interaction in your ANOVA model and assume all assumptions have been reasonably met. )
A)Less than or equal to .01
B)Between .02 and .05
C)Between .06 and .08
D)Greater than .09

Accepted Answer

The answer of A study was done to determine how...

Question 13

An experiment is done as a 3 × 5 ANOVA and the experiment is replicated twice. How large should the F statistic be to reject the null hypothesis for the first main effect at the .05 level?
A)3.68
B)19.43
C)15
D)None of the above

Accepted Answer

The F statistic for the first main effect in a 3 × 5 ANOVA with replication can be found using the F-distribution table. With 2 degrees of freedom for the numerator (3 levels - 1) and a certain number of degrees of freedom for the denominator (depending on the total number of observations), the critical value at the .05 level is approximately 3.68.

Question 14

A researcher wished to compare the effect of the different rates of stepping on volunteers' heart rates in a step-aerobics workout.A collection of thirty adult volunteers,fifteen women and fifteen men,were selected from a local gym.The men were randomly divided into three groups of five subjects each.Each group did a standard step-aerobics workout,with Group 1 at a low rate of stepping,Group 2 at a medium rate of stepping,and Group 3 at a rapid rate.The women were also randomly divided into three groups of five subjects each.As with the men,each group did a standard step-aerobics workout,with Group 1 at a low rate of stepping,Group 2 at a medium rate of stepping,and Group 3 at a rapid rate.The mean heart rate at the end of the workout for all the subjects was determined in beats per minute. A partial ANOVA table for these data is given below.  What can we say about the P-value for the main effect of the stepping rate group?
A)P-value < 0.01
B)0.01 <P-value < 0.05
C)0.05 <P-value < 0.10
D)P-value > 0.10

Accepted Answer

The answer of A researcher wished to compare the effect...

Question 15

A research project studied the physical properties of wood materials constructed by bonding together small flakes of wood.The two factors considered were the size of the flakes and species of tree.The sizes of the flakes were S1: 0.15 inches by 2 inches and S2: 0.25 inches by 2 inches,and the species of tree used were aspen,birch,and maple.For each combination of flake size and tree species,three samples of wood material were constructed.For each sample,the physical property measured was the tension modulus of elasticity in the direction perpendicular to the alignment of the flakes,in pounds per square inch (psi). The table below gives the means for the different size-species groups:  For these data,a two-way ANOVA was run and the partial ANOVA table is given below.  In the ANOVA table,the test for the main effect of size has a P-value of 0.613.What does this indicate?
A)The effect of particle size probably varies considerably for the different species.
B)For about 61.3% of the samples,there was a difference in the effect of particle size.
C)For about 61.3% of the samples,there was no difference in the effect of particle size.
D)None of the above

Accepted Answer

The answer of A research project studied the physical properties...

Question 16

A study was done to determine how gender and stress level affect one's self-rated level of happiness on a scale of 1 to 10.The data are shown below.  What is the P-value for the main effect for gender? (Note: Include the interaction in your ANOVA model and assume all assumptions have been reasonably met. )
A)Less than or equal to .01
B)Between .02 and .05
C)Between .06 and .08
D)Greater than .09

Accepted Answer

The answer of A study was done to determine how...

Question 17

A research project studied the physical properties of wood materials constructed by bonding together small flakes of wood.The two factors considered were the size of the flakes and species of tree.The sizes of the flakes were S1: 0.15 inches by 2 inches and S2: 0.25 inches by 2 inches,and the species of tree used were aspen,birch,and maple.For each combination of flake size and tree species,three samples of wood material were constructed.For each sample,the physical property measured was the tension modulus of elasticity in the direction perpendicular to the alignment of the flakes,in pounds per square inch (psi). The table below gives the means for the different size-species groups:  For these data,a two-way ANOVA was run and the partial ANOVA table is given below.  What is the value of the degrees of freedom for the model,DFM?
A)1
B)2
C)5
D)6

Accepted Answer

The answer of A research project studied the physical properties...

