Deck 13: Experimental Design and Analysis of Variance

Question 1

In an ANOVA procedure, a term that means the same as the term "variable" is

A) factor.
B) treatment.
C) replication.
D) within-variance.

Answer 1

factor.

Question 2

In an analysis of variance problem involving 3 treatments and 10 observations per treatment, SSE = 399.6. The MSE for this situation is

A) 133.2.
B) 33.32.
C) 12.1.
D) 11.1.

Answer 2

12.1.

Question 3

The mean square is the sum of squares divided by

A) the total number of observations.
B) its corresponding degrees of freedom.
C) its corresponding degrees of freedom minus one.
D) the total number of replications.

Answer 3

its corresponding degrees of freedom.

Question 4

The ANOVA procedure is a statistical approach for determining whether or not the means of _____ are equal.

A) two samples
B) two or more samples
C) two populations
D) three or more populations

Question 5

An ANOVA procedure is applied to data obtained from 6 samples where each sample contains 20 observations. The critical value of F occurs with

A) 6 numerator and 20 denominator degrees of freedom.
B) 5 numerator and 20 denominator degrees of freedom.
C) 5 numerator and 114 denominator degrees of freedom.
D) 6 numerator and 114 denominator degrees of freedom.

Question 6

The number of times each experimental condition is observed in a factorial design is known as

A) partition.
B) replication.
C) blocking.
D) factor.

Question 7

In the analysis of variance procedure (ANOVA), "factor" refers to

A) the dependent variable.
B) the independent variable.
C) different levels of a treatment.
D) the critical value of F.

Question 8

The critical F value with 6 numerator and 40 denominator degrees of freedom at α = .05 is

A) 3.77.
B) 2.25.
C) 2.34.
D) 1.96.

Question 9

The independent variable of interest in an ANOVA procedure is called a

A) partition.
B) treatment.
C) response.
D) factor.

Question 10

In order to determine whether or not the means of two populations are equal,

A) a t test must be performed.
B) an analysis of variance must be performed.
C) either a t test or an analysis of variance can be performed.
D) a chi-square test can be performed.

Question 11

In an analysis of variance where the total sample size for the experiment is n_T and the number of populations is k, the mean square due to error is

A) SSE/(n_T - k).
B) SSTR/(n_T - k).
C) SSE/(k - 1).
D) SSTR/k.

Question 12

When an analysis of variance is performed on samples drawn from k populations, the mean square due to treatments (MSTR) is

A) SSTR/n_T.
B) SSTR/(n_T - 1).
C) SSTR/k.
D) SSTR/(k - 1).

Question 13

The F ratio in a completely randomized ANOVA is given by

A) MSTR/MSE.
B) MST/MSE.
C) MSE/MSTR.
D) MSE/MST.

Question 14

In ANOVA, which of the following is not affected by whether or not the population means are equal?

A)
B) between-treatments estimate of σ²
C) within-treatments estimate of σ²
D) ratio of between- and within-treatments estimate of σ²

Question 15

In factorial designs, the response produced when the treatments of one factor interact with the treatments of another in influencing the response variable is known as

A) main effect.
B) replication.
C) interaction.
D) error.

Question 16

The required condition for using an ANOVA procedure on data from several populations is that the

A) selected samples are dependent on each other.
B) response variables from samples are all uniform.
C) sampled populations have equal variances.
D) sampled populations have equal means.

Question 17

An experimental design where the experimental units are randomly assigned to the treatments is known as _____ design.

A) factor block
B) random factor
C) completely randomized
D) randomized treatment

Question 18

In an analysis of variance problem if SST = 120 and SSTR = 90, then SSE is

A) 210.
B) 30.
C) 80.
D) 120.

Question 19

An ANOVA procedure is used for data that was obtained from five sample groups each comprised of six observations. The degrees of freedom for the critical value of F are

A) 5 and 30.
B) 4 and 25.
C) 4 and 30.
D) 4 and 6.

Question 20

In the ANOVA, treatments refer to

A) experimental units.
B) different levels of a factor.
C) the dependent variables.
D) statistical applications.

Question 21

In a completely randomized design involving four treatments, the following information is provided. ​ The overall mean (the grand mean) for all treatments is

A) 40.0.
B) 40.76.
C) 39.25.
D) 41.25.

Question 22

An ANOVA procedure is used for data obtained from five populations. Five samples, each comprised of 25 observations, were taken from the five populations. The numerator and denominator (respectively) degrees of freedom for the critical value of F are

A) 5 and 25.
B) 4 and 25.
C) 4 and 124.
D) 4 and 120.

Question 23

Consider the following information. ​
SSTR = 6750
H₀: μ₁ = μ₂ = μ₃ = μ₄ =μ₅
SSE = 8000
H_a: At least one mean is different
​
The null hypothesis is to be tested at the 5% level of significance. The null hypothesis

A) should be rejected.
B) should not be rejected.
C) was designed incorrectly.
D) cannot be tested.

Question 24

Consider the following ANOVA table. ​ ​
The mean square due to treatments equals

A) 288.
B) 518.4.
C) 1200.
D) 8294.4.

Question 25

An experimental design that permits simultaneous statistical conclusions about two or more factors is a

A) randomized block design.
B) factorial design.
C) completely randomized design.
D) multiple block design.

Question 26

The process of allocating the total sum of squares and degrees of freedom to the various components is called

A) factoring.
B) blocking.
C) replicating.
D) partitioning.

Question 27

The critical F value with 8 numerator and 29 denominator degrees of freedom at α = .05 is

A) 2.28.
B) 3.20.
C) 3.33.
D) 3.64.

Question 28

Consider the following information. ​ ​
If n = 5, the mean square due to error (MSE) equals

A) 400.
B) 500.
C) 1687.5.
D) 2250.

Question 29

Consider the following ANOVA table. ​ ​
The sum of squares due to error equals

A) 13833.6.
B) 2073.6.
C) 5760.
D) 6000.

Question 30

Consider the following information. ​
SSTR = 6750
H₀: μ₁ = μ₂ = μ₃ = μ₄ = μ₅
SSE = 8000
H_a: At least one mean is different
​
The null hypothesis is to be tested at the 5% level of significance. The p-value is

A) less than .01.
B) between .01 and .025.
C) between .025 and .05.
D) greater than .10.

Question 31

In an analysis of variance, one estimate of σ² is based upon the differences between the treatment means and the

A) means of each sample.
B) overall sample mean.
C) sum of observations.
D) population means.

Question 32

An ANOVA procedure is used for data obtained from four populations. Four samples, each comprised of 25 observations, were taken from the four populations. The numerator and denominator (respectively) degrees of freedom for the critical value of F are

A) 3 and 25.
B) 4 and 25.
C) 3 and 99.
D) 3 and 96.

Question 33

Consider the following ANOVA table. ​ ​
The null hypothesis is to be tested at the 1% level of significance. The null hypothesis should

A) be rejected.
B) not be rejected.
C) be revised.
D) not be tested.

Question 34

Consider the following ANOVA table. ​ ​
The null hypothesis for this ANOVA problem is

A) μ₁ = μ₂ = μ₃ = μ₄.
B) μ₁ = μ₂ = μ₃ = μ₄ = μ₅.
C) μ₁ = μ₂ = μ₃ = μ₄ = μ₅ = μ₆.
D) μ₁ = μ₂ = ... = μ₂₀.

Question 35

Consider the following ANOVA table. ​ ​
The null hypothesis is to be tested at the 1% level of significance. The p-value is

A) greater than .10.
B) between .05 to .10.
C) between .025 to .05.
D) less than .01.

Question 36

Consider the following ANOVA table. ​ ​
The test statistic to test the null hypothesis equals

A) .432.
B) 1.8.
C) 4.17.
D) 28.8.

Question 37

Consider the following information. ​
SSTR = 6750
H₀: μ₁ = μ₂ = μ₃ = μ₄ = μ₅
SSE = 8000
H_a: At least one mean is different
​
The test statistic to test the null hypothesis equals

A) .22.
B) .84.
C) 4.22.
D) 4.50.

Question 38

Consider the following information. ​
SSTR = 6750
H₀: μ₁ =μ₂ =μ₃ =μ₄ = μ₅
SSE = 8000
H_a: At least one mean is different
​
The mean square due to treatments (MSTR) equals

A) 400.
B) 500.
C) 1687.5.
D) 1350.

Question 39

Which of the following is not a required assumption for the analysis of variance?

A) The random variable of interest for each population has a normal probability distribution.
B) The variance associated with the random variable must be the same for all populations.
C) At least 2 populations are under consideration.
D) Populations under consideration have equal means.

Question 40

In a completely randomized design involving three treatments, the following information is provided: ​ The overall mean (the grand mean) for all the treatments is

A) 7.33.
B) 7.00.
C) 7.25.
D) 8.55.

Question 41

To test whether or not there is a difference between treatments A, B, and C, a sample of 12 observations has been randomly assigned to the 3 treatments. You are given the results below. ​ ​
The null hypothesis is to be tested at the 1% level of significance. The null hypothesis

A) should be rejected.
B) should not be rejected.
C) should be revised.
D) should not be tested.

