Bluman, Chapter 12

Chapter 12

Analysis of Variance

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Bluman, Chapter 12

Chapter 12 Overview Introduction 12-1 One-Way Analysis of Variance 12-2 The Scheffé Test and the Tukey Test 12-3 Two-Way Analysis of Variance

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Bluman, Chapter 12

Chapter 12 Objectives1. Use the one-way ANOVA technique to

determine if there is a significant difference among three or more means.

2. Determine which means differ, using the Scheffé or Tukey test if the null hypothesis is rejected in the ANOVA.

3. Use the two-way ANOVA technique to determine if there is a significant difference in the main effects or interaction.

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Bluman, Chapter 12

Introduction The F test, used to compare two variances, can

also be used to compare three of more means.

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Bluman, Chapter 12

Introduction The F test, used to compare two variances, can

also be used to compare three of more means. This technique is called analysis of variance

or ANOVA.

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Bluman, Chapter 12

Introduction The F test, used to compare two variances, can

also be used to compare three of more means. This technique is called analysis of variance

or ANOVA. For three groups, the F test can only show

whether or not a difference exists among the three means, not where the difference lies.

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Bluman, Chapter 12

Introduction The F test, used to compare two variances, can

also be used to compare three of more means. This technique is called analysis of variance

or ANOVA. For three groups, the F test can only show

whether or not a difference exists among the three means, not where the difference lies.

Other statistical tests, Scheffé test and the Tukey test, are used to find where the difference exists.

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Bluman, Chapter 12

12-1 One-Way Analysis of Variance

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Bluman, Chapter 12

12-1 One-Way Analysis of Variance When an F test is used to test a

hypothesis concerning the means of three or more populations, the technique is called analysis of variance (commonly abbreviated as ANOVA).

Although the t test is commonly used to compare two means, it should not be used to compare three or more.

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Bluman, Chapter 12

Assumptions for the F Test

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Bluman, Chapter 12

Assumptions for the F TestThe following assumptions apply when using the F test to compare three or more means.

1. The populations from which the samples were obtained must be normally or approximately normally distributed.

2. The samples must be independent of each other.

3. The variances of the populations must be equal.

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Bluman, Chapter 12

The F Test In the F test, two different estimates of the

population variance are made.

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Bluman, Chapter 12

The F Test In the F test, two different estimates of the

population variance are made. The first estimate is called the between-

group variance, and it involves finding the variance of the means.

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Bluman, Chapter 12

The F Test In the F test, two different estimates of the

population variance are made. The first estimate is called the between-

group variance, and it involves finding the variance of the means.

The second estimate, the within-group variance, is made by computing the variance using all the data and is not affected by differences in the means.

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Bluman, Chapter 12

The F Test

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Bluman, Chapter 12

The F Test If there is no difference in the means, the

between-group variance will be approximately equal to the within-group variance, and the F test value will be close to 1—do not reject null hypothesis.

However, when the means differ significantly, the between-group variance will be much larger than the within-group variance; the F test will be significantly greater than 1—reject null hypothesis.

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Bluman, Chapter 12

Chapter 12Analysis of Variance

Section 12-1Example 12-1Page #630

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Bluman, Chapter 12

Example 12-1: Lowering Blood PressureA researcher wishes to try three different techniques to lower the blood pressure of individuals diagnosed with high blood pressure. The subjects are randomly assigned to three groups; the first group takes medication, the second group exercises, and the third group follows a special diet. After four weeks, the reduction in each person’s blood pressure is recorded. At α = 0.05, test the claim that there is no difference among the means.

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Bluman, Chapter 12

Example 12-1: Lowering Blood Pressure

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Bluman, Chapter 12

Example 12-1: Lowering Blood Pressure

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Step 1: State the hypotheses and identify the claim.

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Bluman, Chapter 12

Example 12-1: Lowering Blood Pressure

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Step 1: State the hypotheses and identify the claim.H0: µ1 = µ2 = µ3 (claim)

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Bluman, Chapter 12

Example 12-1: Lowering Blood Pressure

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Step 1: State the hypotheses and identify the claim.H0: µ1 = µ2 = µ3 (claim)H1: At least one mean is different from the others.

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Bluman, Chapter 12

Example 12-1: Lowering Blood Pressure

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Step 2: Find the critical value.