Question 18

A researcher wished to compare the effect of the different rates of stepping on volunteers' heart rates in a step-aerobics workout.A collection of thirty adult volunteers,fifteen women and fifteen men,were selected from a local gym.The men were randomly divided into three groups of five subjects each.Each group did a standard step-aerobics workout,with Group 1 at a low rate of stepping,Group 2 at a medium rate of stepping,and Group 3 at a rapid rate.The women were also randomly divided into three groups of five subjects each.As with the men,each group did a standard step-aerobics workout,with Group 1 at a low rate of stepping,Group 2 at a medium rate of stepping,and Group 3 at a rapid rate.The mean heart rate at the end of the workout for all the subjects was determined in beats per minute. A partial ANOVA table for these data is given below.  What is the value of the degrees of freedom for error?
A)5
B)11
C)24
D)29

Accepted Answer

The degrees of freedom for error can be calculated as (total number of subjects) - (total number of groups) = 30 - 6 = 24. Therefore, the correct answer is C.

Question 19

A study was done to determine whether gender and type of student (graduate students or undergraduate)performed differently on a final exam in a statistics course.The mean scores of each group are shown below.  Create a profile plot.Based on the profile plot,which main effect appears to be the largest?
A)Gender
B)Student

Accepted Answer

The profile plot shows that there is a clear difference in mean scores between male and female students, with male students consistently scoring higher. On the other hand, the difference in mean scores between undergraduate and graduate students is not as pronounced. Therefore, the main effect of gender appears to be the largest.

Question 20

A researcher wished to compare the effect of the different rates of stepping on volunteers' heart rates in a step-aerobics workout.A collection of thirty adult volunteers,fifteen women and fifteen men,were selected from a local gym.The men were randomly divided into three groups of five subjects each.Each group did a standard step-aerobics workout,with Group 1 at a low rate of stepping,Group 2 at a medium rate of stepping,and Group 3 at a rapid rate.The women were also randomly divided into three groups of five subjects each.As with the men,each group did a standard step-aerobics workout,with Group 1 at a low rate of stepping,Group 2 at a medium rate of stepping,and Group 3 at a rapid rate.The mean heart rate at the end of the workout for all the subjects was determined in beats per minute. A partial ANOVA table for these data is given below.  What are the advantages of studying the two factors simultaneously in the same experiment?
A)It is more efficient to study two factors simultaneously rather than separately.
B)We can reduce the residual variation in a model by including a second factor thought to influence the response.
C)We can investigate the interaction between factors.
D)All of the above

Accepted Answer

Studying two factors simultaneously in the same experiment has several advantages. Firstly, it is more efficient as it saves time and resources compared to studying the factors separately. Secondly, by including a second factor, we can reduce the residual variation in a model and improve the accuracy of the results. Finally, studying multiple factors allows us to investigate the interaction between them, which may provide additional insights and help us better understand the relationship between the factors and the response variable.

Quiz 13: Two-Way Analysis of Variance

An experiment is done as a 3 × 5 ANOVA and the experiment is replicated twice. What is the total number of observations in this experiment?

A study was done to determine how gender and stress level affect one's self-rated level of happiness on a scale of 1 to 10.The data are shown below. What is the P-value for the interaction effect? (Note: Assume all assumptions have been reasonably met. )

An experiment is done as a 3 × 5 ANOVA and the experiment is replicated twice. How large should the F statistic be to reject the null hypothesis for the interaction at the .05 level?

An experiment is done as a 3 × 5 ANOVA and the experiment is replicated twice. What are the degrees of freedom of the F statistic associated with the second main effect?

An experiment is done as a 3 × 5 ANOVA and the experiment is replicated twice. What are the degrees of freedom of the F statistic associated with the interaction effect?

An experiment is done as a 3 × 5 ANOVA and the experiment is replicated twice. How large should the F statistic be to reject the null hypothesis for the first main effect at the .05 level?