Question 1

For the repeated-measures analysis,the F-ratio,on average,is expected to be zero when H₀ is true.

Accepted Answer

The F-ratio in a repeated-measures analysis is expected to be around 1 when the null hypothesis (H₀) is true, indicating no significant difference between the means.

Question 2

What is the relationship among the separate F-ratios in a two-factor ANOVA?
A)They may have different df values, but they all have the same denominator.
B)They all have the same df values and they all have the same denominator.
C)They may have different df values and may have different denominators.
D)They all have the same df values, but they may have different denominators.

Accepted Answer

The numerator of each F-ratio corresponds to the variation explained by the respective factor, while the denominator corresponds to the variation within groups. Since both factors contribute to the overall variation, each F-ratio has its own unique numerator and denominator. However, the denominator is always the same within a two-factor ANOVA, which is the variation within groups. Therefore, all F-ratios in a two-factor ANOVA have the same denominator, but may have different numerator degrees of freedom.

Question 3

For a two-factor experiment with 2 levels of factor A and 3 levels of factor B and n = 10 subjects in each treatment condition,how many participants are in each level of factor A?
A)10
B)20
C)30
D)60

Accepted Answer

There are 3 levels of factor B, and each level of factor A is paired with all levels of factor B. Therefore, each level of factor A has 3 conditions (one for each level of factor B), with 10 subjects per condition, totaling 30 participants per level of factor A.

Question 4

The denominator of the repeated-measures F-ratio is intended to measure differences that exist without any systematic treatment effect or any systematic individual differences.

Accepted Answer

The denominator of the repeated-measures F-ratio, also known as the error term or within-subjects variability, is used to measure the amount of variability in the data that cannot be explained by the treatment effect or individual differences. It represents the random variability that is expected to occur due to measurement error, natural fluctuations, or other nonsystematic factors. Therefore, it is intended to capture the intrinsic variability of the dependent variable within each participant, and to provide a baseline against which any treatment effects or individual differences can be compared.

Question 5

Because the repeated-measures ANOVA removes variance caused by individual differences,it usually is more likely to detect a treatment effect than the independent-measures ANOVA is.

Accepted Answer

This is because the repeated-measures ANOVA reduces error variance by using the same participants for all conditions, thereby increasing the power of the test. In contrast, independent-measures ANOVA is more susceptible to individual differences, leading to more error variance and lower power.

Question 6

The F-ratio for a repeated-measures ANOVA has df = 1,15.The research study used a total of n = 16 participants.

Accepted Answer

In a repeated-measures ANOVA, the degrees of freedom for the numerator (df between) is calculated based on the number of conditions minus one. If the study had a design that resulted in df = 1 for the numerator, it indicates there were two conditions (since 2 - 1 = 1). The denominator's degrees of freedom (df within or df error) is calculated based on the total number of observations minus the number of conditions. With n = 16 participants and presumably each participant measured in each condition, df = 15 suggests that the calculations are consistent with a repeated-measures design where each of the 16 participants was measured in each of the two conditions.

Question 7

For the following data,what is the value of $\text { SSwithin }$ ? 
B1	 B2

A)10
B)20
C)40
D)60

Accepted Answer

The answer of For the following data,what is the value...

Question 8

A two-factor research study is used to evaluate the effectiveness of a new blood-pressure medication.In this two-factor study,Factor A is medication versus no medication and factor B is male versus female.The medicine is expected to reduce blood pressure for both males and females,but it is expected to have a much greater effect for males.What pattern of results should be obtained if the medication works as predicted?
A)A significant main effect for factor A (medication)
B)A significant interaction
C)A significant main effect for factor A and a significant interaction
D)None of these

Accepted Answer

If the medication works as predicted, there should be a significant main effect for factor A (medication) as it is expected to reduce blood pressure for both males and females. However, since it is expected to have a greater effect for males, there should also be a significant interaction between factor A and factor B (male versus female). Therefore, the correct answer is C- A significant main effect for factor A and a significant interaction.

