Question 1

A partial ANOVA table in a randomised block design is shown below. $$\begin{array} { | l | c | c | c | c | } 
\hline \text { Source of Variation } & \text { SS } & d f & M S & F \
\hline \text { Treatments } & * & 3 & * & * \
\hline \text { Blocks } & 1256 & 2 & * & * \
\hline \text { Error } & * & * & 67.67 & \
\hline \text { Total } & 2922 & 11 & & \
\hline
\end{array}$$ Can we infer at the 5% significance level that the block means differ?

Accepted Answer

H₀: $\mu$₁= $\mu$₂= $\mu$₃ H₁:At least two means differ. Rejection region: F > F_0.05,2,6 = 5.14. Test statistic: F = 9.28. Conclusion: Reject the null hypothesis. Yes.

Question 2

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).

Accepted Answer

A) True 
 B)False  False

Question 3

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:

Accepted Answer

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

Question 4

The strength of a weld depends to some extent on the metal alloy used in the welding process. A scientist working in the research laboratory of a major automobile manufacturer has developed three new alloys. In order to test their strengths, each alloy is used in several welds. The strengths of the welds are then measured, with the results shown below. Can the scientist conclude at the 5% significance level that differences exist among the strengths of the welds with the different alloys? $\quad $$\quad $$\quad $$ \text { Strength of welds }$
$\begin{array} { | c | c | c | }  
\hline \text { Alloy 1 } & \text { Alloy 2 } & \text { Alloy 3 } \
\hline 15 & 17 & 25 \
\hline 23 & 21 & 27 \
\hline 16 & 19 & 24 \
\hline 29 & 25 & 31 \
\hline & 28 & 23 \
\hline & 19 & \
\hline
\end{array}$

Accepted Answer

The answer of The strength of a weld depends to...

Question 5

The following statistics were calculated based on samples drawn from four normal populations. $$\begin{array} { | c | c | c | c | c | } 
\hline \text { Treatment } & & & & \
\hline \text { Statistic } & 1 & 2 & 3 & 4 \
\hline n _ { j } & 4 & 7 & 5 & 5 \
\hline \sum x _ { j } & 52 & 69 & 71 & 61 \
\hline \sum x _ { j } ^ { 2 } & 753 & 798 & 1248 & 912 \
\hline
\end{array}$$ Test at the 5% level of significance to determine whether differences exist among the population means.

Accepted Answer

The answer of The following statistics were calculated based on...

Question 6

In a completely randomised design, 15 experimental units were assigned to each of four treatments. Fill in the blanks (identified by asterisks) in the partial ANOVA table shown below. $$\begin{array} { | l | c | c | c | c | } 
\hline \text { Source of Variation } & \text { SS } & d f & M S & F \
\hline \text { Treatments } & * & * & 240 & * \
\hline \text { Error } & * & * & * & \
\hline \text { Total } & 2512 & * & & \
\hline
\end{array}$$

Accepted Answer

The answer of In a completely randomised design, 15 experimental...

Question 7

A partial ANOVA table in a randomised block design is shown below. $$\begin{array} { | l | c | c | c | c | } 
\hline \text { Source of Variation } & \text { SS } & d f & M S & F \
\hline \text { Treatments } & * & 3 & * & * \
\hline \text { Blocks } & 1256 & 2 & * & * \
\hline \text { Error } & * & * & 67.67 & \
\hline \text { Total } & 2922 & 11 & & \
\hline
\end{array}$$ Fill in the missing values (identified by asterisks) in the above ANOVA table.

Accepted Answer

The answer of A partial ANOVA table in a randomised...

Question 8

The following statistics were calculated based on samples drawn from three normal populations: $$\begin{array} { | c | c  c  c | } 
\hline & { \text { Treatment } } \
\hline \text { Statistic } & 1 & 2 & 3 \
\hline n & 10 & 10 & 10 \
\hline \bar { x } & 95 & 86 & 92 \
\hline s & 10 & 12 & 15 \
\hline
\end{array}$$ Set up the ANOVA table and test at the 5% level of significance to determine whether differences exist among the population means.

Accepted Answer

The answer of The following statistics were calculated based on...

Question 9

One-way ANOVA is performed on independent samples taken from three normally distributed populations with equal variances. The following summary statistics are calculated: $n _ { 1 } =$ 6, $\bar { x } _ { 1 } =$ 50, $s _ { 1 } =$ 5.2. $n _ { 2 } =$ 8, $\bar { x } _ { 2 } =$ 55, $S _ { 2 } =$ 4.9 . $n _ { 3 } =$ 6, $\bar { x } _ { 3 } =$ 51, $s _ { 3 } =$ 5.4. The grand mean equals:

Accepted Answer

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

Question 10

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

Accepted Answer

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 11

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

Accepted Answer

A) True 
 B)False

Question 12

One-way ANOVA is performed on independent samples taken from three normally distributed populations with equal variances. The following summary statistics were calculated: $n _ { 1 } =$ 7, $\bar { x } _ { 1 } =$ 65, $s _ { 1 } =$ 4.2. $n _ { 2 } =$ 8, $\bar { x } _ { 2 } =$ 59, $S _ { 2 } =$ 4.9. $n _ { 3 } =$ 9, $\bar { x } _ { 3 } =$ 63, $s _ { 3 } =$ 4.6. The value of the test statistics, F, equals:

Accepted Answer

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

Question 13

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

Accepted Answer

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 14

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

Accepted Answer

A) True 
 B)False

Question 15

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

Accepted Answer

A) True 
 B)False

Question 16

A recent college graduate is in the process of deciding which one of three US graduate schools he should apply to. He decides to judge the quality of the schools on the basis of the Graduate Management Admission Test (GMAT) scores of those who are accepted into the school. A random sample of six students in each school produced the following GMAT scores. $\quad $$\quad $$\quad $$\quad $$\quad $$\text { GMAT Scores }$ 
$\begin{array}{|c|c|c|}
\hline \text { School 1 } & \text { School 2 } & \text { School 3 } \
\hline 650 & 510 & 590 \
\hline 620 & 550 & 510 \
\hline 630 & 700 & 520 \
\hline 580 & 630 & 500 \
\hline 710 & 600 & 490 \
\hline 690 & 650 & 530 \
\hline
\end{array}$ Use Tukey's method with $\alpha$ =0.05 to determine which population means differ.

Accepted Answer

The answer of A recent college graduate is in the...

Question 17

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:

Accepted Answer

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

Question 18

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.

Accepted Answer

A) True 
 B)False

Question 19

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.

Accepted Answer

A) True 
 B)False

Question 20

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:

Accepted Answer

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.

Exam 16: Analysis of Variance

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).

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:

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

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

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

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

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

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:

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.

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.

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: