Deck 15: Comparing Means of More Than Two Groups

Question 1

Which of the following best describes the alternative hypothesis is an ANOVA analysis?

A) All the population means are different from at least one other.
B) All the population means are different from one another.
C) At least one population's mean is different from all the others.
D) At least one population's mean is different from at least one other.

Answer 1

D

Question 2

Which of the following best describes when we would reject the null hypothesis when doing an ANOVA analysis?

A) If the variation of group means is less than expected based on the variation between values within each group.
B) If the variation of group means is more than expected based on the variation between values within each group.
C) If the variation of values within each group is less than expected based on the variation between values of the group means.
D) If the variation of values within each group is more than expected based on the variation between values of the group means.

Answer 2

B

Question 3

The error mean square is best described by which of the following?

A) It measures how many values there are.
B) It measures the variation between all values in all of the groups.
C) It measures the variation between the means of the groups.
D) It measures the variation between values within each of the groups.

Answer 3

D

Question 4

Which of the following is what we expect if the samples we use in an ANOVA come from populations with the same means?

A) The error mean square will be identical to the group mean square.
B) The error mean square will be larger than the group mean square.
C) The error mean square will be smaller than the group mean square
D) The error mean square will be very close to than the group mean square.

Question 5

Consider the partially completed ANOVA table shown.
?
$\begin{array}{|lcrcc|}\hline \text { Source of } & \text { Sum of } & & \text { Mean } & \\\text { variation } & \text { squares } & \text { df } & \text { squares } & \text { F-ratio } \\\text { Groups } & 384 & 6 & \ldots & \ldots \\\text { Error } & 676 & 98 & \ldots & \\\text { Total } & 1,060 & 104 & & \\\hline\end{array}$ ?

-How many values are in each group of the data set this is based on?

A) 5
B) 12
C) 14
D) 15

Question 6

Consider the partially completed ANOVA table shown.
?
$\begin{array}{|lcrcc|}\hline \text { Source of } & \text { Sum of } & & \text { Mean } & \\\text { variation } & \text { squares } & \text { df } & \text { squares } & \text { F-ratio } \\\text { Groups } & 384 & 6 & \ldots & \ldots \\\text { Error } & 676 & 98 & \ldots & \\\text { Total } & 1,060 & 104 & & \\\hline\end{array}$ ?

-What is the F-ratio for this ANOVA table?

A) 9.28
B) 10.28
C) 16.33
D) 17.33

Question 7

Consider the data table shown with values for five individuals measured in each of four groups (A, B, C, and D). We will conduct an ANOVA analysis of this.
?
$\begin{array}{c}\text { Groups }\\\begin{array}{rrrr}\text { A } & \text { B } & \text { C } & \text { D } \\6 & 7 & 9 & 11 \\8 & 10 & 11 & 14 \\10 & 11 & 12 & 15 \\12 & 12 & 13 & 16 \\14 & 15 & 15 & 19\end{array}\end{array}$ ?

-What are the means of the groups and overall mean?

A) 8, 11, 12, 15, and 12
B) 9, 11, 12, 15, and 12
C) 10, 11, 12, 15, and 11
D) 10, 11, 12, 15, and 12

Question 8

Consider the data table shown with values for five individuals measured in each of four groups (A, B, C, and D). We will conduct an ANOVA analysis of this.
?
$\begin{array}{c}\text { Groups }\\\begin{array}{rrrr}\text { A } & \text { B } & \text { C } & \text { D } \\6 & 7 & 9 & 11 \\8 & 10 & 11 & 14 \\10 & 11 & 12 & 15 \\12 & 12 & 13 & 16 \\14 & 15 & 15 & 19\end{array}\end{array}$ ?

-What is the SS_groups?

A) 45
B) 60
C) 70
D) 85

Question 9

Consider the data table shown with values for five individuals measured in each of four groups (A, B, C, and D). We will conduct an ANOVA analysis of this.
?
$\begin{array}{c}\text { Groups }\\\begin{array}{rrrr}\text { A } & \text { B } & \text { C } & \text { D } \\6 & 7 & 9 & 11 \\8 & 10 & 11 & 14 \\10 & 11 & 12 & 15 \\12 & 12 & 13 & 16 \\14 & 15 & 15 & 19\end{array}\end{array}$ ?

-What is the group mean square?

A) 17.50
B) 19.33
C) 21.50
D) 23.33

Question 10

Consider the data table shown with values for five individuals measured in each of four groups (A, B, C, and D). We will conduct an ANOVA analysis of this.
?
$\begin{array}{c}\text { Groups }\\\begin{array}{rrrr}\text { A } & \text { B } & \text { C } & \text { D } \\6 & 7 & 9 & 11 \\8 & 10 & 11 & 14 \\10 & 11 & 12 & 15 \\12 & 12 & 13 & 16 \\14 & 15 & 15 & 19\end{array}\end{array}$ ?

-What is the SS_error?

A) 108
B) 128
C) 148
D) 168

Question 11

Consider the data table shown with values for five individuals measured in each of four groups (A, B, C, and D). We will conduct an ANOVA analysis of this.
?
$\begin{array}{c}\text { Groups }\\\begin{array}{rrrr}\text { A } & \text { B } & \text { C } & \text { D } \\6 & 7 & 9 & 11 \\8 & 10 & 11 & 14 \\10 & 11 & 12 & 15 \\12 & 12 & 13 & 16 \\14 & 15 & 15 & 19\end{array}\end{array}$ ?

-What is the error mean square?

A) 8
B) 9
C) 10
D) 11

Question 12

Consider the data table shown with values for five individuals measured in each of four groups (A, B, C, and D). We will conduct an ANOVA analysis of this.
?
$\begin{array}{c}\text { Groups }\\\begin{array}{rrrr}\text { A } & \text { B } & \text { C } & \text { D } \\6 & 7 & 9 & 11 \\8 & 10 & 11 & 14 \\10 & 11 & 12 & 15 \\12 & 12 & 13 & 16 \\14 & 15 & 15 & 19\end{array}\end{array}$ ?

-What is the value of the F-ratio we will use in the analysis?

A) 1.83
B) 2.92
C) 6.34
D) 8.75

Question 13

Consider the data table shown with values for five individuals measured in each of four groups (A, B, C, and D). We will conduct an ANOVA analysis of this.
?
$\begin{array}{c}\text { Groups }\\\begin{array}{rrrr}\text { A } & \text { B } & \text { C } & \text { D } \\6 & 7 & 9 & 11 \\8 & 10 & 11 & 14 \\10 & 11 & 12 & 15 \\12 & 12 & 13 & 16 \\14 & 15 & 15 & 19\end{array}\end{array}$ ?

-What is the value of the R² value we obtain from the analysis?

A) 0.222
B) 0.333
C) 0.444
D) 0.555

Question 14

Consider the data table shown with values for five individuals measured in each of four groups (A, B, C, and D). We will conduct an ANOVA analysis of this.
?
$\begin{array}{c}\text { Groups }\\\begin{array}{rrrr}\text { A } & \text { B } & \text { C } & \text { D } \\6 & 7 & 9 & 11 \\8 & 10 & 11 & 14 \\10 & 11 & 12 & 15 \\12 & 12 & 13 & 16 \\14 & 15 & 15 & 19\end{array}\end{array}$ ?

-To test the significance of the F-ratio value, what degrees of freedom would we use?

A) df = 3 in the numerator, df = 12 in the denominator
B) df = 3 in the numerator, df = 16 in the denominator
C) df = 4 in the numerator, df = 12 in the denominator
D) df = 4 in the numerator, df = 16 in the denominator

Question 15

Use your calculated F-ratio, the degrees of freedom, and your table of critical F values to make a conclusion for this test?

A) P < 0.05, none of the means seem to differ.
B) P < 0.05, at least one of the means seems to differ.
C) P > 0.05, none of the means seem to differ.
D) P > 0.05, at least one of the means seems to differ.

Question 16

Consider the data table shown with values for six individuals measured in each of four groups (A, B, C, and D). We will conduct an ANOVA analysis of this.
?
$\begin{array}{c}\text { Groups }\\\begin{array}{rrrr}\text { A } & \text { B } & \text { C } & \text { D } \\11 & 12 & 11 & 16 \\12 & 13 & 14 & 18 \\14 & 14 & 15 & 19 \\15 & 17 & 16 & 23\end{array}\end{array}$

-What are the means of the groups and overall mean?

A) 12, 13, 13, 19, and 15
B) 13, 13, 14, 19, and 15
C) 13, 14, 14, 19, and 15
D) 14, 14, 15, 19, and 15

Question 17

Consider the data table shown with values for six individuals measured in each of four groups (A, B, C, and D). We will conduct an ANOVA analysis of this.
?
$\begin{array}{c}\text { Groups }\\\begin{array}{rrrr}\text { A } & \text { B } & \text { C } & \text { D } \\11 & 12 & 11 & 16 \\12 & 13 & 14 & 18 \\14 & 14 & 15 & 19 \\15 & 17 & 16 & 23\end{array}\end{array}$

-What is the SS_groups?

A) 72
B) 80
C) 88
D) 96

Question 18

Consider the data table shown with values for six individuals measured in each of four groups (A, B, C, and D). We will conduct an ANOVA analysis of this.
?
$\begin{array}{c}\text { Groups }\\\begin{array}{rrrr}\text { A } & \text { B } & \text { C } & \text { D } \\11 & 12 & 11 & 16 \\12 & 13 & 14 & 18 \\14 & 14 & 15 & 19 \\15 & 17 & 16 & 23\end{array}\end{array}$

-What is the group mean square?

