Question 1

The analysis of variance helps us compare the means of two, or more quantitative populations. Thus, it might be used by the manager of a farm who wants to know whether there is any difference in:
A) the average crop yield associated with the use of five types of fertilizer
B) the average crop yield associated with five alternative quantities of a given type of fertilizer that might be applied
C) the average crop yield associated with five alternative times at which a given quantity of a given type of fertilizer might be applied
D) all of these
E) none of these

Accepted Answer

The analysis of variance can be used to compare the means of multiple populations, regardless of the factor being compared. Therefore, all three scenarios (types of fertilizer, quantities of fertilizer, and times of application) could be compared using ANOVA.

Question 2

In one-way ANOVA, the test statistic is defined as the ratio of the mean square for error (MSE) and the mean square for treatments (MST); namely, F = MSE / MST.

Accepted Answer

In one-way ANOVA, the test statistic is defined as the ratio of the mean square for treatments (MST) to the mean square for error (MSE); namely, F = MST / MSE.

Question 3

In the completely randomized design of ANOVA where there are k treatments and n observations, the degrees of freedom for the F-statistic are equal to:
A) n and k
B) k and n
C) n - k and k - 1
D) k - 1 and n - k
E) n - 1 and k - 1

Accepted Answer

The degrees of freedom for the F-statistic in a completely randomized design of ANOVA are calculated as follows:

- The numerator degrees of freedom is k - 1 because we are comparing the variation between the means of the different treatments (k groups) to the variation within each group. There are k groups, so the degrees of freedom for the numerator is k - 1.
- The denominator degrees of freedom is n - k because we estimate the within-group variance by calculating the variance of each group separately and averaging them. We lose one degree of freedom for each group when calculating within-group variance, so we subtract k from the total sample size n to obtain the degrees of freedom for the denominator, which is n - k.

Thus, the correct answer is D) k - 1 and n - k.

Question 4

The sum of squares for treatments, SST, achieves its smallest value (zero) when all the sample means are equal.

Accepted Answer

This is true by definition. SST represents the variation attributable to differences between treatment means, so if all the treatment means are equal, there will be no variation and SST will be zero.

Question 5

One-way ANOVA is used when:
A) analyzing the difference between more than two population means
B) analyzing the difference between two population variances
C) analyzing the difference between two sample means
D) analyzing the difference between two population proportions
E) none of these

Accepted Answer

One-way ANOVA is used to analyze the difference between more than two population means. It is used when there are three or more groups being compared on the same outcome variable.

Question 6

In a one-way ANOVA, if the null hypothesis that all population means are equal is rejected, then we can conclude that the alternative hypothesis is true and that all population means differ.

Accepted Answer

Rejecting the null hypothesis in a one-way ANOVA only indicates that at least one group mean is different from the others; it does not specify how many or which means differ.

Question 7

The numerator or MST degrees of freedom are 3 and the denominator or MSE degrees of freedom are 18. The total number of observations in the completely randomized design must equal 20.

Accepted Answer

In a completely randomized design, the total number of observations equals the sum of the degrees of freedom for MST (Mean Square Treatment) and MSE (Mean Square Error) plus 1. Given MST degrees of freedom are 3 and MSE degrees of freedom are 18, the total number of observations would be 3 + 18 + 1 = 22.

Question 8

In a completely randomized design for ANOVA, the number of degrees of freedom for the numerator and denominator are 3 and 16, respectively. The total number of observations must equal:
A) 20
B) 48
C) 32
D) 19
E) 24

Accepted Answer

The total number of degrees of freedom is the sum of the degrees of freedom for the numerator and denominator, which is 3 + 16 = 19. The total number of observations is equal to the total number of degrees of freedom plus 1, which is 19 + 1 = 20.

Question 9

The distribution of the test statistic for analysis of variance is the:
A) normal distribution
B) Student t-distribution
C) F-distribution
D) binomial distribution
E) chi-square distribution

Accepted Answer

The distribution of the test statistic for analysis of variance is the F-distribution. This is because the test statistic is calculated by dividing the mean square between groups by the mean square within groups, and this ratio follows an F-distribution under the null hypothesis of no group differences. Therefore, the correct choice is C.

Question 10

Which of the following statements about the ANOVA table is correct?
A) It is a summary table that shows, for each source of variation, the sum of squares, the degrees of freedom, and the ratio of the sum of squares to the associated degrees of freedom (called the mean square).
B) It shows the F statistic.
C) It shows the statistics used to test hypotheses about the population means.
D) Both "It is a summary table that shows, for each source of variation, the sum of squares, the degrees of freedom, and the ratio of the sum of squares to the associated degrees of freedom (called the mean square)" and "It shows the F statistic".
E) Neither "It is a summary table that shows, for each source of variation, the sum of squares, the degrees of freedom, and the ratio of the sum of squares to the associated degrees of freedom (called the mean square)" nor "It shows the F statistic".

Accepted Answer

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

Question 11

Which of the following is not a required condition for one-way ANOVA?
A) The populations are normally distributed.
B) The sample sizes must be equal.
C) The population variances are equal.
D) The samples are selected randomly and independently from their respective populations.
E) None of these.

Accepted Answer

While equal sample sizes are desirable, they are not a required condition for one-way ANOVA. However, the other conditions listed (normality, equal variances, random and independent samples) are necessary for the validity of the analysis.

Question 12

The F-test of the analysis of variance requires that the populations be normally distributed with equal variances.

Accepted Answer

The F-test assumes that the populations being compared are normally distributed with equal variances.

Question 13

Which of the following statements about ANOVA completely randomized design is correct?
A) The ANOVA test assumes that the sampled populations are normally distributed.
B) The ANOVA test assumes that the sampled populations have a common variances.
C) The ANOVA test assumes that the samples are randomly and independently selected from their respective populations.
D) All of these.
E) None of these.

Accepted Answer

A) The ANOVA test assumes that the sampled populations are normally distributed.
B) The ANOVA test assumes that the sampled populations have a common variances.
C) The ANOVA test assumes that the samples are randomly and independently selected from their respective populations. 
All three statements are true and are assumptions for a completely randomized design ANOVA test.

Question 14

The degrees of freedom for a denominator in a one-way ANOVA test including 3 population means with 10 observations sampled from each population is equal to:
A) 30
B) 27
C) 13
D) 12
E) 9

Accepted Answer

For a one-way ANOVA test with k groups and n observations per group, the degrees of freedom for the denominator (error or within-group variability) is (k-1)(n-1). Therefore, in this case, the degrees of freedom for the denominator is (3-1)(10-1) = 18, which corresponds to answer choice B.

Question 15

In one-way ANOVA, suppose that there are three treatments with  ,  and  . Then the rejection region for this test at the 5% level of significance is:
A) F > 3.24
B) F > 4.08
C) F > 3.63
D) F > 3.68
E) F > 3.22

Accepted Answer

The answer of In one-way ANOVA, suppose that there are...

Question 16

An experimental plan that creates one treatment group for each treatment and then randomly assigns each experimental unit to one of these groups is called:
A) a completely randomized design
B) a randomized block design
C) either a completely randomized design or a randomized block design
D) a matched-pairs design
E) none of these

Accepted Answer

A completely randomized design creates one treatment group for each treatment and then randomly assigns each experimental unit to one of these groups.

Question 17

Which of the following is not true of the F-distribution?
A) It is symmetric around zero.
B) It is skewed to the right.
C) Its values are always positive.
D) It is used in ANOVA test.
E) All of these.

Accepted Answer

The F-distribution is not symmetric around zero; it is skewed to the right, its values are always positive, and it is used in ANOVA tests.

Question 18

In one-way ANOVA test, the test statistic is F = 4.25. The rejection region is F > 3.06 for the 5% level of significance, F > 3.8 for the 2.5% level, and F > 4.89 for the 1% level. For this test, the approximate p-value is:
A) greater than 0.05
B) between 0.025 and 0.05
C) between 0.01 and 0.025
D) approximately 0.05
E) approximately 0.01

Accepted Answer

The approximate p-value can be found by comparing the test statistic F = 4.25 to the critical values for the rejection regions. Since F > 4.89 for the 1% level, we can conclude that the p-value is less than 0.01. Since F < 3.8 for the 2.5% level, we can conclude that the p-value is greater than 0.025. Therefore, the best choice is C, between 0.01 and 0.025.

Question 19

The calculated value of F in a one-way ANOVA is 7.88. The numerator and denominator degrees of freedom are 3 and 9, respectively. The most accurate statement to be made about the p-value is that p-value < 0.01.

Accepted Answer

The calculated F-value of 7.88 with a numerator df of 3 and denominator df of 9 corresponds to a p-value less than 0.01, which indicates that the difference between the means of at least two groups is statistically significant at alpha level of 0.01 or lower.

Question 20

The test statistic of the completely randomized ANOVA design equals:
A) sum of squares for treatments / sum of squares for error
B) sum of squares for error / sum of squares for treatments
C) mean square for treatments / mean square for error
D) mean square for error / mean square for treatments
E) sum of squares for error / mean square for error

Accepted Answer

The test statistic for a completely randomized ANOVA design is the F statistic, which is the ratio of the mean square for treatments (the variance between treatment groups) to the mean square for error (the variance within treatment groups). Therefore, the correct choice is C, which is the ratio of the mean square for treatments to the mean square for error.

Quiz 11: The Analysis of Variance

The analysis of variance helps us compare the means of two, or more quantitative populations. Thus, it might be used by the manager of a farm who wants to know whether there is any difference in:

In one-way ANOVA, the test statistic is defined as the ratio of the mean square for error (MSE) and the mean square for treatments (MST); namely, F = MSE / MST.

In the completely randomized design of ANOVA where there are k treatments and n observations, the degrees of freedom for the F-statistic are equal to:

The sum of squares for treatments, SST, achieves its smallest value (zero) when all the sample means are equal.

One-way ANOVA is used when:

In a one-way ANOVA, if the null hypothesis that all population means are equal is rejected, then we can conclude that the alternative hypothesis is true and that all population means differ.

The numerator or MST degrees of freedom are 3 and the denominator or MSE degrees of freedom are 18. The total number of observations in the completely randomized design must equal 20.

In a completely randomized design for ANOVA, the number of degrees of freedom for the numerator and denominator are 3 and 16, respectively. The total number of observations must equal:

The distribution of the test statistic for analysis of variance is the:

Which of the following statements about the ANOVA table is correct?

Which of the following is not a required condition for one-way ANOVA?

The F-test of the analysis of variance requires that the populations be normally distributed with equal variances.

Which of the following statements about ANOVA completely randomized design is correct?

The degrees of freedom for a denominator in a one-way ANOVA test including 3 population means with 10 observations sampled from each population is equal to:

In one-way ANOVA, suppose that there are three treatments with , and . Then the rejection region for this test at the 5% level of significance is:

An experimental plan that creates one treatment group for each treatment and then randomly assigns each experimental unit to one of these groups is called:

Which of the following is not true of the F-distribution?

In one-way ANOVA test, the test statistic is F = 4.25. The rejection region is F > 3.06 for the 5% level of significance, F > 3.8 for the 2.5% level, and F > 4.89 for the 1% level. For this test, the approximate p-value is:

The calculated value of F in a one-way ANOVA is 7.88. The numerator and denominator degrees of freedom are 3 and 9, respectively. The most accurate statement to be made about the p-value is that p-value < 0.01.

The test statistic of the completely randomized ANOVA design equals: