Deck 15: Analysis of Variance

Question 1

In testing the null hypotheses $\mu_{1}=\mu_{2}=\mu_{3}=\mu_{4}$ , the computed $F$ value is found by calculating

A) $\frac{M S T}{M S E}$ .
B) $\frac{M S E}{M S(T r)}$ .
C) $\frac{M S(T r)}{M S T}$ .
D) $\frac{M S(T r)}{M S E}$ .

Answer 1

$\frac{M S(T r)}{M S E}$ .

Question 2

In a complete-block design

A) $S S T=S S(T r)+S S E$ .
B) $S S T=S S(T r)+S S B+S S E$ .
C) $S S T=S S B+S S E$ .
D) $S S T=S S(T r)+S S B$ .

Answer 2

$S S T=S S(T r)+S S B+S S E$ .

Question 3

The degrees of freedom for error in a two-way ANOVA equals

A) $k(n-1)$ .
B) $(k-1)(n-1)$ .
C) $(k-1) n$ .
D) $k-1$ .

Answer 3

$(k-1)(n-1)$ .

Question 4

The degrees of freedom for blocks in a complete-block design equals

A) $k(n-1)$ .
B) $(k-1)(n-1)$ .
C) $k-1$ .
D) $n-1$ .

Question 5

In a one-factor ANOVA having three treatment levels with five observations in each sample, the between samples degrees of freedom is equal to

A) 2 .
B) 4 .
C) 12 .
D) 10 .

Question 6

A blocking factor is

A) the treatment factor in a one-way ANOVA.
B) a treatment factor ANOVA in a two-way ANOVA.
C) a factor which causes variations that are included in SSE.
D) an extraneous variable which causes variations not included in SSE.

Question 7

In a one-factor ANOVA having three treatment levels with five observations in each sample, the within-samples degrees of freedom equals

A) 2 .
B) 4 .
C) 12 .
D) 10

Question 8

It is not true that the $F$ ratio for a one-way ANOVA

A) can be less than one.
B) must be at least zero.
C) measures $\frac{S S(T r)}{S S E}$ .
D) involves two kinds of degrees of freedom.

Question 9

If the null hypothesis in a one-way ANOVA is true, the between-samples variation is probably __________ the within-samples variation.

A) close to
B) significantly larger than
C) significantly smaller than
D) significantly different from (more or less)

Question 10

An experimental design in which the number of rows must equal the number of columns is provided by the

A) Latin square experiment.
B) incomplete block design.
C) complete factorial experiment.
D) two-way analysis of variance.

Question 11

If each kind of treatment appears with each kind of treatment once within the same block, the design is referred to as the __________ design.

A) Latin square
B) complete factorial
C) randomized block
D) balanced incomplete block

Question 12

The ratio which evaluates the significance of an extraneous variable in an analysis of variance is

A) $\frac{M S T}{M S E}$ .
B) $\frac{M S(T r)}{M S E}$ .
C) $\frac{M S(T r)}{M S T}$ .
D) $\frac{M S B}{M S E}$ .

Question 13

Find $F_{0.05}$ if the degrees of freedom for treatments is 4 and the degrees of freedom for error is 12.

Question 14

Find $F_{0.05}$ for 4 treatments, 3 elements per sample.

Question 15

Find $F_{0.01}$ if the degrees of freedom for treatments is 4 the degrees of freedom for error is 12.

Question 16

Find $F_{0.01}$ for 4 treatments, 3 elements per sample.

Question 17

The analysis of variance table below represents part of the calculations for testing the null hypothesis $\mu_{1}=\mu_{2}=\mu_{3}=\mu_{4}=\mu_{5}$ . Complete the table and test the null hypothesis at $\alpha=0.01$ .

Question 18

A job performance evaluator is concerned with how workers' performance ratings are affected by their skill level and their attitude. He obtained the following data:
a. Construct the ANOVA table.
b. Test the two hypotheses at $\alpha=0.05$ .

Question 19

Table 15.1


-A complete-block experiment is used to test the null hypothesis that three teaching methods produce no difference in mean achievement score. The blocking variable consists of four ability levels. Use the data in Table 15.1.
a. Construct the ANOVA table.
b. The null hypothesis of equal population means is tested at $\alpha=0.05$ . Should it be rejected?

Question 20

A complete-block design is being used to test the null hypothesis that mean responses are identical under four treatments, with three levels used for the blocking factor.
The following data are obtained: $S S(T r)=60, S S B=90, S S T=200$ .
a. Construct the ANOVA table.
b. Should the null hypothesis of identical population means be rejected at the $\alpha=0.05$ significance level.

Question 21

A two-factor ANOVA is being used to evaluate two null hypotheses. The first factor has 5 levels and the second has 4 levels.
The following data are obtained: $S S A=60, S S B=24, S S T=124$ .
a. Construct the ANOVA table
b. The two null hypotheses of equal means are tested at $\alpha=0.05$ . What are the conclusions?

Question 22

A new all-purpose cleaner is placed in four different locations in a supermarket. We would like to evaluate whether there is a significant difference in the number of cans sold with regard to location. The sample data below gives the number of cans sold in randomly selected supermarkets during a one-week period.
a. Complete the one-way ANOVA table.
b. Test whether there is a significant difference in sales of the all-purpose cleaner with regard to location. Use $\alpha=0.01$ .

Question 23

An educational researcher wants to compare four different teaching methods A, B, C, and D which she wants to try on freshmen $(\mathrm{F})$ and juniors $(\mathrm{J})$ who have three different ability levels: low (L), medium (M), and high (H). List the 24 tests she must perform so that each teaching method is used once with each combination of grade level and ability level.

Question 24

A marketing researcher wants to evaluate the success (based on resulting sales) of three different marketing strategies (I, II, III) employing three different media (radio, television, and newspapers) in three different cities: Chicago, New York, and Los Angeles.
Analyze this Latin Square using the 0.05 level of significance for each test.

Question 25

Under what circumstances would a Latin Square design be used in an experiment?

Question 26

What is the complete-block experiment?

Question 27

Under what circumstances would a complete-block design be a desirable design for an experiment?

Question 28

Analysis of variance is a method by which we can decide whether or not observed differences among more than two sample variances can be attributed to chance.

Question 29

If the null hypothesis in a one-way analysis of variance is false, then the variance among the sample means is larger than the variation within samples.

Question 30

The treatment sum of squares measures the variation within samples.

Question 31

The formula for SST in a two-factor experiment is the same as for a complete-block experiment.

Question 32

In a one-way analysis of variance, the null hypothesis is rejected if the obtained $F$ value is greater than the tabled $F$ value.

Question 33

The treatment sum of squares is symbolized by SST.

Question 34

A Latin Square experiment is an example of a complete factorial experiment.

Question 35

The method of analysis of variance is not applicable for data in which sample sizes are not all equal.

Question 36

In a two-way ANOVA, the degrees of freedom for treatment is calculated in the same way as that for a one-way ANOVA.

Question 37

In a Latin Square experiment, it is impossible for one factor to have three levels and another to have four levels.

Question 38

In an $F$ ratio for a one-way ANOVA, the numerator degrees of freedom is __________.

Question 39

In an $F$ ratio for a one-way ANOVA, the denominator degrees of freedom is __________.

Question 40

The __________ sum of squares measures the variation within samples.

Question 41

A two-way analysis of variance applied to an experiment in which we want to test both factors is called a __________.

Question 42

In a one-way ANOVA, in terms of other sums of squares, $S S E=$ __________.

Question 43

A two-way ANOVA in which only one variable is of material concern, consists of one treatment factor and one __________.

Question 44

The __________ sum of squares measures the variation between samples.

Question 45

In a complete-block design, the sum of squares for blocks measures the variation __________ samples.

Question 46

In a complete-block design, in terms of squares, $S S E=$ __________.

Question 47

In a two-factor ANOVA, in terms of squares, $S S T=$ __________.

Question 48

When each kind of treatment appears together with each other kind of treatment once within the same block, the design is referred to as __________.

Question 49

In a Latin Square experiment, there are always __________ factors.