Deck 12: Analysis of Variance

Question 1

Use the data in the given table and the corresponding Minitab display to test the hypothesis. The following table shows the mileage for four different cars and three different brands of
gas. Assuming no effect from the interaction between car and brand of gas, test the claim that
the three brands of gas provide the same mean gas mileage. Use a 0.05 significance level.

Answer 1

H₀: The brands of gas provide the same mean mileage. H₁: The brands of gas do not provide the same mean mileage. The P -value is 0.200726 , which is greater than 0.05 . We fail to reject the null hypothesis; it appears that the brands of gas provide the same mean mileage.

Question 2

Test the claim that the samples come from populations with the same mean. Assume that the populations are normally distributed with the same variance. At the 1% significance level, does it appear that a difference exists in the true mean weight
loss produced by the three exercise programs?

Answer 2

Test statistic: F = 1.491. Critical value: F = 6.9266. P-value: p = 0.264.
Fail to reject the claim of equal means. The data do not provide sufficient evidence to conclude that there is a
difference in the true mean weight loss produced by the three exercise programs.

Question 3

Explain the procedure for two-way analysis of variance, and how it varies depending on whether there is an interaction between the two factors or not.

Answer 3

First we test whether there is an effect due to interaction between the two factors. If there is an interaction between
the two factors, we stop. If there is no effect due to interaction between the two factors, we test for effect from the
row factor and we test for effect from the column factor.

Question 4

At the same time each day, a researcher records the temperature in each of three greenhouses. The table shows the temperatures in degrees Fahrenheit recorded for one week.

i) Use a 0.05 significance level to test the claim that the average temperature is the same in each greenhouse.
ii) How are the analysis of variance results affected if the same constant is added to every one of the original sample values?

Question 5

The following data contains task completion times, in minutes, categorized according to the gender of the machine operator and the machine used.

The ANOVA results lead us to conclude that the completion times are not affected by an interaction between machine and gender, and the times are not affected by gender, but they are affected by the machine. Change the table entries so that there is an effect from the interaction between machine and gender.

Question 6

List the assumptions for testing hypotheses that three or more means are equivalent.

Question 7

The data below represent the weight losses for people on three different exercise programs
$\begin{array} { c c c } \text { Exercise A } & \text { Exercise B } & \text { Exercise C } \\\hline 2.5 & 5.8 & 4.3 \\8.8 & 4.9 & 6.2 \\7.3 & 1.1 & 5.8 \\9.8 & 7.8 & 8.1 \\5.1 & 1.2 & 7.9\end{array}$
If we want to test the claim that the three size categories have the same means, why don't we use three separate hypothesis tests for $\mu$ ₁= $\mu$ $\mu$ ₂= $\mu$ ₃ , and $\mu$ ₁ = $\mu$ ₃

Question 8

Draw an example of an F distribution and list the characteristics of the F distribution

Question 9

Suppose you are to test for equality of four different population means, with Write the hypotheses for the paired tests. Use methods of probability to explain why the process of ANOVA has a higher degree of confidence than testing each of the pairs separately.

Question 10

Test the claim that the samples come from populations with the same mean. Assume that the populations are normally distributed with the same variance. At the 0.025 significance level,
test the claim that the three brands have the same mean if the following sample results have
been obtained.
$\begin{array} { c c c } \text { Brand A } & \text { Brand B } & \text { Brand C } \\\hline 32 & 27 & 22 \\34 & 24 & 25 \\37 & 33 & 32 \\33 & 30 & 22 \\36 & & 21 \\39 & &\end{array}$

Question 11

Use the Minitab display to test the indicated claim. A manager records the production output of three employees who each work on three different machines for three different days. The
sample results are given below and the Minitab results follow. Using a 0.05 significance level, test the claim that the interaction between employee and
machine has no effect on the number of items produced.

Question 12

Test the claim that the samples come from populations with the same mean. Assume that the populations are normally distributed with the same variance. At the 0.025 significance level, test the claim that the four brands have the same mean if the following sample results havebeen obtained. $\begin{array} { c c c c } \text { Brand A } & \text { Brand B } & \text { Brand C } & \text { Brand D } \\\hline 17 & 18 & 21 & 22 \\20 & 18 & 24 & 25 \\21 & 23 & 25 & 27 \\22 & 25 & 26 & 29 \\21 & 26 & 29 & 35 \\& & 29 & 36 \\& & & 37\end{array}$

Question 13

Use the data in the given table and the corresponding Minitab display to test the hypothesis. The following table shows the mileage for four different cars and three different brands of
gas. Assuming no effect from the interaction between car and brand of gas, test the claim
that the four cars have the same mean mileage. Use a 0.05 significance level.

Question 14

Fill in the missing entries in the following partially completed one-way ANOVA table

Question 15

The following data contains task completion times, in minutes, categorized according to the gender of the machine operator and the machine used. Assume that two-way ANOVA is used to analyze the data. How are the ANOVA results
affected if 5 minutes is added to each completion time?

Question 16

Why do researchers concentrate on explaining an interaction in a two-way ANOVA rather than the effects of each factor separately?

Question 17

The following data contains task completion times, in minutes, categorized according to the gender of the machine operator and the machine used.

Assume that two-way ANOVA is used to analyze the data. How are the ANOVA results affected if the times are converted to hours?

Question 18

The following results are from a statistics software package in which all of the F values and P-
values are given. Is there a significant effect from the interaction? Should you test to see if
there is a significant effect due to either A or B? If the answer is yes, is there a significant
effect due to either A or B?

Question 19

Use the data in the given table and the corresponding Minitab display to test the hypothesis. The following table entries are test scores for males and females at different times of day.
Assuming no effect from the interaction between gender and test time, test the claim that time
of day does not affect test scores. Use a 0.05 significance level.

Question 20

The following data contains task completion times, in minutes, categorized according to the gender of the machine operator and the machine used.

The ANOVA results lead us to conclude that the completion times are not affected by an interaction between machine and gender, and the times are not affected by gender, but they are affected by the machine. Change the table entries so that there is no effect from the interaction between machine and gender, but there is an effect from the gender of the operator.

Question 21

Suppose you are to test for equality of four different population means, with Write the hypotheses for the paired tests. Use methods of probability to explain why the process of ANOVA has a higher degree of confidence than testing each of the pairs separately.

Question 22

Use the Minitab display to test the indicated claim. A manager records the production output of three employees who each work on three different machines for three different days. The
sample results are given below and the Minitab results follow.

Question 23

The data below represent the weight losses for people on three diets If we want to test the claim that the three size categories have the same means, why don't we
use three separate hypothesis tests for

Question 24

Use the Minitab display to test the indicated claim. A manager records the production output of three employees who each work on three different machines for three different days. The
sample results are given below and the Minitab results follow. Assume that the number of items produced is not affected by an interaction between
employee and machine. Using a 0.05 significance level, test the claim that the choice of
employee has no effect on the number of items produced.

Question 25

The following data shows the yield, in bushels per acre, categorized according to three varieties of corn and three different soil conditions. Assume that yields are not affected by an
interaction between variety and soil conditions, and test the null hypothesis that variety has no
effect on yield. Use a 0.05 significance level.

Question 26

Test the claim that the samples come from populations with the same mean. Assume that the populations are normally distributed with the same variance. The data below represent the
weight losses for people on three different exercise programs. At the 1% significance level, does it appear that a difference exists in the true mean weight
loss produced by the three exercise programs?

Question 27

Fill in the missing entries in the following partially completed one-way ANOVA table

Question 28

The test statistics for one-way ANOVA is
Describe variance within samples and variance between samples. What relationship does variance within samples and variance between samples would result in the conclusion that the value of F is significant?

Question 29

Explain the procedure for two-way analysis of variance varies depending on whether there is an interaction between the two factors or not.

Question 30

Test the claim that the samples come from populations with the same mean. Assume that the populations are normally distributed with the same variance. Given the sample data below,
test the claim that the populations have the same mean. Use a significance level of 0.05.

Question 31

Why do researchers concentrate on explaining an interaction in a two -way ANOVA rather than the effects of each factor separately?

Question 32

The following data shows annual income, in thousands of dollars, categorized according to the two factors of gender and level of education. Assume that incomes are not affected by an
interaction between gender and level of education, and test the null hypothesis that level of
education has no effect on income. Use a 0.05 significance level.

Question 33

Use the Minitab display to test the indicated claim. A manager records the production output of three employees who each work on three different machines for three different days. The
sample results are given below and the Minitab results follow. Assume that the number of items produced is not affected by an interaction between
employee and machine. Using a 0.05 significance level, test the claim that the machine has no
effect on the number of items produced.

Question 34

Use the data in the given table and the corresponding Minitab display to test the hypothesis. The following Minitab display results from a study in which three different teachers taught calculus
classes of five different sizes. The class average was recorded for each class. Assuming no
effect from the interaction between teacher and class size, test the claim that the teacher has no
effect on the class average. Use a 0.05 significance level.

Question 35

Assume that the number of items produced is not affected by an interaction between employee and machine. Using a 0.05 significance level, test the claim that the machine has no effect on
the number of items produced. Using a 0.05 significance level, test the claim that the interaction between employee and
machine has no effect on the number of items produced.

Question 36

Use the data in the given table and the corresponding Minitab display to test the hypothesis. The following table shows the mileage for four different cars and three different brands of
gas. Assuming no effect from the interaction between car and brand of gas, test the claim that
the four cars have the same mean mileage. Use a 0.05 significance level.

Question 37

Why is it unnecessary to conduct multiple comparison tests after a nonsignificant F test statistic results?

Question 38

List the assumptions for testing hypotheses that three or more means are equivalent.

Question 39

Test the claim that the samples come from populations with the same mean. Assume that the populations are normally distributed with the same variance. At the 0.025 significance level,test the claim that the four brands have the same mean if the following sample results have been obtained.
$\begin{array} { c c c c } \text { Brand A } & \text { Brand B } & \text { Brand C } & \text { Brand D } \\\hline 17 & 18 & 21 & 22 \\20 & 18 & 24 & 25 \\21 & 23 & 25 & 27 \\22 & 25 & 26 & 29 \\21 & 26 & 29 & 35 \\& & 29 & 36 \\& & & 37\end{array}$

Question 40

Test the claim that the samples come from populations with the same mean. Assume that the populations are normally distributed with the same variance. At the 0.025 significance level, test the claim that the three brands have the same mean if the following sample results have been obtained.
$\begin{array} { c c c } \text { Brand A } & \text { Brand B } & \text { Brand C } \\\hline 32 & 27 & 22 \\34 & 24 & 25 \\37 & 33 & 32 \\33 & 30 & 22 \\36 & & 21 \\39 & &\end{array}$

Question 41

Given below are the analysis of variance results from a Minitab display. to use a 0.05 significance level in testing the null hypothesis that the different samples come from populations with the same mean. Identify the P-value.

A) 0.264
B) 10.00
C) 6.25
D) 1.6

Question 42

There is a(n)____________________ between two factors if the effect of one of the factors changes for different categories of the other factor.

A) variation
B) interaction
C) category effect
D) transformation

Question 43

Given below are the analysis of variance results from a Minitab display. to use a 0.05 significance level in testing the null hypothesis that the different samples come from populations with the same mean. What can you conclude about the equality of the
Population means?

A) Reject the null hypothesis since the P-value is greater than the significance level.
B) Accept the null hypothesis since the P-value is less than the significance level.
C) Accept the null hypothesis since the P-value is greater than the significance level.
D) Reject the null hypothesis since the P-value is less than the significance level.

Question 44

The data below represent the weight losses for people on three different exercise programs $$\begin{array} { c c c } \text { Exercise A } & \text { Exercise B } & \text { Exercise C } \\\hline 2.5 & 5.8 & 4.3 \\8.8 & 4.9 & 6.2 \\7.3 & 1.1 & 5.8 \\9.8 & 7.8 & 8.1 \\5.1 & 1.2 & 7.9\end{array}$$
If we want to test the claim that the three size categories have the same means, why don't we use three separate hypothesis tests for $\mu$ ₁= $\mu$ ₂, $\mu$ = $\mu$ ₃ , and $\mu$ ₁= $\mu$ ₃ ?

A) Actually, we do want to use three separate hypothesis tests.
B) The risk of type I error increases and becomes too high.
C) The risk of type II error increases and becomes too high.
D) A hypothesis test for comparing two means does not exist.

Question 45

ANOVA requires usage of the _______ distribution

A) Normal
B) t
C) Chi-square
D) F

Question 46

Use the Minitab display to test the indicated claim. A manager records the production output of three employees who each work on three different machines for three different days. The sample results are given below and the Minitab results follow.
Assume that the number of items produced is not affected by an interaction between employee and machine. Using a 0.05 significance level, test the claim that the choice of employee has no effect on the number of items produced. What is the value of the test statistic, F ?

A) F=0.1898
B) F=0.2687
C) F=5.2692
D) F=53.7172

Question 47

Identify the value of the test statistic.

A) 30
B) 1.6
C) 0.264
D) 10.00

Question 48

Fill in the missing entries in the following partially completed one -way ANOVA table.

A)
B)
C)
D)

Question 49

At the same time each day, a researcher records the temperature in each of three greenhouses. The table shows the temperatures in degrees Fahrenheit recorded for one week.
What type of test would you use to test the claim that the average temperature is the same in each greenhouse at a 0.05 significance level?

A) One-way ANOVA
B) Two-way ANOVA
C) Goodness-of-Fit
D) Test of Independence

Question 50

Given below are the analysis of variance results from a Minitab display. Assume that you want to use a 0.05 significance level in testing the null hypothesis that the different samples come from populations with the same mean. Find the critical value.

A) 4.07
B) 8.85
C) 7.59
D) 1.6

Question 51

Use the Minitab display to test the indicated claim. A manager records the production output of three employees who each work on three different machines for three different days. The sample results are given below and the Minitab results follow.
Using a 0.05 significance level, test the claim that the interaction between employee and machine has no effect on the number of items produced. State your conclusion about the null hypothesis and the claim.

A) Reject the null hypothesis. There does not appear to be an interaction effect.
B) Reject the null hypothesis. There appears to be an interaction effect.
C) Fail to reject the null hypothesis. There does not appear to be an interaction effect.
D) Fail to reject the null hypothesis. There appears to be an interaction effect.

Question 52

Given below are the analysis of variance results from a Minitab display. to use a 0.05 significance level in testing the null hypothesis that the different samples come from populations with the same mean. Identify the p-value.

A) 10.00
B) 6.25
C) 1.6
D) 0.264

Question 53

Select a property of the F distribution

A) It is symmetric.
B) The exact shape of the F distribution depends on two different degrees of freedom.
C) Values of the F distribution can be negative.
D) The total area under the curve is equal to 2.

Question 54

Given below are the analysis of variance results from a Minitab display. to use a 0.05 significance level in testing the null hypothesis that the different samples come from populations with the same mean. Identify the value of the test statistic.

A) 5.17
B) 4.500
C) 13.500
D) 0.011

Question 55

The test statistic for one-way ANOVA is equal to _________________.

A)
B)
C)
D)

Question 56

Select an appropriate null hypothesis for a one way analysis of variance test.

A) H0: $\mu$ ₁ = $\mu$ ₂ = $\mu$ ₃ = $\mu$ ₄
B) H0: $\mu$ ₁ ≠ $\mu$ ₂ ≠ $\mu$ ₃ ≠ $\mu$ ₄
C) H0: p₁ ≠ p₂ ≠ p₃ ≠ p₄
D) H0 : σ1 = σ2 = σ3 = σ4

Question 57

Given below are the analysis of variance results from a Minitab display. to use a 0.05 significance level in testing the null hypothesis that the different samples come from populations with the same mean. Identify the P-value.

A) 4.500
B) 5.17
C) 0.870
D) 0.011

Question 58

At the same time each day, a researcher records the temperature in each of three classrooms. The table shows the temperatures in degrees Fahrenheit recorded for one week.
How are the analysis of variance results affected if the same constant is added to every one of the original sample values?

A) The test statistic increases by the value of the constant.
B) The test statistic increases by a factor of the constant.
C) The test statistic is divided by the value of the constant.
D) The results are not affected.

Question 59

Which of the following is NOT a requirement for one-way ANOVA?

A) The populations have distributions that are approximately normal.
B) The samples are simple random samples.
C) The samples are independent of each other.
D) The data is categorical data.