Question 42

Part of an ANOVA table is shown below. ​ The mean square due to treatments (MSTR) is

A) 36.
B) 16.
C) 64.
D) 15.

Question 43

Part of an ANOVA table is shown below. ​ The number of degrees of freedom corresponding to between-treatments is

A) 18.
B) 2.
C) 4.
D) 3.

Question 44

In a completely randomized experimental design involving five treatments, 13 observations were recorded for each of the five treatments (a total of 65 observations). Also, the design provided the following information. ​
SSTR = 300 (Sum of Squares Due to Treatments)
SST = 800 (Total Sum of Squares)
​
The number of degrees of freedom corresponding to within-treatments is

A) 60.
B) 59.
C) 5.
D) 4.

Question 45

To test whether or not there is a difference between treatments A, B, and C, a sample of 12 observations has been randomly assigned to the 3 treatments. You are given the results below. ​ ​
The mean square due to treatments (MSTR) equals

A) 1.872.
B) 5.86.
C) 34.
D) 36.

Question 46

To test whether or not there is a difference between treatments A, B, and C, a sample of 12 observations has been randomly assigned to the 3 treatments. You are given the results below. ​ ​
The test statistic to test the null hypothesis equals

A) .944.
B) 1.06.
C) 3.13.
D) 19.231.

Question 47

In a completely randomized experimental design involving five treatments, 13 observations were recorded for each of the five treatments (a total of 65 observations). Also, the design provided the following information. ​
SSTR = 300 (Sum of Squares Due to Treatments)
SST = 800 (Total Sum of Squares)
​
The mean square due to treatments (MSTR) is

A) 60.00.
B) 10.00.
C) 75.00.
D) 12.00.

Question 48

In a completely randomized experimental design involving five treatments, 13 observations were recorded for each of the five treatments (a total of 65 observations). Also, the design provided the following information. ​
SSTR = 300 (Sum of Squares Due to Treatments)
SST = 800 (Total Sum of Squares)
​
The number of degrees of freedom corresponding to between-treatments is

A) 60.
B) 59.
C) 5.
D) 4.

Question 49

Part of an ANOVA table is shown below. ​ The mean square due to treatments (MSTR) is

A) 20.
B) 60.
C) 18.
D) 15.

Question 50

In a completely randomized experimental design involving five treatments, 13 observations were recorded for each of the five treatments (a total of 65 observations). Also, the design provided the following information. ​
SSTR = 300 (Sum of Squares Due to Treatments)
SST = 800 (Total Sum of Squares)
​
The mean square due to error (MSE) is

A) 75.
B) 8.3.
C) 200.
D) 500.

Question 51

Part of an ANOVA table is shown below. ​ The test statistic is

A) 2.25.
B) 6.00.
C) 2.67.
D) 3.00.

Question 52

Part of an ANOVA table is shown below. ​ The mean square due to error (MSE) is

A) 60.
B) 15.
C) 18.
D) 20.

Question 53

In a completely randomized experimental design involving five treatments, 13 observations were recorded for each of the five treatments (a total of 65 observations). Also, the design provided the following information. ​
SSTR = 300 (Sum of Squares Due to Treatments)
SST = 800 (Total Sum of Squares)
​
If, at a 5% level of significance, we want to determine whether or not the means of the five populations are equal, the critical value of F is

A) 2.53.
B) 19.48.
C) 4.98.
D) 5.69.

Question 54

In a completely randomized experimental design involving five treatments, 13 observations were recorded for each of the five treatments (a total of 65 observations). The following information is provided. ​
SSTR = 300 (Sum of Squares Due to Treatments)
SST = 800 (Total Sum of Squares)
​
If we want to determine whether or not the means of the five populations are equal, the p-value is

A) greater than .10.
B) between .025 to .05.
C) between .01 to .025.
D) less than .01.

Question 55

To test whether or not there is a difference between treatments A, B, and C, a sample of 12 observations has been randomly assigned to the 3 treatments. You are given the results below. ​ ​
The null hypothesis for this ANOVA problem is

A) μ₁ = μ₂.
B) μ₁ = μ₂ = μ₃.
C) μ₁ = μ₂ = μ₃ = μ₄.
D) μ₁ = μ₂ = ... = μ₁₂.

Question 56

In a completely randomized experimental design involving five treatments, 13 observations were recorded for each of the five treatments (a total of 65 observations). The following information is provided. ​
SSTR = 300 (Sum of Squares Due to Treatments)
SST = 800 (Total Sum of Squares)
​
The test statistic is

A) .11.
B) 9.04.
C) 3.75.
D) 15.

Question 57

To test whether or not there is a difference between treatments A, B, and C, a sample of 12 observations has been randomly assigned to the 3 treatments. You are given the results below. ​ The mean square due to error (MSE) equals

A) 1.872.
B) 5.86.
C) 34.
D) 36.

Question 58

To test whether or not there is a difference between treatments A, B, and C, a sample of 12 observations has been randomly assigned to the 3 treatments. You are given the results below. ​ ​
The null hypothesis is to be tested at the 1% level of significance. The p-value is

A) greater than .1.
B) between .05 to .10.
C) less than .01.
D) between .01 to .025.

Question 59

Part of an ANOVA table is shown below. ​ The number of degrees of freedom corresponding to within-treatments is

A) 22.
B) 4.
C) 5.
D) 18.

Question 60

In a completely randomized experimental design involving five treatments, 13 observations were recorded for each of the five treatments (a total of 65 observations). Also, the design provided the following information. ​
SSTR = 300 (Sum of Squares Due to Treatments)
SST = 800 (Total Sum of Squares)
​
The sum of squares due to error (SSE) is

A) 1100.
B) 500.
C) 300.
D) 1600.

Question 61

The heating bills for a selected sample of houses in a city using various forms of heating are given below (values are in dollars). ​
At α = .05, test to see if there is a significant difference among the average bills of all the homes (in a city) using three forms of heating. Use both the critical and p-value approaches.

Question 62

Three universities in your state decided to administer the same comprehensive examination to the recipients of MBA degrees from the three institutions. From each institution, MBA recipients were randomly selected and were given the test. The following table shows the scores of the students tested by each university. ​
At α = .01, test to see if there is any significant difference in the average scores of all the students who took the exam from the three universities. (Note that the sample sizes are not equal.) Use both the critical and p-value approaches.

Question 63

Individuals were randomly assigned to three different production processes. The hourly units of production for the three processes are shown below. ​
Use the analysis of variance procedure with α = .05 to determine if there is a significant difference in the mean hourly units of production for the three types of production processes. Use both the critical and p-value approaches.

Question 64

At α = .01, test to determine if the means of the three populations (from which the following samples are selected) are equal. Use both the critical and p-value approaches.

Question 65

In testing for the equality of k population means, the number of treatments is​

A) ​k.
B) ​k - 1.
C) ​n_T.
D) ​n_T - k.

Question 66

In a factorial experiment, if there are x levels of factor A and y levels of factor B, there is a total of​

A) ​x + y treatment combinations​.
B) ​(x + y)/2 treatment combinations​.
C) ​2(x + y) treatment combinations.
D) ​xy treatment combinations​.

Question 67

MNM, Inc. has three stores located in three different areas. Random samples of the daily sales of the three stores (in $1000) are shown below. ​
At the 1% level of significance, test to see if there is a significant difference in the average sales of the three stores. Use both the critical and p-value approaches.

Question 68

For four populations, the population variances are assumed to be equal. Random samples from each population provide the following data. ​
Using a .05 level of significance, test to see if the means for all four populations are the same.

Question 69

A completely randomized design is useful when the experimental units are

A) homogeneous.
B) heterogeneous.
C) ​clustered.
D) stratified.

Question 70

If we are testing for the equality of three population means, we should use the​

A) ​test statistic t.
B) ​test statistic z.
C) ​test statistic F.
D) ​test statistic χ².

Question 71

Random samples of employees from three different departments of MNM Corporation showed the following yearly incomes (in $1000). ​
At α = .05, test to determine if there is a significant difference among the average incomes of the employees from the three departments. Use both the critical and p-value approaches.

Question 72

The test scores for selected samples of sociology students who took the course from three different instructors are shown below. ​
At α = .05, test to see if there is a significant difference among the averages of the three groups. Use both the critical value and p-value approaches.

Question 73

Three major automobile manufacturers have entered their cars in the Indianapolis 500 race. The speeds (in miles per hour) of the tested cars are given below. Please note the sample sizes are not equal. ​
At α = .05, test to see if there is a significant difference in the average racing speeds of the cars of the three auto manufacturers. Use both the critical and p-value approaches.

Question 74

The process of using the same or similar experimental units for all treatments is called

A) factoring.
B) blocking.
C) replicating.
D) partitioning.

Question 75

Part of an ANOVA table is shown below. ​ If we want to determine whether or not the means of the populations are equal, the p-value is

A) greater than .1.
B) between .05 to .1.
C) between .025 to .05.
D) less than .01.

Question 76

Part of an ANOVA table is shown below. ​ At a 5% level of significance, if we want to determine whether or not the means of the populations are equal, the conclusion of the test is that

A) all means are equal.
B) some means may be equal.
C) not all means are equal.
D) some means will never be equal.