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Bluman, Chapter 12

Example 12-1: Lowering Blood Pressure

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Step 2: Find the critical value.Since k = 3, N = 15, and α = 0.05,

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Bluman, Chapter 12

Example 12-1: Lowering Blood Pressure

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Step 2: Find the critical value.Since k = 3, N = 15, and α = 0.05,d.f.N. = k – 1 = 3 – 1 = 2

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Bluman, Chapter 12

Example 12-1: Lowering Blood Pressure

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Step 2: Find the critical value.Since k = 3, N = 15, and α = 0.05,d.f.N. = k – 1 = 3 – 1 = 2d.f.D. = N – k = 15 – 3 = 12

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Bluman, Chapter 12

Example 12-1: Lowering Blood Pressure

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Step 2: Find the critical value.Since k = 3, N = 15, and α = 0.05,d.f.N. = k – 1 = 3 – 1 = 2d.f.D. = N – k = 15 – 3 = 12The critical value is 3.89, obtained from Table H.

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Bluman, Chapter 12

Example 12-1: Lowering Blood Pressure

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Step 3: Compute the test value.a. Find the mean and variance of each sample

(these were provided with the data).

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Bluman, Chapter 12

Example 12-1: Lowering Blood Pressure

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Step 3: Compute the test value.a. Find the mean and variance of each sample

(these were provided with the data).

b. Find the grand mean, the mean of all

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Bluman, Chapter 12

Example 12-1: Lowering Blood Pressure

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Step 3: Compute the test value.a. Find the mean and variance of each sample

(these were provided with the data).

b. Find the grand mean, the mean of all

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Bluman, Chapter 12

Example 12-1: Lowering Blood Pressure

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Step 3: Compute the test value.a. Find the mean and variance of each sample

(these were provided with the data).

b. Find the grand mean, the mean of all values in the samples.

c. Find the between-group variance, .

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Bluman, Chapter 12

Example 12-1: Lowering Blood Pressure

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Step 3: Compute the test value.a. Find the mean and variance of each sample

(these were provided with the data).

b. Find the grand mean, the mean of all values in the samples.

c. Find the between-group variance, .

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Bluman, Chapter 12

Example 12-1: Lowering Blood Pressure

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Bluman, Chapter 12

Example 12-1: Lowering Blood Pressure

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Bluman, Chapter 12

Example 12-1: Lowering Blood Pressure

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Step 3: Compute the test value. (continued)c. Find the between-group variance, .

d. Find the within-group variance, .

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Bluman, Chapter 12

Example 12-1: Lowering Blood Pressure

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Step 3: Compute the test value. (continued)c. Find the between-group variance, .

d. Find the within-group variance, .

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Bluman, Chapter 12

Example 12-1: Lowering Blood Pressure

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Step 3: Compute the test value. (continued)c. Find the between-group variance, .

d. Find the within-group variance, .

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Bluman, Chapter 12

Step 3: Compute the test value. (continued)e. Compute the F value.

Step 4: Make the decision.Reject the null hypothesis, since 9.17 > 3.89.

Step 5: Summarize the results.There is enough evidence to reject the claim and conclude that at least one mean is different from the others.

Example 12-1: Lowering Blood Pressure

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Bluman, Chapter 12

Step 3: Compute the test value. (continued)e. Compute the F value.

Step 4: Make the decision.Reject the null hypothesis, since 9.17 > 3.89.

Step 5: Summarize the results.There is enough evidence to reject the claim and conclude that at least one mean is different from the others.

Example 12-1: Lowering Blood Pressure

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Bluman, Chapter 12

Step 3: Compute the test value. (continued)e. Compute the F value.

Step 4: Make the decision.Reject the null hypothesis, since 9.17 > 3.89.

Step 5: Summarize the results.There is enough evidence to reject the claim and conclude that at least one mean is different from the others.

Example 12-1: Lowering Blood Pressure

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Bluman, Chapter 12

Step 3: Compute the test value. (continued)e. Compute the F value.

Example 12-1: Lowering Blood Pressure

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Bluman, Chapter 12

Step 3: Compute the test value. (continued)e. Compute the F value.

Example 12-1: Lowering Blood Pressure

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Bluman, Chapter 12

Step 3: Compute the test value. (continued)e. Compute the F value.

Example 12-1: Lowering Blood Pressure

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Bluman, Chapter 12

Step 3: Compute the test value. (continued)e. Compute the F value.

Example 12-1: Lowering Blood Pressure

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Bluman, Chapter 12

Step 3: Compute the test value. (continued)e. Compute the F value.

Step 4: Make the decision.

Example 12-1: Lowering Blood Pressure

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Bluman, Chapter 12

Step 3: Compute the test value. (continued)e. Compute the F value.

Step 4: Make the decision.Reject the null hypothesis, since 9.17 > 3.89.

Example 12-1: Lowering Blood Pressure

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Bluman, Chapter 12

Step 3: Compute the test value. (continued)e. Compute the F value.

Step 4: Make the decision.Reject the null hypothesis, since 9.17 > 3.89.

Example 12-1: Lowering Blood Pressure

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Bluman, Chapter 12

Step 3: Compute the test value. (continued)e. Compute the F value.

Step 4: Make the decision.Reject the null hypothesis, since 9.17 > 3.89.

Step 5: Summarize the results.

Example 12-1: Lowering Blood Pressure

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15Friday, January 25, 13

Bluman, Chapter 12

Step 3: Compute the test value. (continued)e. Compute the F value.

Step 4: Make the decision.Reject the null hypothesis, since 9.17 > 3.89.

Step 5: Summarize the results.There is enough evidence to reject the claim and conclude that at least one mean is different from the others.

Example 12-1: Lowering Blood Pressure

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Bluman, Chapter 12

ANOVA The between-group variance is sometimes

called the mean square, MSB.

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Bluman, Chapter 12

ANOVA The between-group variance is sometimes

called the mean square, MSB. The numerator of the formula to compute MSB

is called the sum of squares between groups, SSB.

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16Friday, January 25, 13

Bluman, Chapter 12

ANOVA The between-group variance is sometimes

called the mean square, MSB. The numerator of the formula to compute MSB

is called the sum of squares between groups, SSB.

The within-group variance is sometimes called the mean square, MSW.

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16Friday, January 25, 13

Bluman, Chapter 12

ANOVA The between-group variance is sometimes

called the mean square, MSB. The numerator of the formula to compute MSB

is called the sum of squares between groups, SSB.

The within-group variance is sometimes called the mean square, MSW.

The numerator of the formula to compute MSW is called the sum of squares within groups, SSW.

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Bluman, Chapter 12

ANOVA Summary Table

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Source Sum of Squares

d.f. MeanSquares

F

Between

Within (error)

SSB

SSW

k – 1

N – k

MSB

MSW

Total

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Bluman, Chapter 12

ANOVA Summary Table for Example 12-1

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Source Sum of Squares

d.f. MeanSquares

F

Between

Within (error)

160.13

104.80

2

12

80.07

8.73

9.17

Total 264.93 14

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Bluman, Chapter 12

Chapter 12Analysis of Variance

Section 12-1Example 12-2Page #632

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Bluman, Chapter 12

Example 12-2: Toll Road EmployeesA state employee wishes to see if there is a significant difference in the number of employees at the interchanges of three state toll roads. The data are shown. At α = 0.05, can it be concluded that there is a significant difference in the average number of employees at each interchange?

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Bluman, Chapter 12

Example 12-2: Toll Road Employees

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Bluman, Chapter 12

Example 12-2: Toll Road Employees

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Step 1: State the hypotheses and identify the claim.

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Bluman, Chapter 12

Example 12-2: Toll Road Employees

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Step 1: State the hypotheses and identify the claim.H0: µ1 = µ2 = µ3

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Bluman, Chapter 12

Example 12-2: Toll Road Employees

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Step 1: State the hypotheses and identify the claim.H0: µ1 = µ2 = µ3 H1: At least one mean is different from the others

(claim).

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Bluman, Chapter 12

Example 12-2: Toll Road Employees

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Bluman, Chapter 12

Example 12-2: Toll Road Employees

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Step 2: Find the critical value.

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Bluman, Chapter 12

Example 12-2: Toll Road Employees

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Step 2: Find the critical value.Since k = 3, N = 18, and α = 0.05,

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Bluman, Chapter 12

Example 12-2: Toll Road Employees

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Step 2: Find the critical value.Since k = 3, N = 18, and α = 0.05,d.f.N. = 2, d.f.D. = 15

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Bluman, Chapter 12

Example 12-2: Toll Road Employees

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Step 2: Find the critical value.Since k = 3, N = 18, and α = 0.05,d.f.N. = 2, d.f.D. = 15The critical value is 3.68, obtained from Table H.

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Bluman, Chapter 12

Example 12-2: Toll Road Employees

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Step 3: Compute the test value.a. Find the mean and variance of each sample

(these were provided with the data).

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Bluman, Chapter 12

Example 12-2: Toll Road Employees

23

Step 3: Compute the test value.a. Find the mean and variance of each sample

(these were provided with the data).

b. Find the grand mean, the mean of all

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Bluman, Chapter 12

Example 12-2: Toll Road Employees

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Step 3: Compute the test value.a. Find the mean and variance of each sample

(these were provided with the data).

b. Find the grand mean, the mean of all

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Bluman, Chapter 12

Example 12-2: Toll Road Employees

23

Step 3: Compute the test value.a. Find the mean and variance of each sample

(these were provided with the data).

b. Find the grand mean, the mean of all values in the samples.

c. Find the between-group variance, .

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Bluman, Chapter 12

Example 12-2: Toll Road Employees

23

Step 3: Compute the test value.a. Find the mean and variance of each sample

(these were provided with the data).

b. Find the grand mean, the mean of all values in the samples.

c. Find the between-group variance, .

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Bluman, Chapter 12

Example 12-2: Toll Road Employees

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Bluman, Chapter 12

Example 12-2: Toll Road Employees

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Bluman, Chapter 12

Example 12-2: Toll Road Employees

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Step 3: Compute the test value. (continued)c. Find the between-group variance, .

d. Find the within-group variance, .

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Bluman, Chapter 12

Example 12-2: Toll Road Employees

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Step 3: Compute the test value. (continued)c. Find the between-group variance, .

d. Find the within-group variance, .

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Bluman, Chapter 12

Example 12-2: Toll Road Employees

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Step 3: Compute the test value. (continued)c. Find the between-group variance, .

d. Find the within-group variance, .

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Bluman, Chapter 12

Step 3: Compute the test value. (continued)e. Compute the F value.

Step 4: Make the decision.Reject the null hypothesis, since 5.05 > 3.68.

Step 5: Summarize the results.There is enough evidence to support the claim that there is a difference among the means.

Example 12-2: Toll Road Employees

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Bluman, Chapter 12

Step 3: Compute the test value. (continued)e. Compute the F value.

Step 4: Make the decision.Reject the null hypothesis, since 5.05 > 3.68.

Step 5: Summarize the results.There is enough evidence to support the claim that there is a difference among the means.

Example 12-2: Toll Road Employees

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Bluman, Chapter 12

Step 3: Compute the test value. (continued)e. Compute the F value.

Step 4: Make the decision.Reject the null hypothesis, since 5.05 > 3.68.

Step 5: Summarize the results.There is enough evidence to support the claim that there is a difference among the means.

Example 12-2: Toll Road Employees

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Bluman, Chapter 12

Step 3: Compute the test value. (continued)e. Compute the F value.

Example 12-2: Toll Road Employees

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Bluman, Chapter 12

Step 3: Compute the test value. (continued)e. Compute the F value.

Example 12-2: Toll Road Employees

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Bluman, Chapter 12

Step 3: Compute the test value. (continued)e. Compute the F value.

Example 12-2: Toll Road Employees

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Bluman, Chapter 12

Step 3: Compute the test value. (continued)e. Compute the F value.

Example 12-2: Toll Road Employees

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Bluman, Chapter 12

Step 3: Compute the test value. (continued)e. Compute the F value.

Step 4: Make the decision.

Example 12-2: Toll Road Employees

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Bluman, Chapter 12

Step 3: Compute the test value. (continued)e. Compute the F value.

Step 4: Make the decision.Reject the null hypothesis, since 5.05 > 3.68.

Example 12-2: Toll Road Employees

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Bluman, Chapter 12

Step 3: Compute the test value. (continued)e. Compute the F value.

Step 4: Make the decision.Reject the null hypothesis, since 5.05 > 3.68.

Example 12-2: Toll Road Employees

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Bluman, Chapter 12

Step 3: Compute the test value. (continued)e. Compute the F value.

Step 4: Make the decision.Reject the null hypothesis, since 5.05 > 3.68.

Step 5: Summarize the results.

Example 12-2: Toll Road Employees

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Bluman, Chapter 12

Step 3: Compute the test value. (continued)e. Compute the F value.

Step 4: Make the decision.Reject the null hypothesis, since 5.05 > 3.68.

Step 5: Summarize the results.There is enough evidence to support the claim that there is a difference among the means.

Example 12-2: Toll Road Employees

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Bluman, Chapter 12

ANOVA Summary Table for Example 12-2

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Source Sum of Squares

d.f. MeanSquares

F

Between

Within (error)

459.18

682.5

2

15

229.59

45.5

5.05

Total 1141.68 17

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