Question 1

Based on the results of a two-factor factorial experiment,the ANOVA table showed that SSE = 5.5.If we ignore one of the factors and perform a one-way ANOVA with the remaining factor using the same data,the SSE will always be larger than 5.5.

Accepted Answer

Ignoring one of the factors in a two-factor factorial experiment and performing a one-way ANOVA with the remaining factor generally leads to a loss of information about the interaction between the two factors. This typically results in a less accurate model, which is reflected in a larger Sum of Squares for Error (SSE) because the model can't account for as much variability in the data.

Question 2

After rejecting the null hypothesis of equal means between three treatment groups,a researcher computes Tukey simultaneous 95% confidence intervals for the difference between each pair of treatments.If all of the confidence intervals exclude the value zero,then at $\alpha$ 
= 0.05 we can conclude that there are significant differences between all pairs of treatment group means.

Accepted Answer

If all the Tukey simultaneous 95% confidence intervals for the differences between each pair of treatments exclude zero, it indicates that each pair of treatment group means is significantly different from each other at the $\alpha = 0.05$ level.

Question 3

In one-way ANOVA,a large value of F results when the within-treatment variability is large compared to the between-treatment variability.

Accepted Answer

A large value of F results when the between-treatment variability is large compared to the within-treatment variability.

Question 4

In a two-factor factorial experiment,the interaction effect between two factors can only be measured if there are multiple observations in each cell (combination of levels of two factors).

Accepted Answer

The interaction effect between two factors can only be measured if there are multiple observations in each cell. If there is only one observation in each cell, it is not possible to distinguish the effects of each factor from the interaction effect between them. Therefore, multiple observations are necessary to estimate the interaction effect accurately.

Question 5

Suppose that a teacher wants to study how four different study methods affect a student's final grade.The study method is the dependent variable.

Accepted Answer

The study method is the independent variable because it is what the teacher manipulates to observe its effect on the dependent variable, which in this case would be the student's final grade.

Question 6

When using a randomized block design,each block is used exactly once to measure the effect of each and every treatment.

Accepted Answer

In a randomized block design, each block is used to test each treatment exactly once to control for block-to-block variability and more accurately measure the effect of the treatments.

Question 7

The 95% individual confidence interval for $\mu _ { 1 } - \mu _ { 2 }$ 
will always be narrower than the Tukey 95% simultaneous confidence interval for $\mu _ { 1 } - \mu _ { 2 }$ 
.

Accepted Answer

The Tukey 95% simultaneous confidence interval accounts for multiple comparisons and thus is wider to maintain the overall confidence level across all comparisons, whereas the 95% individual confidence interval for $\mu _ { 1 } - \mu _ { 2 }$ is calculated for a single comparison and does not adjust for multiple comparisons, making it narrower.

Question 8

In a one-way ANOVA for a completely randomized design,all other things being held constant,as two sample treatment group means get closer further apart the probability of rejecting the null hypothesis increases.

Accepted Answer

As the treatment group means get further apart, the variability between the groups increases, which results in a larger F-statistic and a lower p-value. This leads to an increased probability of rejecting the null hypothesis. Therefore, the statement is true.

Question 9

If the total sum of squares in a one-way analysis of variance is 25 and the between-groups sum of squares is 17,then the error sum of squares is?

Accepted Answer

The error sum of squares is calculated by subtracting the between-groups sum of squares from the total sum of squares: 25 - 17 = 8.

Question 10

In a one-way ANOVA for a completely randomized design, $q _ { \alpha }$ 
is always less than $t _ { \alpha / 2 }$ 
.

Accepted Answer

In a one-way ANOVA for a completely randomized design, $q _ { \alpha }$ (the critical value from the Tukey's range test) and $t _ { \alpha / 2 }$ (the critical value from the Student's t-test) are not directly comparable in this manner. The value of $q _ { \alpha }$ depends on the number of groups and the total number of observations, while $t _ { \alpha / 2 }$ depends on the degrees of freedom for the error term and the chosen level of significance. Their relative sizes can vary depending on these parameters.

Question 11

In one-way ANOVA,the denominator of the F statistic is an estimate of the constant population variance of the treatment groups.

Accepted Answer

The denominator of the F statistic is calculated using the sample variances of each treatment group, which provides an estimate of the constant population variance of the treatment groups.

Question 12

In a completely randomized design,different levels of the factor are called _____.

Accepted Answer

In a completely randomized design, the factor being studied is manipulated by applying different levels of treatment to the groups being tested. These different levels of treatment are commonly referred to as "treatments."

Question 13

If a researcher cannot control the factor being studied,then the researcher is carrying out a designed experiment.

Accepted Answer

If a researcher cannot control the factor being studied, then the researcher is carrying out an observational study, not a designed experiment. In a designed experiment, researchers have control over the factor being studied and manipulate it in some way, such as by administering a treatment or intervention.

Question 14

In a one-way ANOVA for a completely randomized design,all other things being held constant,as the sample treatment group means get closer to each other the probability of rejecting the null hypothesis decreases.

Accepted Answer

As the sample treatment group means get closer to each other, the variation within groups relative to the variation between groups decreases, making it less likely to find statistically significant differences, thus decreasing the probability of rejecting the null hypothesis.

Question 15

In a one-way analysis of variance for a completely randomized design with three treatments groups,each with five measurements,what are the degrees of freedom associated with the error sum of squares?

Accepted Answer

The degrees of freedom associated with the error sum of squares is calculated as follows:
Degrees of Freedom (Error) = Total Degrees of Freedom - Degrees of Freedom (Treatments)
Total Degrees of Freedom = (Number of Observations) - 1 = 15 - 1 = 14
Degrees of Freedom (Treatments) = (Number of Treatment Groups) - 1 = 3 - 1 = 2
Therefore, Degrees of Freedom (Error) = 14 - 2 = 12. So, the best choice is D.

Question 16

In a one-way analysis of variance for a completely randomized design with three treatments groups,each with five measurements,what are the degrees of freedom associated with the between-groups sum of squares?

Accepted Answer

The degrees of freedom associated with the between-groups sum of squares in a one-way ANOVA is equal to the number of treatment groups minus 1, which in this case is 3-1 = 2. Therefore, the correct choice is B.

Question 17

A one-way analysis of variance is a method that allows us to estimate and compare the effects of several treatments on a response variable.

Accepted Answer

This statement is true. A one-way ANOVA is a statistical method used to compare the means of three or more groups to determine whether there is a statistically significant difference between them.

Question 18

In one-way ANOVA,the numerator degrees of freedom equals the number of treatment groups.

Accepted Answer

The numerator degrees of freedom in one-way ANOVA is the number of treatment groups minus one.

Question 19

In a one-way ANOVA for a completely randomized design,all other things being held constant,as the between-treatment variation decreases,the probability of rejecting the null hypothesis increases.

Accepted Answer

In a one-way ANOVA, as the between-treatment variation decreases, the F-ratio decreases, making it less likely to reject the null hypothesis, which states that all group means are equal.

Question 20

The error sum of squares measures the between-treatment variability.

Accepted Answer

The error sum of squares measures the within-treatment variability, not the between-treatment variability.

Quiz 10: Experimental Design and Analysis of Variance