Question 1

A researcher wanted to examine whether people remembered information better with auditory cues or with visual cues. Ten participants were randomly assigned to each of the two groups: one group heard a list of words from headphones, and the other group saw the same list of words from a screen. The participants were then asked to write down as many words as they could remember. The dependent variable was the number of words remembered.
-For which of the following situations is it appropriate to use ANCOVA?
A) In each group, half of the participants had a time limit to write the words down. Whether a time limit was imposed was also included in the model.
B) The participants' socioeconomic status (low, middle, or high) was expected to have an effect on their performance and was also included in the model.
C) Participants took a vocabulary test before the experiment and the scores of the test were included in the model.
D) Participants took a vocabulary test after the experiment and the scores of the test were included in the model.

Accepted Answer

ANCOVA (Analysis of Covariance) is appropriate when controlling for pre-existing differences among participants that might affect the outcome. In this case, including participants' pre-experiment vocabulary test scores (C) as a covariate helps to control for their initial vocabulary knowledge, which could influence how many words they remember, making the comparison between auditory and visual cues more accurate.

Question 2

A researcher wanted to examine whether people remembered information better with auditory cues or with visual cues. Ten participants were randomly assigned to each of the two groups: one group heard a list of words from headphones, and the other group saw the same list of words from a screen. The participants were then asked to write down as many words as they could remember. The dependent variable was the number of words remembered.
-Suppose the researcher used IQ as the covariate. He then found that the mean IQ is slightly different across two groups, and people with higher IQ tended to remember more words than people with lower IQ. It seems likely that
A) the assumption of independence was violated.
B) the assumption of homogeneity of variance was violated.
C) the assumption of independence of covariate and treatment was violated.
D) the assumption of homogeneity of slopes was violated.
E) there is no indication of assumption violation.

Accepted Answer

E

Question 3

A researcher wanted to examine whether people remembered information better with auditory cues or with visual cues. Ten participants were randomly assigned to each of the two groups: one group heard a list of words from headphones, and the other group saw the same list of words from a screen. The participants were then asked to write down as many words as they could remember. The dependent variable was the number of words remembered.
-Suppose out of budget concerns, the researcher included only five participants in each group, tested each participant twice, and used each time as separate observations. It seems likely that
A) the assumption of independence was violated.
B) the assumption of homogeneity of variance was violated.
C) the assumption of independence of covariate and treatment was violated.
D) the assumption of homogeneity of slopes was violated.
E) there is no indication of assumption violation.

Accepted Answer

Testing each participant twice and using each time as separate observations violates the assumption of independence because the scores from the same participant are likely to be correlated, not independent.

Question 4

based on the following scenario:
A researcher wanted to examine if soil type had any effects on the heights of daylilies. Ten bulbs of daylilies were planted in each of the three different types of soil. The thickness of the soil (X) was also measured for each pot. After three months, the heights of the plants (Y) were measured. Below are the group means of the study.
 $\begin{array}{cccc}
\hline \text { soil type } & n & \text { heights of daylilies (Y) } & \text { thickness of soil }(X) \
\hline 1 & 10 & \bar{Y}_{1}=20 & \bar{X}_{1}=21 \
2 & 10 & \bar{Y}_{2}=25 & \bar{X}_{2}=18 \
3 & 10 & \bar{Y}_{3}=30 & \bar{X}_{2}=15 \
\hline
\end{array}$
-If ANCOVA is used, which group will have the highest adjusted mean height?
A) Group 1 ( $\bar{Y}_{1}^{\prime}$ )
B) Group 2 ( $\bar{Y}_{2}^{\prime}$ )
C) Group 3 ( $\bar{Y}_{3}^{\prime}$ )
D) All groups would have the same adjusted mean.
E) Any of the above situations is possible (not enough information provided).

Accepted Answer

ANCOVA (Analysis of Covariance) adjusts the means based on covariates, in this case, the thickness of the soil (X). Without knowing the relationship between the thickness of the soil and the heights of the daylilies, or the specific adjustment formula used in the ANCOVA, it's impossible to determine which group will have the highest adjusted mean height.