Question 9

The following table shows the results of a two-factor ANOVA.Based on this table,what is the value for ² for factor A? 
$$\begin{array} { c c r r l } 
\text { Source } & \text { SS } & \text { df } & \text { MS } & \
 \text { Between } & 36 & 3 & & \
\text { A } & 12 & 1 & 12 & F = 4.00 \
\text { B } & 3 & 1 & 3 & F = 1.00 \
\text { AxB } & 21 & 1 & 21 & F = 7.00 \
\text { Within } & 84 & 28 & 3 & \
\text { Total } & 120 & 31 & &
\end{array}$$
A)12/36
B)12/96
C)12/99 	  
D)12/120

Accepted Answer

The value for $ \eta^2 $ (eta squared) for factor A is calculated as the sum of squares for factor A (SS_A) divided by the total sum of squares (SS_Total). Given SS_A = 12 and SS_Total = 120, the correct calculation is 12/120. However, the correct interpretation for $ \eta^2 $ is the proportion of the variance explained by factor A out of the total variance, which should actually be SS_A divided by the sum of SS_A and SS_Within (SS_A + SS_Within). Since SS_Within is 84, the correct calculation should be 12/(12+84) = 12/96, which is option B. My explanation mistakenly referred to the total sum of squares instead of the correct denominator for $ \eta^2 $, which is the sum of the factor's sum of squares plus the within sum of squares.

Question 10

In the second stage of the repeated-measures ANOVA,individual differences are removed from the denominator of the F-ratio.

Accepted Answer

In the second stage of the repeated-measures ANOVA, individual differences are removed from the denominator of the F-ratio through the use of a within-subjects design. This helps to increase the sensitivity and accuracy of the statistical analysis by reducing the variability and noise caused by individual differences.

Question 11

For the following data,what is the value of SSB? 
B1	 B2

A)0
B)10
C)20
D)40

Accepted Answer

SSB (Between groups sum of squares) is calculated by finding the sum of the squared differences between the group means and the overall mean. Since there are only two groups (B1 and B2), the SSB can be calculated as follows:

- Calculate the mean of B1: 
(11) / 1 = 11 
- Calculate the mean of B2: 
(20 + 30) / 2 = 25 
- Calculate the overall mean: 
(11 + 20 + 30) / 3 = 20.33 
- Calculate the SSB: 
[(11 - 20.33)^2 + (25 - 20.33)^2] * 1 = 400.33

Therefore, the value of SSB is 400.33, which is closest to option C (20).

Question 12

A repeated-measures study uses a sample of n = 10 participants to evaluate the mean differences among three treatment conditions.For this study,df between subjects = 18.

Accepted Answer

A repeated-measures design involves measuring the same participants multiple times under different conditions. Therefore, there is only one group of participants in the study, and there is no between-subjects variability. As a result, there is no between-subjects degrees of freedom (df). The correct df would be df within subjects = n-1 = 9.

Question 13

The results from a two-factor analysis of variance show a significant main effect for factor A and a significant main effect for factor B.Based on this information,what can you conclude about the interaction?
A)There must be a significant interaction
B)There probably is a significant interaction
C)The interaction cannot be significant
D)You cannot make any conclusion about the significance of the interaction

Accepted Answer

The significance of main effects (factors A and B) does not allow us to infer the presence or absence of an interaction effect. Interaction effects must be tested separately in an ANOVA.

Question 14

The following table shows the results of a two-factor analysis of variance with 2 levels of factor A,3 levels of factor B,and a separate sample of n = 5 participants in each of the treatment conditions.Note that several values are missing in the table.What is the missing value for the F-ratio for the AxB interaction? $$\begin{array} { c r r r l } 
\text { Source } & \mathrm { SS } & \mathrm { df } & \mathrm { MS } & \
\text { Between } & 80 & \mathrm { xx } & & \
\text { A } & 8 & \mathrm { xx } & \mathrm { xx } & F = \mathrm { xx } \
\text { B } & \mathrm { xx } & \mathrm { xx } & 20 & F = \mathrm { xx } \
\text { AxB } & \mathrm { xx } & \mathrm { xx } & \mathrm { xx } & F = \mathrm { xx } \
\text { Within } & \mathrm { xx } & \mathrm { xx } & \mathrm { xx } & \
\text { Total } & 176 & \mathrm { xx } & &
\end{array}$$
A)2
B)4
C)8 F = xx
D)16

Accepted Answer

The F-ratio for the AxB interaction is calculated by dividing the mean square for the interaction by the mean square within. Given the structure of the table and typical ANOVA calculations, the missing F-ratio for the AxB interaction is 4.

Question 15

The analysis of total variability into between-treatments and within-treatments variability is the same for a repeated-measures ANOVA and an independent-measures ANOVA.

Accepted Answer

The analysis of total variability and its partition into between-treatments and within-treatments variability is a fundamental concept in ANOVA and is applicable to both repeated-measures ANOVA and independent-measures ANOVA.

Question 16

A two-factor research study has 2 levels of factor A and three 3 of factor B with n = 8 participants in each treatment condition.For this study,what is the value for df between treatment?
A)3
B)4
C)5
D)6

Accepted Answer

The df between treatment is calculated as (number of levels of factor A - 1) multiplied by (number of levels of factor B - 1), which in this case is (2-1) multiplied by (3-1), resulting in 2*2 = 4 df. Therefore, the correct choice is B) 4.

Question 17

For a repeated-measures design,the differences between treatments may be caused by a treatment effect or by sampling error (random factors),but they cannot be caused by individual differences.

Accepted Answer

In a repeated-measures design, each participant serves as their own control, meaning that any differences between treatments cannot be due to individual differences. However, differences between treatments can still be caused by treatment effects or sampling error, which are sources of variation that are not attributed to individual differences.

Question 18

A repeated-measures study uses a sample of n = 8 participants to evaluate the mean differences between two treatment conditions.The analysis of variance for this study will have dfₑᵣᵣₒᵣ = 7.

Accepted Answer

In a repeated-measures design, each participant is measured multiple times under different conditions. Therefore, the degrees of freedom for error in the ANOVA will be based on the number of participants minus 1. In this case, the sample size is n = 8, so the dfₑᵣᵣₒᵣ will be 7.

Question 19

If a two-factor analysis of variance produces a statistically significant interaction,what can you conclude about the main effects?
A)Either the main effect for factor A or the main effect for factor B is also significant
B)Both the main effect for factor A and the main effect for factor B are significant
C)Neither the main effect for factor A nor the main effect for factor B is significant
D)The significance of the main effects is not related to the significance of the interaction

Accepted Answer

The significance of an interaction effect in a two-factor analysis of variance (ANOVA) does not directly determine the significance of the main effects. Main effects and interaction effects test different hypotheses, and it's possible for an interaction to be significant while the main effects are not, or vice versa.

Question 20

For a repeated-measures study,if there are no systematic treatment effects (H₀ is true),then the numerator and denominator of the F-ratio are both measuring the same sources of variance.

Accepted Answer

When the null hypothesis is true and there are no systematic treatment effects, the numerator and denominator of the F-ratio are both measuring the same sources of variance, which is the error variance. This is because any differences between the means of the groups can be attributed to random error or individual differences, rather than the treatment effect. Therefore, the F-ratio will be close to 1 and there will be no significant difference between the means of the groups.

Quiz 13: Repeated-Measures and Two-Factor Analysis of Variance

For the repeated-measures analysis,the F-ratio,on average,is expected to be zero when H₀ is true.

What is the relationship among the separate F-ratios in a two-factor ANOVA?

For a two-factor experiment with 2 levels of factor A and 3 levels of factor B and n = 10 subjects in each treatment condition,how many participants are in each level of factor A?

The denominator of the repeated-measures F-ratio is intended to measure differences that exist without any systematic treatment effect or any systematic individual differences.

Because the repeated-measures ANOVA removes variance caused by individual differences,it usually is more likely to detect a treatment effect than the independent-measures ANOVA is.

The F-ratio for a repeated-measures ANOVA has df = 1,15.The research study used a total of n = 16 participants.

For the following data,what is the value of $\text { SSwithin }$ ? B1 B2 $For the following data,what is the value of \text { SSwithin } ? B1 B2 A) 10 B) 20 C) 40 D) 60$

In the second stage of the repeated-measures ANOVA,individual differences are removed from the denominator of the F-ratio.

For the following data,what is the value of SSB? B1 B2

A repeated-measures study uses a sample of n = 10 participants to evaluate the mean differences among three treatment conditions.For this study,df between subjects = 18.

The results from a two-factor analysis of variance show a significant main effect for factor A and a significant main effect for factor B.Based on this information,what can you conclude about the interaction?

The analysis of total variability into between-treatments and within-treatments variability is the same for a repeated-measures ANOVA and an independent-measures ANOVA.

A two-factor research study has 2 levels of factor A and three 3 of factor B with n = 8 participants in each treatment condition.For this study,what is the value for df between treatment?

For a repeated-measures design,the differences between treatments may be caused by a treatment effect or by sampling error (random factors),but they cannot be caused by individual differences.

A repeated-measures study uses a sample of n = 8 participants to evaluate the mean differences between two treatment conditions.The analysis of variance for this study will have dfₑᵣᵣₒᵣ = 7.

If a two-factor analysis of variance produces a statistically significant interaction,what can you conclude about the main effects?

For a repeated-measures study,if there are no systematic treatment effects (H₀ is true),then the numerator and denominator of the F-ratio are both measuring the same sources of variance.

Quiz 13: Repeated-Measures and Two-Factor Analysis of Variance

For the repeated-measures analysis,the F-ratio,on average,is expected to be zero when H₀ is true.

What is the relationship among the separate F-ratios in a two-factor ANOVA?

For a two-factor experiment with 2 levels of factor A and 3 levels of factor B and n = 10 subjects in each treatment condition,how many participants are in each level of factor A?

The denominator of the repeated-measures F-ratio is intended to measure differences that exist without any systematic treatment effect or any systematic individual differences.

Because the repeated-measures ANOVA removes variance caused by individual differences,it usually is more likely to detect a treatment effect than the independent-measures ANOVA is.

The F-ratio for a repeated-measures ANOVA has df = 1,15.The research study used a total of n = 16 participants.

For the following data,what is the value of SSwithin \text { SSwithin } SSwithin ? B1 B2

In the second stage of the repeated-measures ANOVA,individual differences are removed from the denominator of the F-ratio.

For the following data,what is the value of SSB? B1 B2

A repeated-measures study uses a sample of n = 10 participants to evaluate the mean differences among three treatment conditions.For this study,df between subjects = 18.

The results from a two-factor analysis of variance show a significant main effect for factor A and a significant main effect for factor B.Based on this information,what can you conclude about the interaction?

The analysis of total variability into between-treatments and within-treatments variability is the same for a repeated-measures ANOVA and an independent-measures ANOVA.

A two-factor research study has 2 levels of factor A and three 3 of factor B with n = 8 participants in each treatment condition.For this study,what is the value for df between treatment?

For a repeated-measures design,the differences between treatments may be caused by a treatment effect or by sampling error (random factors),but they cannot be caused by individual differences.

A repeated-measures study uses a sample of n = 8 participants to evaluate the mean differences between two treatment conditions.The analysis of variance for this study will have dfₑᵣᵣₒᵣ = 7.

If a two-factor analysis of variance produces a statistically significant interaction,what can you conclude about the main effects?

For a repeated-measures study,if there are no systematic treatment effects (H₀ is true),then the numerator and denominator of the F-ratio are both measuring the same sources of variance.

For the following data,what is the value of $\text { SSwithin }$ ? B1 B2 $For the following data,what is the value of \text { SSwithin } ? B1 B2 A) 10 B) 20 C) 40 D) 60$