Question 1

Find the test statistic F to test the hypothesis that the populations have the same mean. Use α = 0.05. $$\begin{array} { l l l } 
\hline \text { Brand 1 } & \text { Brand 2 } & \text { Brand 3 } \
\hline \mathrm { n } _ { 1 } = 8 & \mathrm { n } _ { 2 } = 8 & \mathrm { n } _ { 3 } = 8 \
\overline { \mathrm { x } } 1 = 3.0 & \overline { \mathrm { x } } 2 = 2.6 & \overline { \mathrm { x } } 3 = 2.6 \
\mathrm {~s} _ { 1 } = 0.50 & \mathrm {~s} _ { 2 } = 0.60 & \mathrm {~s} 3 = 0.55
\end{array}$$
A) 1.403
B) 1.021
C) 1.182
D) 0.832

Accepted Answer

The F-test statistic is calculated using the formula for ANOVA: $ F = \frac{MS_{between}}{MS_{within}} $. Here, $ MS_{between} $ and $ MS_{within} $ are calculated based on the sample means and variances. After performing the calculations, the F statistic is found to be approximately 1.403.

Question 2

Test the hypothesis that the populations have the same mean. Use α= 0.05. $$\begin{array} { l l l } 
\hline \text { Brand 1 } & \text { Brand 2 } & \text { Brand 3 } \
\hline \mathrm { n } _ { 1 } = 8 & \mathrm { n } _ { 2 } = 8 & \mathrm { n } _ { 3 } = 8 \
\overline { \mathrm { x } } 1 = 3.0 & \overline { \mathrm { x } } 2 = 2.6 & \overline { \mathrm { x } } 3 = 2.6 \
\mathrm {~s} _ { 1 } = 0.50 & \mathrm {~s} _ { 2 } = 0.60 & \mathrm {~s} 3 = 0.55 \
\hline
\end{array}$$

Accepted Answer

critical value $\mathrm { F } _ { 0.05,2,21 } \approx 3.49$; test statistic $\mathrm { F } \approx 1.403$; fail to reject $\mathrm { H } _ { 0 } ;$ The data does not provide enough evidence to indicate that the means are unequal.

Question 3

The analysis of variance is an inferential method that is used to test the equality of
A) three or more population means.
B) three or more population variances.
C) two population means.
D) two population variances.

Accepted Answer

The analysis of variance (ANOVA) is specifically designed to test the equality of three or more population means.

Question 4

A realtor wishes to compare the square footage of houses in 4 different towns, all of which are pricedapproximately the same. The data are listed below. Can the realtor conclude that the mean square footage inthe four towns are equal? Use α = 0.01. $$\begin{array} { c c c c } 
\hline \text { Town 1 } & \text { Town 2 } & \text { Town } 3 & \text { Town 4 } \
\hline 2150 & 1780 & 1530 & 2400 \
2000 & 1540 & 1670 & 2350 \
1980 & 1690 & 1580 & 2600 \
2210 & 1650 & 1600 & 2150 \
1900 & & 1500 & 2000 \
& & 1750 & 2200 \
& & & 2350 \
\hline
\end{array}$$

Accepted Answer

The answer of A realtor wishes to compare the square...

Question 5

Four different types of insecticides are used on strawberry plants. The number of strawberries on eachrandomly selected plant is given below. Test the hypothesis that the type of insecticide makes no difference inthe mean number of strawberries per plant. Use α = 0.01. $$\begin{array} { c c c c } 
\hline \text { Insecticide } 1 & \text { Insecticide 2 } & \text { Insecticide 3 } & \text { Insecticide 4 } \
\hline 6 & 5 & 6 & 3 \
7 & 8 & 3 & 5 \
6 & 5 & 3 & 3 \
5 & 5 & 4 & 4 \
7 & 5 & 2 & 4 \
6 & 6 & 3 & 5 \
\hline
\end{array}$$

Accepted Answer

The answer of Four different types of insecticides are used...

Question 6

A researcher wishes to determine whether there is a difference in the average age of middle school, highschool, and college teachers. Teachers are randomly selected. Their ages are recorded. State the requirementsthat must be satisfied in order to use the one-way ANOVA procedure.

Accepted Answer

The answer of A researcher wishes to determine whether there...

Question 7

Four different types of insecticides are used on strawberry plants. The number of strawberries on eachrandomly selected plant is given below. Find the critical value F-value to test the hypothesis that the type ofinsecticide makes no difference in the mean number of strawberries per plant. Use α = 0.01. $$\begin{array} { c c c c } 
\hline \text { Insecticide } 1 & \text { Insecticide } 2 & \text { Insecticide } 3 & \text { Insecticide 4 } \
\hline 6 & 8 & 6 & 3 \
6 & 5 & 3 & 5 \
5 & 5 & 4 & 3 \
7 & 5 & 2 & 4 \
7 & 5 & 3 & 5 \
6 & 6 & 3 & 4 \
\hline
\end{array}$$
A) 4.94
B) 4.43
C) 26.69
D) 4.18

Accepted Answer

The critical F-value is determined by the degrees of freedom for the numerator (k-1, where k is the number of groups) and the denominator (N-k, where N is the total number of observations). With 4 groups and 24 total observations, the degrees of freedom are 3 and 20. Using an F-distribution table at α = 0.01, the critical F-value is approximately 4.94.

Question 8

Four different types of insecticides are used on strawberry plants. The number of strawberries on eachrandomly selected plant is recorded. State the requirements that must be satisfied in order to use the one-wayANOVA procedure.

Accepted Answer

The answer of Four different types of insecticides are used...

Question 9

The grade point averages of students participating in sports at a local college are to be compared. The data arelisted below. Test the hypothesis that there is no difference in the mean grade point averages of the 3 groups.Use α = 0.05. $$\begin{array} { c | c | c } 
\text { Hockey } & \text { Track } & \text { Basketball } \
\hline 2.6 & 1.8 & 2.7 \
3.2 & 1.9 & 2.5 \
2.5 & 2.1 & 2.8 \
3.5 & 3.3 & 3.0 \
3.1 & & 2.5 \
2.1 & &
\end{array}$$

Accepted Answer

The answer of The grade point averages of students participating...

Question 10

How, in general, is the requirement of equal population variances to allow the use of analysis of variancemethod verified?
A) The largest sample standard deviation must be less than or equal to twice the smallest sample standard deviation.
B) All three sample standard deviations must be equal.
C) The largest sample standard deviation is equal to the sum of the other standard deviations.
D) The largest sample standard deviation is less than or equal to the variance of the smallest sample.

Accepted Answer

The requirement of equal population variances (homogeneity of variances) for the use of analysis of variance (ANOVA) is often verified using the rule of thumb that the largest sample standard deviation should be no more than twice the smallest sample standard deviation. This is a practical guideline rather than a strict statistical test, but it helps to ensure that the assumption of equal variances is not grossly violated.

Question 11

The times (in minutes) to assemble a component for 3 different cell phones are listed below. Workers arerandomly selected. Test the hypothesis that there is no difference in the mean time for each cell phone. Use α =0.01. $$\begin{array} { c | c | c } 
\text { Phone 1 } & \text { Phone 2 } & \text { Phone 3 } \
\hline 32 & 40 & 28 \
32 & 29 & 25 \
31 & 38 & 29 \
30 & 33 & 31 \
33 & 32 & \
31 & 36 & \
& 35 &
\end{array}$$

Accepted Answer

The answer of The times (in minutes) to assemble a...

Question 12

A researcher wishes to determine whether there is a difference in the average age of middle school, highschool, and college teachers. Teachers are randomly selected. Their ages are recorded below. Find the teststatistic F to test the hypothesis that there is no difference in the average age of each group. Use α = 0.01. $$\begin{array} { c c c } 
\hline \text { Middle School Teachers } & \text { High School Teachers } & \text { College Teachers } \
\hline 23 & 36 & 45 \
28 & 38 & 45 \
27 & 41 & 36 \
37 & 47 & 61 \
52 & 42 & 39 \
25 & 31 & 35 \
\hline
\end{array}$$
A) 2.517
B) 2.913
C) 3.189
D) 4.312

Accepted Answer

The correct answer is A because the test statistic F is 2.517, which is the ratio of the between-group variance to the within-group variance. The between-group variance is the variance of the means of the three groups, and the within-group variance is the average of the variances of the three groups. The test statistic F is used to test the null hypothesis that there is no difference in the average age of each group. The p-value for the test statistic F is 0.01, which is less than the alpha level of 0.05. Therefore, we reject the null hypothesis and conclude that there is a difference in the average age of each group.

Question 13

A researcher wishes to determine whether there is a difference in the average age of middle school, highschool, and college teachers. Teachers are randomly selected. Their ages are recorded below. Test thehypothesis that there is no difference in the average age of each group. Use α = 0.01. $$\begin{array} { c c c } 
\hline \text { Middle School Teachers } & \text { High School Teachers } & \text { College Teachers } \
\hline 23 & 41 & 39 \
28 & 38 & 45 \
27 & 36 & 45 \
52 & 47 & 61 \
25 & 42 & 36 \
37 & 31 & 35 \
\hline
\end{array}$$

Accepted Answer

The answer of A researcher wishes to determine whether there...

Question 14

A medical researcher wishes to try three different techniques to lower cholesterol levels of patients with highcholesterol levels. The subjects are randomly selected and assigned to one of three groups. Group 1 is givenmedication, Group 2 is given an exercise program, and Group 3 is assigned a diet program. At the end of sixweeks, each subject?s cholesterol level is recorded. Find the critical value F-value to test the hypothesis thatthere is no difference among the means. Use ? = 0.05. $$\begin{array} { c c c } 
\hline \text { Group 1 }&\text { Group 2 }&\text { Group 3 } \ 
\hline 9 & 8 & 6 \
12 & 2 & 12 \
11 & 3 & 4 \
15 & 5 & 4 \
13 & 4 & 9 \
8 & 0 & 8 \
\hline
\end{array}$$
A) 3.68
B) 19.43
C) 4.77
D) 39.43

Accepted Answer

The critical F-value for a one-way ANOVA test with 2 and 15 degrees of freedom at a significance level of 0.05 is approximately 3.68.

Question 15

A medical researcher wishes to try three different techniques to lower cholesterol levels of patients with highcholesterol levels. The subjects are randomly selected and assigned to one of three groups. Group 1 is givenmedication, Group 2 is given an exercise program, and Group 3 is assigned a diet program. At the end of sixweeks, each subjectʹs cholesterol level is recorded. State the requirements that must be satisfied in order to usethe one-way ANOVA procedure.

Accepted Answer

The answer of A medical researcher wishes to try three...

Question 16

A medical researcher wishes to try three different techniques to lower cholesterol levels of patients with highcholesterol levels. The subjects are randomly selected and assigned to one of three groups. Group 1 is givenmedication, Group 2 is given an exercise program, and Group 3 is assigned a diet program. At the end of sixweeks, each subjectʹs cholesterol level is recorded. Test the hypothesis that there is no difference among themeans. Use α = 0.05. $$\begin{array} { c c c } 
\hline \text { Group 1 }&\text { Group 2 }&\text { Group 3 } \  
\hline 9 & 8 & 6 \
12 & 5 & 12 \
11 & 2 & 4 \
15 & 3 & 8 \
13 & 4 & 9 \
8 & 0 & 4 \
\hline
\end{array}$$

Accepted Answer

The answer of A medical researcher wishes to try three...

Question 17

A researcher wishes to determine whether there is a difference in the average age of middle school, highschool, and college teachers. Teachers are randomly selected. Their ages are recorded below. Find the criticalvalue F-value to test the hypothesis that there is no difference in the average age of each group. Useα = 0.01. $$\begin{array} { c c c } 
\hline \text { Middle School Teachers } & \text { High School Teachers } & \text { College Teachers } \
\hline 23 & 38 & 39 \
28 & 36 & 45 \
27 & 41 & 36 \
52 & 47 & 61 \
25 & 42 & 45 \
37 & 31 & 35
\end{array}$$
A) 6.36
B) 5.09
C) 9.43
D) 5.42

Accepted Answer

The critical F-value at α = 0.01 for (2, 15) degrees of freedom (between groups and within groups) is approximately 6.36.

Question 18

Find the critical F0-value to test the hypothesis that the populations have the same mean. Use α= 0.05. $$\begin{array} { l c c } 
\hline \text { Brand 1 } & \text { Brand 2 } & \text { Brand 3 } \
\hline \mathrm { n } _ { 1 } = 8 & \mathrm { n } _ { 2 } = 8 & \mathrm { n } _ { 3 } = 8 \
\overline { \mathrm { x } } 1 = 3.0 & \overline { \mathrm { x } } 2 = 2.6 & \overline { \mathrm { x } } 3 = 2.6 \
\mathrm {~s} _ { 1 } = 0.50 & \mathrm {~s} _ { 2 } = 0.60 & \mathrm {~s} _ { 3 } = 0.55 \
\hline
\end{array}$$
A) 3.49
B) 3.210
C) 2.819
D) 1.892

Accepted Answer

The critical F0-value is determined by the degrees of freedom for the numerator (k-1) and the denominator (N-k), where k is the number of groups and N is the total number of observations. Here, k=3 and N=24, so the degrees of freedom are 2 and 21. Using an F-distribution table at α=0.05, the critical value is 3.49.

Question 19

A medical researcher wishes to try three different techniques to lower cholesterol levels of patients with highcholesterol levels. The subjects are randomly selected and assigned to one of three groups. Group 1 is givenmedication, Group 2 is given an exercise program, and Group 3 is assigned a diet program. At the end of sixweeks, each subject?s cholesterol level is recorded. Find the test statistic F to test the hypothesis that there is nodifference among the means. Use ? = 0.05. $$\begin{array} { c c c } 
\hline \text { Group 1 }&\text { Group 2 }&\text { Group 3 } \  
\hline 13 & 8 & 4 \
12 & 2 & 12 \
9 & 3 & 4 \
15 & 5 & 6 \
11 & 4 & 9 \
8 & 0 & 8 \
\hline
\end{array}$$
A) 11.095
B) 9.812
C) 8.369
D) 12.162

Accepted Answer

The F-test statistic is calculated using ANOVA. After computing the sum of squares between groups and within groups, and dividing by their respective degrees of freedom, the F-value is found to be approximately 11.095.

Question 20

Four different types of insecticides are used on strawberry plants. The number of strawberries on eachrandomly selected plant is given below. Find the test statistic F to test the hypothesis that the type of insecticidemakes no difference in the mean number of strawberries per plant. Use α = 0.01. $$\begin{array} { c c c c } 
\hline \text { Insecticide } 1 & \text { Insecticide } 2 & \text { Insecticide } 3 & \text { Insecticide } 4 \
\hline 6 & 5 & 6 & 3 \
5 & 8 & 3 & 5 \
6 & 5 & 4 & 3 \
7 & 5 & 3 & 4 \
7 & 5 & 2 & 5 \
6 & 6 & 3 & 4 \
\hline
\end{array}$$
A) 8.357
B) 8.123
C) 7.123
D) 6.912

Accepted Answer

The test statistic F is calculated using ANOVA (Analysis of Variance) to compare the means of the four groups. Given the data, the correct calculation results in an F value of 8.357, indicating the variance between the groups relative to the variance within the groups.

Quiz 13: Comparing Three or More Means

The analysis of variance is an inferential method that is used to test the equality of

A researcher wishes to determine whether there is a difference in the average age of middle school, highschool, and college teachers. Teachers are randomly selected. Their ages are recorded. State the requirementsthat must be satisfied in order to use the one-way ANOVA procedure.

Four different types of insecticides are used on strawberry plants. The number of strawberries on eachrandomly selected plant is recorded. State the requirements that must be satisfied in order to use the one-wayANOVA procedure.

How, in general, is the requirement of equal population variances to allow the use of analysis of variancemethod verified?