Question 1

The Bonferroni adjustment decreases the experimentwise Type I error rate, but it increases the probability of a Type II error.

Accepted Answer

The Bonferroni adjustment decreases the experimentwise Type I error rate by controlling the familywise error rate (FWER), but this comes at the expense of increasing the probability of a Type II error. This is because by applying the Bonferroni correction, we are reducing the significance level of each individual hypothesis test, which makes it more difficult to reject the null hypothesis and increases the likelihood of failing to detect a true effect.

Question 2

A randomized block experiment having five treatments and six blocks produced the following values: SST = 252, SS(Total) = 1,545, SSE = 198. The value of SSB must be 1095.

Accepted Answer

SS(Total) = SSB + SSW + SSE

where SSW is the sum of squares within the blocks. 
Since there are 5 treatments and 6 blocks, each treatment appears in each block exactly once. Thus, the number of observations per treatment is 6.

SSE = (n-1) * MSW = (6-1) * MSW = 5 * MSW 
where MSW is the mean sum of squares within the blocks. 
Therefore, MSW = SSE/5 = 198/5 = 39.6

SST = SSB + SSW + SSE 
1545 = 1095 + 5*6*39.6 + 198 
1095 = SST - SSW - SSE 
Substituting the value of SSE and MSW, we get: 
1095 = SST - (5*6*39.6) - 5*39.6 
Solving for SST, we get: 
SST = 1095 + 5*6*39.6 + 5*39.6 
SST = 252

Therefore, the statement is true and SSB must be 1095.

Question 3

SSE in the independent samples design is equal to the sum of SSB and SSE in the randomized block design.

Accepted Answer

The answer of SSE in the independent samples design is...

Question 4

In a one-way ANOVA, error variability is computed as the sum of the squared errors, SSE, for all values of the response variable. This variability is the:
A) the total variation.
B) within-treatments variation.
C) between-treatments variation.
D) None of these choices.

Accepted Answer

Error variability is the within-treatments variation, which represents the variation in the response variable that cannot be explained by the treatment factors. The between-treatments variation represents the variation that can be attributed to the treatment factors, and the total variation is the sum of both within-treatments and between-treatments variation. Therefore, option B is the correct answer. Option A is incorrect because total variation includes between-treatments variation as well. Option C is incorrect because between-treatments variation represents the treatment effects. Option D is incorrect because there is a correct answer.

Question 5

Tukey's multiple comparison method is based on the Studentized range statistic.

Accepted Answer

Tukey's multiple comparison method does use the Studentized range statistic as the basis for its calculations.

Question 6

In a single-factor analysis of variance, MST is the mean square for treatments and MSE is the mean square for error. The null hypothesis of equal population means is rejected if:
A) MST is much larger than MSE.
B) MST is much smaller than MSE.
C) MST is equal to MSE.
D) None of these choices.

Accepted Answer

In a single-factor analysis of variance, the null hypothesis of equal population means is rejected if the F-ratio (MST/MSE) is significant. A significant F-ratio occurs when MST is much larger than MSE, indicating that the treatment means significantly differ from each other, and therefore the null hypothesis is rejected.

Question 7

The purpose of designing a randomized block experiment is to reduce the between-treatments variation (SST) to more easily detect differences between the treatment means.

Accepted Answer

The purpose of designing a randomized block experiment is to reduce the within-treatment variation by accounting for variability due to other factors (blocks), thereby making it easier to detect differences between treatment means. It does not directly aim to reduce the between-treatments variation (SST), but rather to control for extraneous variation.

Question 8

The F-test of the randomized block design of the analysis of variance has the same requirements as the independent samples design; that is, the random variable must be normally distributed and the population variances must be equal.

Accepted Answer

The F-test in both randomized block design and independent samples design of analysis of variance requires that the data be normally distributed and that the variances across groups (or blocks) are homogenous (equal). These assumptions ensure the validity of the F-test results.

Question 9

The test statistic of the single-factor ANOVA 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.

Accepted Answer

The test statistic of the single-factor ANOVA is the F-statistic, which is calculated by taking the mean square for treatments divided by the mean square for error. Therefore, the correct choice is C.

Question 10

A randomized block design with 4 treatments and 5 blocks produced the following sum of squares values: SS(Total) = 2000, SST = 400, SSE = 200. The value of MSB must be 350.

Accepted Answer

The answer of A randomized block design with 4 treatments...

Question 11

When the variation associated with blocks is removed from the data, SSE increases.

Accepted Answer

When the variation associated with blocks is removed from the data, the within-group variation (SSE) decreases because there is less error attributed to the variability between blocks.

Question 12

In employing the randomized block design, the primary interest lies in reducing sum of squares for blocks (SSB).

Accepted Answer

The primary interest in employing the randomized block design is to reduce the sum of squares due to error (SSE), not the sum of squares for blocks (SSB), by accounting for variability due to the blocks.

Question 13

Which of the following is not a required condition for one-way ANOVA?
A) The sample sizes must be equal.
B) The populations must all be normally distributed.
C) The population variances must be equal.
D) The samples for each treatment must be selected randomly and independently.

Accepted Answer

The requirement for equal sample sizes is not a strict condition for conducting a one-way ANOVA. While equal sample sizes can increase the power of the test, ANOVA can still be performed with unequal sample sizes, provided the other assumptions (normality, equal variances, and independent random sampling) are met.

Question 14

In one-way ANOVA, the amount of total variation that is unexplained is measured by the:
A) sum of squares for treatments.
B) degrees of freedom.
C) total sum of squares.
D) sum of squares for error.

Accepted Answer

The sum of squares for error measures the amount of variation that is unexplained or random in one-way ANOVA. The other options measure the variation that can be explained by the treatment effects (sum of squares for treatments) or the overall variation in the data (total sum of squares). Degrees of freedom is a measure of the number of independent pieces of information used to estimate a parameter and is not specifically related to measuring unexplained variation.

Question 15

When the problem objective is to compare more than two populations, the experimental design that is the counterpart of the matched pairs experiment is called the randomized block design.

Accepted Answer

This is a true statement. In a randomized block design, the subjects are divided into groups or blocks based on certain characteristics (such as age, sex, or health status), and then each block is randomly assigned to one of the treatment groups. This design helps control for variability introduced by the differences between the blocks, making it easier to detect differences between the treatment groups. The matched pairs experiment is used when comparing two populations and involves pairing subjects based on some characteristic (such as age, sex, or weight) and randomly assigning one member of each pair to each treatment group.

Question 16

Tukey's multiple comparison method is more powerful than Fisher's LSD Method at finding differences in pairwise population means.

Accepted Answer

Tukey's method is designed to control for the familywise error rate when making multiple pairwise comparisons, while Fisher's LSD method does not take into account the number of comparisons being made. Thus, Tukey's method is generally more powerful at detecting significant differences while controlling for the probability of making a Type I error.

Question 17

Fisher's least significant difference method (LSD) substitutes the pooled variance estimator from the equal variances t-test with the MSE from ANOVA.

Accepted Answer

Fisher's LSD method does use the MSE from ANOVA as the estimator for the error variance, replacing the pooled variance estimator from the equal variances t-test.

Question 18

The Bonferroni adjustment to Fisher's Least Significant Difference (LSD) multiple comparison method is made by dividing the specified experimentwise Type I error rate by the number of pairs of population means.

Accepted Answer

The Bonferroni adjustment involves dividing the experimentwise Type I error rate by the number of comparisons made, not just the number of pairs of population means.

Question 19

Multiple comparison methods are used to determine whether or not any differences occur amongst a group of population means.

Accepted Answer

The answer of Multiple comparison methods are used to determine...

Question 20

One example of a blocking variable is the dosage level that each subject is assigned to in a randomized experiment.

Accepted Answer

A blocking variable is a variable that is not of primary interest but is controlled or accounted for to reduce variation, not the primary variable being manipulated like dosage level in an experiment. Dosage level is typically a treatment variable, not a blocking variable.

Quiz 14: Analysis of Variance