Deck 13: Experimental Design and Analysis of Variance

Question 1

The F ratio in a completely randomized ANOVA is given by

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

Answer 1

MSTR/MSE.

Question 2

In an analysis of variance problem if SST = 120 and SSTR = 80, then SSE is

A) 200.
B) 40.
C) 80.
D) 120.

Answer 2

40.

Question 3

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 3

factor.

Question 4

In the ANOVA, treatments refer to

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

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

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 7

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.

Question 8

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

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

Question 9

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 10

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 11

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) 13.32.
C) 14.8.
D) 30.0.

Question 12

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

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

Question 13

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 14

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 15

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

A) $\chi$
B) between-treatments estimate of $\sigma$ ²
C) within-treatments estimate of $\sigma$ ²
D) ratio of between- and within-treatments estimate of $\sigma$ ²

Question 16

The critical F value with 6 numerator and 60 denominator degrees of freedom at α = .05 is

A) 3.74.
B) 2.25.
C) 2.37.
D) 1.96.

Question 17

The ANOVA procedure is a statistical approach for determining whether or not the means of

A) two samples are equal.
B) two or more samples are equal.
C) two populations are equal.
D) three or more populations are equal.

Question 18

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 19

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 20

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

A) 3 and 20.
B) 3 and 16.
C) 4 and 17.
D) 3 and 19.

Question 21

Consider the following ANOVA table. $\begin{array}{llll}\begin{array}{l}\text { Source } \\\text { of Variation }\end{array} & \begin{array}{l}\text { Sum } \\\text { of Squares }\end{array} & \begin{array}{l}\text { Degrees } \\\text { of Freed om }\end{array} & \begin{array}{l}\text { Mean } \\\text { Square }\end{array}&F \\\text { Between Treatments } & 2073.6 & 4 & \\\text { Between Blocks } & 6000 & 5 & 1200 \\\text { Error } & & 20 & 288 \\\text { Total } & & 29 &\end{array}$
The sum of squares due to error equals

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

Question 22

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 23

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 24

Consider the following information. $\begin{array}{ll}\mathrm{SSTR}=6750 & H 0: \mu 1=\mu 2=\mu 3=\mu 4 \\\mathrm{SSE}=8000 & H_{\mathrm{a}}: \text { At least one mean is different }\end{array}$
The mean square due to treatments (MSTR) equals

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

Question 25

In a completely randomized design involving three treatments, the following information is provided: $$\begin{array} { l l l l } & \text { Treatment 1 } & \text { Treatment 2 } & \text { Treatment 3 } \\\text { Sample Size } & 5 & 10 & 5 \\\text { Sample Mean } & 4 & 8 & 9\end{array}$$
The overall mean (the grand mean) for all the treatments is

A) 7.00.
B) 6.67.
C) 7.25.
D) 4.89.

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

An ANOVA procedure is used for data obtained from four populations.Four samples, each comprised of 30 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 30.
B) 4 and 30.
C) 3 and 119.
D) 3 and 116.

Question 28

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 29

Consider the following ANOVA table. $\begin{array}{llll}\begin{array}{l}\text { Source } \\\text { of Variation }\end{array} & \begin{array}{l}\text { Sum } \\\text { of Squares }\end{array} & \begin{array}{l}\text { Degrees } \\\text { of Freed om }\end{array} & \begin{array}{l}\text { Mean } \\\text { Square }\end{array}&F \\\text { Between Treatments } & 2073.6 & 4 & \\\text { Between Blocks } & 6000 & 5 & 1200 \\\text { Error } & & 20 & 288 \\\text { Total } & & 29 &\end{array}$
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) should be revised.
D) should not be tested.

Question 30

Consider the following ANOVA table. $\begin{array}{llll}\begin{array}{l}\text { Source } \\\text { of Variation }\end{array} & \begin{array}{l}\text { Sum } \\\text { of Squares }\end{array} & \begin{array}{l}\text { Degrees } \\\text { of Freed om }\end{array} & \begin{array}{l}\text { Mean } \\\text { Square }\end{array}&F \\\text { Between Treatments } & 2073.6 & 4 & \\\text { Between Blocks } & 6000 & 5 & 1200 \\\text { Error } & & 20 & 288 \\\text { Total } & & 29 &\end{array}$
The mean square due to treatments equals

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

Question 31

An ANOVA procedure is used for data obtained from five populations.Five samples, each comprised of 20 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 20.
B) 4 and 20.
C) 4 and 99.
D) 4 and 95.

Question 32

Consider the following information. ​
SSTR = 6750
H₀: μ₁ = μ₂ = μ₃ = μ₄
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 33

In a completely randomized design involving four treatments, the following information is provided. $$\begin{array}{ l cccc } &\text { Treatment } 1 &\text { Treatment } 2&\text { Treatment } 3& \text { Treatment } 4\\\text { Sample Size } & 50 & 18 & 15 & 17 \\\text { Sample Mean } & 32 & 38 & 42 & 48\end{array}$$
The overall mean (the grand mean) for all treatments is

A) 40.0.
B) 37.3.
C) 48.3.
D) 37.0.