A) 19.33
B) 24.50
C) 29.33
D) 34.50

Question 19

Consider the data table shown with values for six individuals measured in each of four groups (A, B, C, and D). We will conduct an ANOVA analysis of this.
?
$\begin{array}{c}\text { Groups }\\\begin{array}{rrrr}\text { A } & \text { B } & \text { C } & \text { D } \\11 & 12 & 11 & 16 \\12 & 13 & 14 & 18 \\14 & 14 & 15 & 19 \\15 & 17 & 16 & 23\end{array}\end{array}$

-What is the SS_error?

A) 64.00
B) 72.00
C) 80.00
D) 88.00

Question 20

Consider the data table shown with values for six individuals measured in each of four groups (A, B, C, and D). We will conduct an ANOVA analysis of this.
?
$\begin{array}{c}\text { Groups }\\\begin{array}{rrrr}\text { A } & \text { B } & \text { C } & \text { D } \\11 & 12 & 11 & 16 \\12 & 13 & 14 & 18 \\14 & 14 & 15 & 19 \\15 & 17 & 16 & 23\end{array}\end{array}$

-What is the error mean square?

A) 4.66
B) 5.00
C) 5.33
D) 5.66

Question 21

Consider the data table shown with values for six individuals measured in each of four groups (A, B, C, and D). We will conduct an ANOVA analysis of this.
?
$\begin{array}{c}\text { Groups }\\\begin{array}{rrrr}\text { A } & \text { B } & \text { C } & \text { D } \\11 & 12 & 11 & 16 \\12 & 13 & 14 & 18 \\14 & 14 & 15 & 19 \\15 & 17 & 16 & 23\end{array}\end{array}$

-What is the value of the F-ratio we will use in the analysis?

A) 5.00
B) 5.25
C) 5.50
D) 5.75

Question 22

Consider the data table shown with values for six individuals measured in each of four groups (A, B, C, and D). We will conduct an ANOVA analysis of this.
?
$\begin{array}{c}\text { Groups }\\\begin{array}{rrrr}\text { A } & \text { B } & \text { C } & \text { D } \\11 & 12 & 11 & 16 \\12 & 13 & 14 & 18 \\14 & 14 & 15 & 19 \\15 & 17 & 16 & 23\end{array}\end{array}$

-To test the significance of the F-ratio value, what degrees of freedom would we use?

A) df = 2 in the numerator, df = 12 in the denominator.
B) df = 2 in the numerator, df = 15 in the denominator.
C) df = 3 in the numerator, df = 12 in the denominator.
D) df = 3 in the numerator, df = 15 in the denominator.

Question 23

Consider the data table shown with values for six individuals measured in each of four groups (A, B, C, and D). We will conduct an ANOVA analysis of this.
?
$\begin{array}{c}\text { Groups }\\\begin{array}{rrrr}\text { A } & \text { B } & \text { C } & \text { D } \\11 & 12 & 11 & 16 \\12 & 13 & 14 & 18 \\14 & 14 & 15 & 19 \\15 & 17 & 16 & 23\end{array}\end{array}$

-Use your calculated F-ratio, the degrees of freedom, and your table of critical F values to make a conclusion for this test?

A) P < 0.05, none of the means seem to differ.
B) P < 0.05, at least one of the means seems to differ.
C) P > 0.05, none of the means seem to differ.
D) P > 0.05, at least one of the means seems to differ.

Question 24

Which of the following is not an assumption of the ANOVA?

A) The populations exhibit normal distributions.
B) The populations have equal variances.
C) The values are randomly chosen from the populations.
D) The values are randomly chosen from the samples.

Question 25

Usually for an ANOVA the variances in all the groups should be the same, but when can this assumption be relaxed?

A) The samples are large, the same size, and the variances differ by less than tenfold.
B) The samples are large, the same size, and the variances differ by less than threefold.
C) The samples are small, the same size, and the variances differ by less than tenfold.
D) The samples are small, the same size, and the variances differ by less than threefold.

Question 26

Planned comparisons after an ANOVA using the t-test procedure tend to have higher precisions than regular two-sample t-tests. Why is that?

A) The chosen pair of groups will always have the largest difference from among all the groups in the ANOVA.
B) The degrees of freedom value used to define the confidence intervals is always larger.
C) The MS_error is always less than the pooled variance.
D) The standard error is always less using all the groups rather than just two.

