Deck 16: Analysis of Variance

Question 1

Which of the following best describes an experimental design model where the treatments are defined as the levels of one factor, and the experimental design specifies independent samples?

A) Randomised block design
B) Two way ANOVA
C) One way ANOVA
D) All of these choices are correct.

Answer 1

C

Question 2

In an ANOVA test, the test statistic is F = 3.08. The rejection region is F > 3.07 for the 5% level of significance, F > 3.82 for the 2.5% level, and F > 4.87 for the 1% level. For this test, the p-value is:

A) greater than 0.05.
B) between 0.01 and 0.025.
C) between 0.025 and 0.05.
D) smaller than 0.01.

Answer 2

C

Question 3

One-way ANOVA is applied to three independent samples having means 10, 13 and 18, respectively. If each observation in the first sample were decreased by 5, the value of the F-statistic would:

A) increase.
B) decrease.
C) remain unchanged.
D) decrease by 5.

Answer 3

B

Question 4

Which of the following is true of the F-distribution?

A) It is skewed to the right.
B) Its values are always positive.
C) It is used in the ANOVA test.
D) All of these choices are correct

Question 5

In one-way ANOVA, suppose that there are four treatments with n₁ = 7, n₂ = 6, n₃ = 5, and n₄ = 7. Then the rejection region for this test at the 1% level of significance is:

A) F > F_0.05,3,24.
B) F > F_0.025,3,21.
C) F > F_0.025,3,24.
D) F > F_0.01,3,21.

Question 6

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.

Question 7

Which of the following is compared in ANOVA ?

A) Two or more population standard deviations.
B) Two or more population means.
C) Two or more population variances.
D) All of these choices are correct.

Question 8

Two independent samples of size 30 each have been selected at random from the female and male students of a university. To test whether there is any difference in the grade point average between female and male students, an equal-variances t-test will be considered. Another test to consider is ANOVA. Which of the following is the most likely ANOVA to fit this test situation?

A) Randomised block design.
B) Two-way ANOVA.
C) One-way ANOVA.
D) Chi-squared test.

Question 9

In one-way ANOVA, the amount of total variation that is unexplained is measured by the:

A) sum of squares for treatments.
B) sum of squares for error.
C) total sum of squares.
D) number of degrees of freedom.

Question 10

In a two-tailed pooled-variance t-test (equal-variances t-test), the null and alternative hypotheses are exactly the same as in one-way ANOVA with:

A) exactly one treatment.
B) exactly two treatments.
C) exactly three treatments.
D) any number of treatments.

Question 11

In a one-way ANOVA where there are k treatments and n observations, the numbers of degrees of freedom for the F-statistic are equal to:

A) k - 1 and n - 1.
B) k - 1 and n - k.
C) n - k and k - 1.
D) nk - 1 and k - 1.

Question 12

The following equation applies to which ANOVA model? SS(Total) = SST + SSE.

A) One-way ANOVA.
B) Two-way ANOVA.
C) Completely randomised design.
D) Randomised block design.

Question 13

In 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) total variation.
B) within-group variation.
C) between-groups variation.
D) None of these choices are correct.

Question 14

Which of the following statements is true?

A) The sum of squares for treatments (SST) explains some of the variation.
B) The sum of squares for error (SSE) measures the amount of variation that is unexplained.
C) SS(Total) = SST + SSE
D) All of these choices are correct.

Question 15

The F-statistic in a one-way ANOVA represents the variation:

A) between the treatments plus the variation within the treatments.
B) within the treatments minus the variation between the treatments.
C) between the treatments divided by the variation within the treatments.
D) variation within the treatments divided by the variation between the treatments.

Question 16

Which of the following is the primary interest of designing a randomised block experiment?

A) Reduce the variation among blocks.
B) Reduce the within-treatments variation to more easily detect differences among the treatment means.
C) Increase the between-treatments variation to more easily detect differences among the treatment means.
D) Increase the total sum of squares.

Question 17

Which of the following is a required condition for one-way ANOVA?

A) The populations must all be normally distributed.
B) The population variances must be equal.
C) The samples for each treatment must be selected randomly and independently.
D) All of these choices are correct.

Question 18

Which of the following best describes the distribution of the test statistic for ANOVA?

A) Normal distribution and the F distribution.
B) Student t distribution.
C) Chi-squared distribution.
D) F distribution.

Question 19

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 likely false if:

A) MST is much larger than MSE.
B) MST is much smaller than MSE.
C) MST is equal to MSE.
D) MST is zero.

Question 20

If we want to conduct a test to determine whether a population mean is greater than another population mean, we:

A) can use the analysis of variance.
B) must use the independent-samples t-test for the difference between two means.
C) must use the chi-squared test.
D) can use the analysis of variance or can use the independent-samples t-test for the difference between two means.

Question 21

When the effect of a level for one factor depends on which level of another factor is present, the most appropriate ANOVA design to use in this situation is the:

A) one-way ANOVA.
B) two-way ANOVA.
C) randomised block design.
D) matched pairs design.

Question 22

Which of the following best describes the between-treatments in single-factor analysis of variance?

A) Sum of squares for error.
B) Total sum of squares.
C) Sum of squares for treatments.
D) All of these choices are correct.

Question 23

In the one-way ANOVA where k is the number of treatments and n is the number of observations in all samples, the number of degrees of freedom for treatments is given by:

A) k - 1.
B) n - k.
C) n - 1.
D) n - k + 1.

Question 24

A professor of statistics at Wayne State University in the US wants to determine whether the average starting salaries among graduates of the 15 universities in Michigan are equal. A sample of 25 recent graduates from each university is randomly taken. The appropriate critical value for the ANOVA test is obtained from the F-distribution with number so of degrees of freedom equal to:

A) 15 and 25.
B) 14 and 360.
C) 360 and 14.
D) 25 and 15.

Question 25

In single-factor analysis of variance, if large differences exist among the sample means, it is then reasonable to:

A) reject the null hypothesis.
B) reject the alternative hypothesis.
C) fail to reject the null hypothesis.
D) None of these choices are correct.

Question 26

A survey will be conducted to compare the grade point averages of US high-school students from four different school districts. Students are to be randomly selected from each of the four districts and their grade point averages recorded. The ANOVA model most likely to fit this situation is:

A) one-way ANOVA.
B) two-way ANOVA.
C) randomised block design.
D) complete 4 * 4 factorial design.

Question 27

One-way ANOVA is applied to independent samples taken from four normally distributed populations with equal variances. If the null hypothesis is rejected, then we can infer that:

A) all population means are equal.
B) all population means differ.
C) at least two population means are equal.
D) at least two population means differ.

Question 28

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

A) The populations are normally distributed.
B) The population variances are equal.
C) The samples are selected independently of each other.
D) The population means are equal.

Question 29

In ANOVA, the F-test is the ratio of two sample variances. In the one-way ANOVA (completely randomised design), the variance used as the denominator of the ratio is the:

A) mean square for treatments.
B) mean square for error.
C) mean square for blocks.
D) total sum of squares.

Question 30

In the one-way ANOVA where k is the number of treatments and n is the number of observations in all samples, the number of degrees of freedom for error is given by:

A) k - 1.
B) n - k.
C) n - 1.
D) n - k + 1.

Question 31

In one-way analysis of variance, within-treatments variation stands for the:

A) sum of squares for error.
B) sum of squares for treatments.
C) total sum of squares.
D) None of these choices are correct.

Question 32

One-way ANOVA is performed on independent samples taken from three normally distributed populations with equal variances. The following summary statistics were calculated: 7, 65, 4.2. 8, 59, 4.9. 9, 63, 4.6. The value of the test statistics, F, equals:

A) 71.250.
B) 0.322.
C) 21.104.
D) 3.376.

Question 33

Consider the following partial ANOVA table: The numbers of degrees of freedom for numerator and denominator, respectively, (identified by asterisks) are:

A) 4 and 15.
B) 3 and 16.
C) 15 and 4.
D) 16 and 3.

Question 34

Which of the following is a correct formulation for the null hypothesis in one-way ANOVA?

A) .
B) .
C) .
D) .

Question 35

Which of the following is not true of Tukey's multiple comparison method?

A) It is based on the studentised range statistic q to obtain the critical value needed to construct individual confidence intervals.
B) It requires that all sample sizes are equal, or at least similar.
C) It can be employed instead of the analysis of variance.
D) All of these choices are correct.

Question 36

The randomised block design with exactly two treatments is equivalent to a two-tailed:

A) independent samples z-test.
B) independent-samples equal-variances t-test.
C) independent-samples unequal-variances t-test.
D) matched pairs t-test.

Question 37

One-way ANOVA is performed on independent samples taken from three normally distributed populations with equal variances. The following summary statistics are calculated: 6, 50, 5.2. 8, 55, 4.9 . 6, 51, 5.4. The grand mean equals:

A) 50.0.
B) 52.0.
C) 52.3.
D) 53.0.

Question 38

Three tennis players, one a beginner, one intermediate and one advanced, have been randomly selected from the membership of a club in a large city. Using the same tennis ball, each player hits ten serves, one with each of three racquet models, with the three racquet models selected randomly. The speed of each serve is measured with a machine and the result recorded. Among the ANOVA models listed below, the most likely model to fit this situation is the:

A) one-way ANOVA.
B) two-way ANOVA.
C) randomised block design.
D) matched-pairs model.

Question 39

One-way ANOVA is performed on three independent samples with n₁ = 10, n₂ = 8 and n₃ = 9. The critical value obtained from the F-table for this test at the 5% level of significance equals:

A) 39.46.
B) 3.40.
C) 4.32.
D) 19.45.

Question 40

In the randomised block design ANOVA, the sum of squares for error equals:

A) SS(Total) - SST.
B) SS(Total) - SSB.
C) SS(Total) - SST - SSB.
D) SS(Total) - SS(A) - SS(B) - SS(AB).

Question 41

If we examine two or more independent samples to determine if their population means could be equal, we are performing one-way analysis of variance (ANOVA).

Question 42

Given the significance level 0.05, the F-value for the numbers of degrees of freedom d.f. = (9, 6) is 4.10.

Question 43

When the data are obtained through a controlled experiment in the single-factor ANOVA, we call the experimental design the completely randomised design of the analysis of variance.

Question 44

Statistics practitioners use the analysis of variance (ANOVA) technique to compare two or more populations of interval data.

Question 45

Three tennis players, one a beginner, one experienced and one a professional, have been randomly selected from the membership of a large city tennis club. Using the same ball, each person hits four serves with each of five racquet models, with the five racquet models selected randomly. Each serve is clocked with a radar gun and the result recorded. Among ANOVA models, this setup is most like the simple regression model.

Question 46

The analysis of variance (ANOVA) technique analyses the variance of the data to determine whether differences exist between the population means.

Question 47

Conceptually and mathematically, the F-test of the independent-samples single-factor ANOVA is an extension of the t-test of .

Question 48

Two samples of 10 each have been taken from the male and female workers of a large company. The data involve the wage rate of each worker. To test whether there is any difference in the average wage rate between male and female workers, a pooled-variances t-test will be considered. Another test option to consider is ANOVA. The most likely ANOVA to fit this test situation is one way ANOVA.

Question 49

The sum of squares for treatments stands for the between-treatments variation.

Question 50

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 one-way analysis of variance.

Question 51

In ANOVA, the between-treatments variation is denoted by SST, which stands for sum of squares of treatments.

Question 52

In one-way ANOVA, the total variation SS(Total) is partitioned into three sources of variation: the sum of squares for treatments (SST), the sum of squares for blocks (SSB) and the sum of squares for error (SSE).

Question 53

In ANOVA, a factor is an independent variable.

Question 54

A study is to be undertaken to examine the effects of two kinds of background music and of two assembly methods on the output of workers at a fitness shoe factory. Two workers will be randomly assigned to each of four groups, for a total of eight in the study. Each worker will be given a headphone set so that the music type can be controlled. The number of shoes completed by each worker will be recorded. Does the kind of music or the assembly method or a combination of music and method affect output? The ANOVA model most likely to fit this situation is the two-way analysis of variance.

Question 55

In employing the randomised block design, the primary interest lies in reducing the within-treatments variation in order to make easier to detect differences between the treatment means.

Question 56

In one-way ANOVA, suppose that there are five treatments with and . Then the mean square for error, MSE, equals:

A) SSE / 4.
B) SSE / 29.
C) SSE / 24.
D) SSE / 5.

Question 57

Given the significance level 0.025, the F-value for the numbers of degrees of freedom d.f. = (8, 10) is 3.85.

Question 58

Three tennis players, one a beginner, one experienced and one a professional, have been randomly selected from the membership of a large city tennis club. Using the same ball, each person hits four serves with each of five racquet models, with the five racquet models selected randomly. Each serve is clocked with a radar gun and the result recorded. Among ANOVA models, this setup is most like the single-factor analysis of variance: independent samples.

Question 59

We do not need the t-test of , since the analysis of variance can be used to test the difference between the two population means.

Question 60

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

Question 61

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

Question 62

The distribution of the test statistic for analysis of variance is the F-distribution.

Question 63

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

Question 64

A randomised block experiment having 5 treatments and 6 blocks produced the following values:
SST = 252, SSB = 1095, SSE = 198. The value of SS(Total) must be 645.

Question 65

Given the significance level 0.025, the F-value for the numbers of degrees of freedom d.f. = (4, 8) is 5.05.

Question 66

A survey is to be conducted to compare the superannuation contributions made by employees from three Victorian universities. Employees are to be randomly selected from each of the three universities and the dollar amounts of their contributions recorded. The ANOVA model most likely to fit this situation is the randomised block design.

Question 67

The sum of squares for treatments (SST) is the variation attributed to the differences between the treatment means, while the sum of squares for error (SSE) measures the variation within the samples.

Question 68

The F-test of the randomised 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.

Question 69

One-way ANOVA is applied to three independent samples having means 12, 15 and 20, respectively. If each observation in the third sample were increased by 40, the value of the F-statistic would increase by 40.

Question 70

The sum of squares for error (SSE) explains some of the total variation, while the sum of squares for treatments (SST) measures the amount of variation that is unexplained.

Question 71

The randomised block design is also called the two-way analysis of variance.

Question 72

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

Question 73

A randomised 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.

Question 74

The objective of designing a randomized block experiment is to decrease the within-treatments variation to detect differences between the treatment means.

Question 75

If we first arrange test units into similar groups before assigning treatments to them, the test design we should use is the randomised block design.

Question 76

The F-statistic in a one-way ANOVA represents the variation between the treatments divided by the variation within the treatments.

Question 77

The number of degrees of freedom for the numerator or MST is 3 and that for the denominator or MSE is 18. The total number of observations in the completely randomised design must equal 22.

Question 78

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

Question 79

When the response is not normally distributed, we can replace the randomised block ANOVA with its non-parametric counterpart; the Friedman test.

Question 80

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