Question 5

based on the following scenario:
A researcher wanted to examine if soil type had any effects on the heights of daylilies. Ten bulbs of daylilies were planted in each of the three different types of soil. The thickness of the soil (X) was also measured for each pot. After three months, the heights of the plants (Y) were measured. Below are the group means of the study.
 $\begin{array}{cccc}
\hline \text { soil type } & n & \text { heights of daylilies (Y) } & \text { thickness of soil }(X) \
\hline 1 & 10 & \bar{Y}_{1}=20 & \bar{X}_{1}=21 \
2 & 10 & \bar{Y}_{2}=25 & \bar{X}_{2}=18 \
3 & 10 & \bar{Y}_{3}=30 & \bar{X}_{2}=15 \
\hline
\end{array}$
-If there is a substantial, positive correlation between the thickness of soil (X) and the height of daylilies (Y), which group will have the highest adjusted mean height?
A) Group 1 ( $\bar{Y}_{1}^{\prime}$ )
B) Group 2 ( $\bar{Y}_{2}^{\prime}$ )
C) Group 3 ( $\bar{Y}_{3}^{\prime}$ )
D) All groups would have the same adjusted mean.
E) Any of the above situation is possible.

Accepted Answer

The answer of based on the following scenario:
A researcher wanted...

Question 6

based on the following scenario: A researcher wanted to examine if soil type had any effects on the heights of daylilies. Ten bulbs of daylilies were planted in each of the three different types of soil. The thickness of the soil (X) was also measured for each pot. After three months, the heights of the plants (Y) were measured. Below are the group means of the study. $\begin{array}{cccc} \hline \text { soil type } & n & \text { heights of daylilies (Y) } & \text { thickness of soil }(X) \ \hline 1 & 10 & \bar{Y}_{1}=20 & \bar{X}_{1}=21 \ 2 & 10 & \bar{Y}_{2}=25 & \bar{X}_{2}=18 \ 3 & 10 & \bar{Y}_{3}=30 & \bar{X}_{2}=15 \ \hline \end{array}$ -If there is no correlation between X and Y, the MS_with for ANCOVA as compared to that for ANOVA will be A) less B) the same C) greater D) unpredictably different

Accepted Answer

The answer of based on the following scenario:
A researcher wanted...

Question 7

based on the following scenario:
A researcher wanted to examine if soil type had any effects on the heights of daylilies. Ten bulbs of daylilies were planted in each of the three different types of soil. The thickness of the soil (X) was also measured for each pot. After three months, the heights of the plants (Y) were measured. Below are the group means of the study.
 $\begin{array}{cccc}
\hline \text { soil type } & n & \text { heights of daylilies (Y) } & \text { thickness of soil }(X) \
\hline 1 & 10 & \bar{Y}_{1}=20 & \bar{X}_{1}=21 \
2 & 10 & \bar{Y}_{2}=25 & \bar{X}_{2}=18 \
3 & 10 & \bar{Y}_{3}=30 & \bar{X}_{2}=15 \
\hline
\end{array}$
-If there is a substantial negative correlation between X and Y, the error variation for ANCOVA as compared to that for ANOVA will be
A) less
B) the same
C) greater
D) unpredictably different

Accepted Answer

When there is a substantial negative correlation between the covariate (X) and the dependent variable (Y), using ANCOVA (Analysis of Covariance) helps in adjusting for the effects of the covariate, thereby reducing the error variation compared to ANOVA (Analysis of Variance), which does not adjust for covariates.

Question 8

based on the following scenario:
A researcher wanted to examine if soil type had any effects on the heights of daylilies. Ten bulbs of daylilies were planted in each of the three different types of soil. The thickness of the soil (X) was also measured for each pot. After three months, the heights of the plants (Y) were measured. Below are the group means of the study.
 $\begin{array}{cccc}
\hline \text { soil type } & n & \text { heights of daylilies (Y) } & \text { thickness of soil }(X) \
\hline 1 & 10 & \bar{Y}_{1}=20 & \bar{X}_{1}=21 \
2 & 10 & \bar{Y}_{2}=25 & \bar{X}_{2}=18 \
3 & 10 & \bar{Y}_{3}=30 & \bar{X}_{2}=15 \
\hline
\end{array}$
-If the correlation between X and Y is 0.5 for soil type 1, 0.1 for soil type 2, and -0.5 for soil type 3, it seems likely that
A) the assumption of normality is violated.
B) the assumption of homogeneity of variance is violated.
C) the assumption of linearity is violated.
D) the assumption of homogeneity of slopes is violated.
E) there is no indication of assumption violation.

Accepted Answer

The varying correlations between X and Y across the different soil types suggest that the relationship between soil thickness and daylily height differs by soil type, indicating a violation of the homogeneity of slopes assumption.

Question 9

Matt has generated an ANCOVA. In testing the assumptions, he reviews the scatterplot of the covariate (X) and the dependent variable (Y) for each group and overall. For which assumption(s) is Matt likely reviewing evidence?
A) Homogeneity of regression slopes.
B) Homogeneity of variance.
C) Linearity.
D) All of the above.

Accepted Answer

Matt is likely reviewing evidence for all of the assumptions: homogeneity of regression slopes, homogeneity of variance, and linearity.

Question 10

In ANCOVA, suppose Y is the dependent variable, X is the covariate, and the factor has two levels. Also all assumptions of ANCOVA are met. Which of the following situations is the most desirable? A) r_XY = 0.5; $\bar{X}_{1}$ =10, $\bar{X}_{2}$ =12, $\bar{X}$ =11. B) r_XY = 0.1; $\bar{X}_{1}$ =10, $\bar{X}_{2}$ =12, $\bar{X} $ =11. C) r_XY = -0.1; $\bar{X}_{1}$ =10, $\bar{X}_{2}$ =20, $\bar{X} $ =15. D) r_XY = -0.5; $\bar{X}_{1}$ =10, $\bar{X}_{2}$ =20, $\bar{X} $ =15.

Accepted Answer

The answer of In ANCOVA, suppose Y is the dependent...

Question 11

In an one-factor ANCOVA, suppose the factor has two levels (groups). Scatterplots of the dependent variable (Y) and the covariate (X) are generated (where group 1 is indicated by "•" and group 2 by "o"). Which of the following graphs shows the most desirable situation?
a.  b.  c.  d.

Accepted Answer

The answer of In an one-factor ANCOVA, suppose the factor...

Question 12

Which of the following conditions about the dependent variable (Y) and the covariate (X) is required for ANCOVA?
A) Both X and Y are measured on interval or ratio scale.
B) Both X and Y are measured in the same unit.
C) Both X and Y are independent of the independent variable.
D) Both X and Y are measured without error.

Accepted Answer

The answer of Which of the following conditions about the...

Question 13

An experiment was conducted to compare four different weight loss programs. Twenty subjects participated in each program, and their weights were recorded before and after the program. If both the pretest weight and age are used as covariates, what are the degrees of freedom for the error term?
A) 74
B) 75
C) 76
D) 79

Accepted Answer

The degrees of freedom for the error term in an ANCOVA (Analysis of Covariance) can be calculated by subtracting the number of groups and the number of covariates from the total number of observations, then subtracting 1 more for the overall mean. Here, there are 4 groups (the weight loss programs), 2 covariates (pretest weight and age), and 80 observations (20 subjects in each of the 4 programs). Thus, the degrees of freedom for the error term are 80 - 4 - 2 - 1 = 73. However, since the question seems to imply a calculation error by providing options not directly aligning with this calculation, and given the closest option is 74, it suggests a possible misinterpretation in the calculation instructions or an error in the question's formulation. Normally, the correct calculation would indeed result in 73, but given the options, 74 is chosen based on the closest match principle, acknowledging a potential discrepancy in the question's structure or a typo in the provided options.

Question 14

A study was conducted to compare five new textbooks. Thirty teachers were asked to rate the textbooks they used in the previous semester, and then they were randomly assigned to one of five new textbook groups (n = 6). At the end of the semester, teachers' ratings of the new textbooks (Y) were collected. What may be an appropriate covariate for this study?
A) Students' achievement at the end of the semester.
B) Students' rating of the new textbook at the end of the semester.
C) Teachers' rating of the old textbook.
D) Teachers' ethnicity.

Accepted Answer

Teachers' rating of the old textbook could serve as an appropriate covariate because it provides a baseline measure of the teachers' perceptions or biases towards textbooks in general, which can be controlled for when analyzing the effect of the new textbooks on their ratings. This helps in isolating the effect of the new textbooks from teachers' inherent rating tendencies.

Question 15

In the study described in Question 14, suppose an appropriate covariate has been chosen. What are the degrees of freedom for the error term?
A) 4
B) 24
C) 25
D) 29

Accepted Answer

The degrees of freedom for the error term in an analysis of covariance (ANCOVA) are calculated by subtracting the number of parameters estimated (including the covariate) from the total number of observations. Without specific details from Question 14, a typical scenario might involve adjusting for group sizes and the inclusion of a covariate. If there were 30 observations and parameters estimated include group means, the covariate, and the overall mean, the degrees of freedom for error would be reduced accordingly. Assuming one covariate and groups are accounted for, and without exact numbers from the original question, the choice closest to a common scenario for an ANCOVA with a moderate sample size and one covariate is 24, suggesting some parameters (including groups and the covariate) have been accounted for, leaving 24 degrees of freedom for error.

Question 16

If the dependent variable, math score (Y), is about the same for persons with higher scores in writing (X) and for those with lower scores in writing, one would expect that
A) adding the covariate will not substantially reduce error variation.
B) the adjusted mean for each group will be the same.
C) ANCOVA will be more powerful than ANOVA.
D) the covariate and factor will be independent of each other.

Accepted Answer

Adding a covariate that does not vary systematically with the dependent variable will not substantially reduce error variation in the analysis. Since the math score is about the same for persons with higher and lower scores in writing, the writing score (as a covariate) is not expected to explain a significant portion of the variance in math scores, thus not reducing error variation substantially.

Question 17

In which of the following situations will the adjusted means always equal the unadjusted means in ANCOVA? A) All groups have the same unadjusted mean. B) All groups have the same mean on the covariate. C) The dependent variable and covariate are highly correlated. D) Regression of the dependent variable on the covariate with b_w = 0.6.

Accepted Answer

When all groups have the same mean on the covariate, the adjustment made by ANCOVA (which corrects group means based on their covariate values) does not change the means from their original values, because the covariate does not differentiate between the groups.

Question 18

In reviewing the assumptions, Susan runs an ANOVA using the covariate X as the dependent variable and the factor A as the independent variable. For this test, which of the following situations is the most desirable?
A) p = 0.01
B) p = 0.05
C) p = 0.1
D) p = 0.5

Accepted Answer

In the context of ANOVA, when using a covariate, the goal is to control for its effect. A high p-value (like 0.5) when testing the covariate against the factor indicates that the covariate does not significantly vary with the factor, which is desirable because it suggests that the covariate's effect is consistent across levels of the factor, allowing for a clearer analysis of the factor's effect on the dependent variable.

Question 19

Now using Y as the dependent variable, Susan runs another ANOVA to evaluate if there is any interaction between the covariate X and the factor ِ. For $F_{A \times X}$ , which of the following situations is the most desirable?
A) p = 0.01
B) p = 0.05
C) p = 0.1
D) p = 0.5

Accepted Answer

In ANOVA, a high p-value (such as 0.5) for the interaction term $F_{A \times X}$ suggests that there is no significant interaction between the covariate X and the factor, which is often desirable as it indicates that the effect of the covariate is consistent across levels of the factor.

Question 20

In which of the following situations is the assumption of linearity violated?
A) Trend analysis shows that there exists a significant quadratic trend.
B) There is no association between the independent variable and the covariate.
C) The distribution of the covariate is seriously skewed in all groups.
D) The scatterplot of the dependent variable and the covariate is U shaped for all groups.

Accepted Answer

The answer of In which of the following situations is...

Quiz 14: One-Factor Fixed-Effects Ancova With Single Covariate

Matt has generated an ANCOVA. In testing the assumptions, he reviews the scatterplot of the covariate (X) and the dependent variable (Y) for each group and overall. For which assumption(s) is Matt likely reviewing evidence?

In ANCOVA, suppose Y is the dependent variable, X is the covariate, and the factor has two levels. Also all assumptions of ANCOVA are met. Which of the following situations is the most desirable?

In an one-factor ANCOVA, suppose the factor has two levels (groups). Scatterplots of the dependent variable (Y) and the covariate (X) are generated (where group 1 is indicated by "•" and group 2 by "o"). Which of the following graphs shows the most desirable situation? a. b. c. d.

Which of the following conditions about the dependent variable (Y) and the covariate (X) is required for ANCOVA?

An experiment was conducted to compare four different weight loss programs. Twenty subjects participated in each program, and their weights were recorded before and after the program. If both the pretest weight and age are used as covariates, what are the degrees of freedom for the error term?

In the study described in Question 14, suppose an appropriate covariate has been chosen. What are the degrees of freedom for the error term?

If the dependent variable, math score (Y) , is about the same for persons with higher scores in writing (X) and for those with lower scores in writing, one would expect that

In which of the following situations will the adjusted means always equal the unadjusted means in ANCOVA?

In reviewing the assumptions, Susan runs an ANOVA using the covariate X as the dependent variable and the factor A as the independent variable. For this test, which of the following situations is the most desirable?

Now using Y as the dependent variable, Susan runs another ANOVA to evaluate if there is any interaction between the covariate X and the factor ِ. For $F_{A \times X}$ , which of the following situations is the most desirable?

In which of the following situations is the assumption of linearity violated?