Question 27

Consider the figure showing the results of a Tukey-Kramer procedure on the means of individuals collected from four different environments. If the bars represent 95% confidence intervals, what letter or combination of letters would the blank have?
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A) a
B) ab
C) b
D) c

Question 28

Consider the figure showing the results of a Tukey-Kramer procedure on the means of individuals collected from four different environments. If the bars represent 95% confidence intervals, what letter or combination of letters would the blank have?
​
​

A) a
B) ab
C) b
D) c

Question 29

If groups are predefined and of interest themselves, the explanatory variable is called a(n) ____ effect, but if the groups are randomly sampled from a population of possible groups, the explanatory variable is called a(n) ____ effect.

A) fixed; random
B) focal; independent
C) independent; focal
D) random; fixed

Question 30

When studying randomly chosen groups, we don't tend to do which of the following?

A) Create an ANOVA table.
B) Compare the within-group variance to the between-group variance.
C) Estimate variance components.
D) Test a null hypothesis.

Question 31

Based on the ANOVA table shown, what is the $S_{A}^{2}$ value for this data?
?
$\begin{array}{|lcrcc|}\hline \text { Source of } & \text { Sum of } & & {\text { Mean }} \\\text { variation } & \text { squares } & \text { df } & \text { squares } & \text { F-ratio } \\\text { Groups } & 406 & 7 & 58 & 5.27 \\\text { Error } & 704 & 64 & 11 & \\\text { Total } & 1,110 & 71 & & \\\hline\end{array}$ ?

A) 4.55
B) 4.77
C) 4.99
D) 5.22

Question 32

Based on the ANOVA table shown, what is the repeatability for this data?
?
$\begin{array}{|lcrcc|}\hline \text { Source of } & \text { Sum of } & & {\text { Mean }} \\\text { variation } & \text { squares } & \text { df } & \text { squares } & \text { F-ratio } \\\text { Groups } & 406 & 7 & 58 & 5.27 \\\text { Error } & 704 & 64 & 11 & \\\text { Total } & 1,110 & 71 & & \\\hline\end{array}$ ?

A) 0.190
B) 0.322
C) 0.345
D) 0.408

Question 33

In an ANOVA analysis, the R² measures the total variation in the data.

Question 34

In an ANOVA analysis, the R² is likely to be larger when the P-value is smaller and vice-versa.

Question 35

The ANOVA technique requires more than two groups.

Question 36

The ANOVA techniques is robust to violations of its assumptions when sample sizes are large.

Question 37

If the normality assumption of the single-factor ANOVA is violated, then the Kruskal-Wallis test is a good alternative as long as the sample sizes are large and the distributions are similar.

Question 38

A planned comparison when doing an ANOVA must be specified before the analysis, never afterwards.

Question 39

A planned comparison is really just a t-test, but using a standard error based on the variance from all the groups.

Question 40

Planned comparisons are slightly more robust to violations of the assumptions of the ANOVA than the ANOVA they follow.

Question 41

When using the Tukey-Kramer method, the probability of making a Type I error is α for each pairwise comparison made.

Question 42

When using the Tukey-Kramer method, the P-value is exact when the sample sizes in all groups are the same.

Question 43

A study examining the heights of single individuals from each of a series of families based on whether the family was vegetarian or not would be analyzed using a random-effects ANOVA.

Question 44

A study examining the heights of multiple individuals in each of a series of families chosen at random from a population would be analyzed using a random-effects ANOVA.

Question 45

The repeatability measures the overall similarity of repeat measurements made on the same group.

Question 46

An experimental mistake is always equivalent to either a Type I or Type II error.

Question 47

Describe the reason we need a technique like the ANOVA. Illustrate the problem with an example using t-tests to study several groups. Make reference to the appropriate type of statistical error and use an example calculation to show the problem the ANOVA addresses.

Question 48

Consider a study involving six populations from which we draw samples of 12 values each in which the group mean square is 426 and the error mean square is 1416.
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(a) Complete the ANOVA table.
(b) Does it appear that any of the population the samples are drawn from have means that differ from one another?

Question 49

Consider a study involving seven populations from which we draw samples of 13 values each in which the group mean square is 315 and the error mean square is 2020.
​
(a) Complete the ANOVA table.
(b) Does it appear that any of the population the samples are drawn from have means that differ from one another?
​

Question 50

If the data intended for an ANOVA does not meet the assumptions, there are three options. Describe each of these options and what you would do to pursue each of them.

Question 51

Describe the difference between ANOVA analyses using planned and unplanned comparisons. Describe how the analysis is done differently.

Question 52

Describe the difference between ANOVA analyses examined fixed and random effects. Describe how the interpretation of the same numerical values is done differently.

Question 53

Using the data set shown, calculate all the values you need to complete the ANOVA table.
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Question 54

Using the data set shown, calculate all the values you need to complete an ANOVA table.
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