Question 34

Consider the following ANOVA table. $\begin{array}{llll}\begin{array}{l}\text { Source } \\\text { of Variation }\end{array} & \begin{array}{l}\text { Sum } \\\text { of Squares }\end{array} & \begin{array}{l}\text { Degrees } \\\text { of Freed om }\end{array} & \begin{array}{l}\text { Mean } \\\text { Square }\end{array}&F \\\text { Between Treatments } & 2073.6 & 4 & \\\text { Between Blocks } & 6000 & 5 & 1200 \\\text { Error } & & 20 & 288 \\\text { Total } & & 29 &\end{array}$
The test statistic to test the null hypothesis equals

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

Question 35

Consider the following information. $\begin{array}{ll}\mathrm{SSTR}=6750 & H 0: \mu 1=\mu 2=\mu 3=\mu 4 \\\mathrm{SSE}=8000 & H_{\mathrm{a}}: \text { At least one mean is different }\end{array}$
If n = 5, the mean square due to error (MSE) equals

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

Question 36

Consider the following ANOVA table. $$\begin{array} { l l l l } \text { Source } & \text { Sum } & \text { Degrees } & \text { Mean } \\\text { of Variation } & \text { of Squares } & \text { of Freedom } & \text { Square } \\\text { Between Treatments } & 2073.6 & 4 & \\\text { Between Blocks } & 6000 & 5 & 1200 \\\text { Error } & & 20 & 288 \\\text { Total } & & 29 &\end{array}$$
The null hypothesis for this ANOVA problem is

A) $\mu$ ₁ = $\mu$ ₂ = $\mu$ ₃ = $\mu$ ₄.
B) $\mu$ ₁ = $\mu$ ₂ = $\mu$ ₃ = $\mu$ ₄ = $\mu$ ₅.
C) $\mu$ ₁ = $\mu$ ₂ = $\mu$ ₃ = $\mu$ ₄ = $\mu$ ₅ = $\mu$ ₆.
D) $\mu$ ₁ = $\mu$ ₂ = ... = $\mu$ ₂₀.

Question 37

Consider the following information. $\begin{array}{ll}\mathrm{SSTR}=6750 & H 0: \mu 1=\mu 2=\mu 3=\mu 4 \\\mathrm{SSE}=8000 & H_{\mathrm{a}}: \text { At least one mean is different }\end{array}$
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
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 39

The critical F value with 8 numerator and 29 denominator degrees of freedom at α = .01 is

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

Question 40

Consider the following ANOVA table. $\begin{array}{llll}\begin{array}{l}\text { Source } \\\text { of Variation }\end{array} & \begin{array}{l}\text { Sum } \\\text { of Squares }\end{array} & \begin{array}{l}\text { Degrees } \\\text { of Freed om }\end{array} & \begin{array}{l}\text { Mean } \\\text { Square }\end{array}&F \\\text { Between Treatments } & 2073.6 & 4 & \\\text { Between Blocks } & 6000 & 5 & 1200 \\\text { Error } & & 20 & 288 \\\text { Total } & & 29 &\end{array}$
The null hypothesis is to be tested at the 5% 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 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. $\begin{array}{lllll}\text { Treatment } &&{\text { Observations }} \\\text { A } & 20 & 30 & 25 & 33 \\\text { B } & 22 & 26 & 20 & 28 \\\mathrm{C} & 40 & 30 & 28 & 22\end{array}$
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

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. $\begin{array}{lllll}\text { Treatment } &&{\text { Observations }} \\\text { A } & 20 & 30 & 25 & 33 \\\text { B } & 22 & 26 & 20 & 28 \\\mathrm{C} & 40 & 30 & 28 & 22\end{array}$
The null hypothesis for this ANOVA problem is

A) $\mu$ ₁ = $\mu$ ₂.
B) $\mu$ ₁ = $\mu$ ₂ = $\mu$ ₃.
C) $\mu$ ₁ = $\mu$ ₂ = $\mu$ ₃ = $\mu$ ₄.
D) $\mu$ ₁ = $\mu$ ₂ = ... = $\mu$ ₁₂.

Question 43

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 = 200 (Sum of Squares Due to Treatments)
SST = 800 (Total Sum of Squares)
​
The mean square due to error (MSE) is

A) 50.
B) 10.
C) 200.
D) 600.

Question 44

Part of an ANOVA table is shown below. $$\begin{array} { l l l l l } \text { Source of } & \text { Sum of } & \text { Degrees of } & \text { Mean } \\\text { Variation } & \text { Squares } & \text { Freedom } & \text { Square } & F \\\text { Between Treatments } & 64 & &&8 \\\text { Within Treatments (Error) } & &&2 & \\\text { TOTAL } &100\end{array}$$ The number of degrees of freedom corresponding to between-treatments is

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

Question 45

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 = 200 (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 46

Part of an ANOVA table is shown below. $$\begin{array} { l l l l l } \text { Source of } & \text { Sum of } & \text { Degrees of } & \text { Mean } \\\text { Variation } & \text { Squares } & \text { Freedom } & \text { Square } & F \\\text { Between Treatments } & 180 & 3 & \\\text { Within Treatments (Error) } & & & \\\text { TOTAL } & 480 & 18 &\end{array}$$ The mean square due to error (MSE) is

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

Question 47

Part of an ANOVA table is shown below. $$\begin{array} { l l l l l } \text { Source of } & \text { Sum of } & \text { Degrees of } & \text { Mean } \\\text { Variation } & \text { Squares } & \text { Freedom } & \text { Square } & F \\\text { Between Treatments } & 64 & &&8 \\\text { Within Treatments (Error) } & &&2 & \\\text { TOTAL } &100\end{array}$$ The number of degrees of freedom corresponding to within-treatments is

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

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).The following information is provided. ​
SSTR = 200 (Sum of Squares Due to Treatments)
SST = 800 (Total Sum of Squares)
​
The test statistic is

A) .2.
B) 5.0.
C) 3.75.
D) 15.

Question 49

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 = 200 (Sum of Squares Due to Treatments)
SST = 800 (Total Sum of Squares)
​
The mean square due to treatments (MSTR) is

A) 40.00.
B) 10.00.
C) 50.00.
D) 12.00.

Question 50

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. $\begin{array}{lllll}\text { Treatment } &&{\text { Observations }} \\\text { A } & 20 & 30 & 25 & 33 \\\text { B } & 22 & 26 & 20 & 28 \\\mathrm{C} & 40 & 30 & 28 & 22\end{array}$
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 51

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. $\begin{array}{lllll}\text { Treatment } &&{\text { Observations }} \\\text { A } & 20 & 30 & 25 & 33 \\\text { B } & 22 & 26 & 20 & 28 \\\mathrm{C} & 40 & 30 & 28 & 22\end{array}$
The mean square due to treatments (MSTR) equals

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

Question 52

Part of an ANOVA table is shown below. $$\begin{array} { l l l l l } \text { Source of } & \text { Sum of } & \text { Degrees of } & \text { Mean } \\\text { Variation } & \text { Squares } & \text { Freedom } & \text { Square } & F \\\text { Between Treatments } & 180 & 3 & \\\text { Within Treatments (Error) } & & & \\\text { TOTAL } & 480 & 18 &\end{array}$$ The test statistic is

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

Question 53

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. $\begin{array}{lllll}\text { Treatment } &&{\text { Observations }} \\\text { A } & 20 & 30 & 25 & 33 \\\text { B } & 22 & 26 & 20 & 28 \\\mathrm{C} & 40 & 30 & 28 & 22\end{array}$
The test statistic to test the null hypothesis equals

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

Question 54

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. $\begin{array}{lllll}\text { Treatment } &&{\text { Observations }} \\\text { A } & 20 & 30 & 25 & 33 \\\text { B } & 22 & 26 & 20 & 28 \\\mathrm{C} & 40 & 30 & 28 & 22\end{array}$
The mean square due to error (MSE) equals

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

Question 55

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 $= 200$ (Sum of Squares Due to Treatments)
SST $= 800$ (Total Sum of Squares)
The sum of squares due to error (SSE) is

A) 1000.
B) 600.
C) 200.
D) 1600.

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).Also, the design provided the following information. ​
SSTR = 200 (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 57

Part of an ANOVA table is shown below. $$\begin{array} { l l l l l } \text { Source of } & \text { Sum of } & \text { Degrees of } & \text { Mean } \\\text { Variation } & \text { Squares } & \text { Freedom } & \text { Square } & F \\\text { Between Treatments } & 180 & 3 & \\\text { Within Treatments (Error) } & & & \\\text { TOTAL } & 480 & 18 &\end{array}$$ The mean square due to treatments (MSTR) is

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

Question 58

Part of an ANOVA table is shown below. $$\begin{array} { l l l l l } \text { Source of } & \text { Sum of } & \text { Degrees of } & \text { Mean } \\\text { Variation } & \text { Squares } & \text { Freedom } & \text { Square } & F \\\text { Between Treatments } & 64 & &&8 \\\text { Within Treatments (Error) } & &&2 & \\\text { TOTAL } &100\end{array}$$ The mean square due to treatments (MSTR) is

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

Question 59

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 = 200 (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 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 = 200 (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 61

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

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

Question 62

Part of an ANOVA table is shown below. $$\begin{array} { l l l l l } \text { Source of } & \text { Sum of } & \text { Degrees of } & \text { Mean } \\\text { Variation } & \text { Squares } & \text { Freedom } & \text { Square } & F \\\text { Between Treatments } & 64 & &&8 \\\text { Within Treatments (Error) } & &&2 & \\\text { TOTAL } &100\end{array}$$ 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.

Question 63

Part of an ANOVA table is shown below. $$\begin{array} { l l l l l } \text { Source of } & \text { Sum of } & \text { Degrees of } & \text { Mean } \\\text { Variation } & \text { Squares } & \text { Freedom } & \text { Square } & F \\\text { Between Treatments } & 64 & &&8 \\\text { Within Treatments (Error) } & &&2 & \\\text { TOTAL } &100\end{array}$$ 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 64

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 65

A completely randomized design is useful when the experimental units are

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

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

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.