1. 10 Chapter Chi-Square Tests and the F-Distribution Chapter 10
Larson/Farber 4th ed

2. Chapter Outline 10.1 Goodness of Fit 10.2 Independence
10.3 Comparing Two Variances
10.4 Analysis of Variance .

3. Section 10.4
Analysis of Variance .

4. Section 10.4 Objectives
How to use one-way analysis of variance to test claims involving three or more means
An introduction to two-way analysis of variance .

5. One-Way ANOVA One-way analysis of variance
Chapter 10
One-Way ANOVA
One-way analysis of variance
A hypothesis-testing technique that is used to compare means from three or more populations.
Analysis of variance is usually abbreviated ANOVA.
Hypotheses:
H0: μ1 = μ2 = μ3 =…= μk (all population means are equal)
Ha: At least one of the means is different from the others. .
Larson/Farber 4th ed

6. One-Way ANOVA In a one-way ANOVA test, the following must be true.
Each sample must be randomly selected from a normal, or approximately normal, population.
The samples must be independent of each other.
Each population must have the same variance. .

7. One-Way ANOVA
The variance between samples MSB measures the differences related to the treatment given to each sample and is sometimes called the mean square between.
The variance within samples MSW measures the differences related to entries within the same sample. This variance, sometimes called the mean square within, is usually due to sampling error. .

8. One-Way Analysis of Variance Test
If the conditions for a one-way analysis of variance are satisfied, then the sampling distribution for the test is approximated by the F-distribution.
The test statistic is
The degrees of freedom for the F-test are
d.f.N = k – 1 and d.f.D = N – k
where k is the number of samples and N is the sum of the sample sizes. .

9. Test Statistic for a One-Way ANOVA
Chapter 10
Test Statistic for a One-Way ANOVA
In Words In Symbols
Find the mean and variance of each sample.
Find the mean of all entries in all samples (the grand mean).
Find the sum of squares between the samples.
Find the sum of squares within the samples. .
Larson/Farber 4th ed

10. Test Statistic for a One-Way ANOVA
Chapter 10
Test Statistic for a One-Way ANOVA
In Words In Symbols
Find the variance between the samples.
Find the variance within the samples
Find the test statistic. .
Larson/Farber 4th ed

11. Performing a One-Way ANOVA Test
Chapter 10
Performing a One-Way ANOVA Test
In Words In Symbols
Verify that the samples are random and independent, the populations have normal distributions, and the population variances are equal.
Identify the claim. State the null and alternative hypotheses.
Specify the level of significance.
State H0 and Ha.
Identify α. .
Larson/Farber 4th ed

12. Performing a One-Way ANOVA Test
Chapter 10
Performing a One-Way ANOVA Test
In Words In Symbols
Determine the degrees of freedom for the numerator and the denominator.
Determine the critical value.
Determine the rejection region.
d.f.N = k – 1 d.f.D = N – k
Use Table 7 in Appendix B. .
Larson/Farber 4th ed

13. Performing a One-Way ANOVA Test
Chapter 10
Performing a One-Way ANOVA Test
In Words In Symbols
Calculate the test statistic and sketch the sampling distribution.
Make a decision to reject or fail to reject the null hypothesis.
Interpret the decision in the context of the original claim.
If F is in the rejection region, reject H0. Otherwise, fail to reject H0. .
Larson/Farber 4th ed

14. Chapter 10
ANOVA Summary Table
A table is a convenient way to summarize the results in a one-way ANOVA test.
d.f.D
SSW
Within
d.f.N
SSB
Between F
Mean squares
Degrees of freedom
Sum of squares
Variation .
Larson/Farber 4th ed

15. Example: Performing a One-Way ANOVA
A medical researcher wants to determine whether there is a difference in the mean length of time it takes three types of pain relievers to provide relief from headache pain. Several headache sufferers are randomly selected and given one of the three medications. Each headache sufferer records the time (in minutes) it takes the medication to begin working. The results are shown on the next slide. At α = 0.01, can you conclude that the mean times are different? Assume that each population of relief times is normally distributed and that the population variances are equal. .

16. Example: Performing a One-Way ANOVA
Medication 1
Medication 2
Medication 3 12 16 14 15 17 21 20 19
Solution:
k = 3 (3 samples)
N = n1 + n2 + n3 = = 13 (sum of sample sizes) .

17. Solution: Performing a One-Way ANOVA
H0:
Ha:
α =
d.f.N=
d.f.D=
Rejection Region:
μ1 = μ2 = μ3
At least one mean is different. (Claim)
Test Statistic:
Decision:
0. 01
3 – 1 = 2
13 – 3 = 10 .

18. Solution: Performing a One-Way ANOVA
To find the test statistic, the following must be calculated. .

19. Solution: Performing a One-Way ANOVA
To find the test statistic, the following must be calculated. .

20. Solution: Performing a One-Way ANOVA
H0:
Ha:
α =
d.f.N=
d.f.D=
Rejection Region:
μ1 = μ2 = μ3
At least one mean is different. (Claim)
Test Statistic:
Decision:
0.01
3 – 1 = 2
13 – 3 = 10
Fail to Reject H0
There is not enough evidence at the 1% level of significance to conclude that there is a difference in the mean length of time it takes the three pain relievers to provide relief from headache pain.
1.50
7.56 .

21. Example: Using the TI-83/84 to Perform a One-Way ANOVA
A researcher believes that the mean earnings of top-paid actors, athletes, and musicians are the same. The earnings (in millions of dollars) for several randomly selected from each category are shown in the table in next slide. Assume that the populations are normally distributed, the samples are independent, and the population variances are equal. At α = 0.10, can you conclude that the mean earnings are the same for the three categories? Use a technology tool. (Source: Forbes.com LLC) .

22. Example: Using the TI-83/84 to Perform a One-Way ANOVA.

23. Solution: Using the TI-83/84 to Perform a One-Way ANOVA
Ha:
Store data into lists L1, L2, and L3
μ1 = μ2 = μ3 (Claim)
At least one mean is different.
Decision:
P ≈ 0.06 so P < α Reject H0
There is enough evidence at the 10% level of significance to reject the claim that the mean earnings are the same. .

24. Two-Way ANOVA Two-way analysis of variance
A hypothesis-testing technique that is used to test the effect of two independent variables, or factors, on one dependent variable. .

25. Two-Way ANOVA Example:
Suppose a medical researcher wants to test the effect of gender and type of medication on the mean length of time it takes pain relievers to provide relief.
Gender
Male
Female
Type of medication I
Males taking type I
Females taking type I
Males taking type II
Females taking type II
Males taking type III
Females taking type III II
III .

26. Two-Way ANOVA Hypotheses
Main effect
The effect of one independent variable on the dependent variable.
Interaction effect
The effect of both independent variables on the dependent variable. .

27. Two-Way ANOVA Hypotheses
Hypotheses for main effects:
H0: Gender has no effect on the mean length of time it takes a pain reliever to provide relief.
Ha: Gender has an effect on the mean length of time it takes a pain reliever to provide relief.
H0: Type of medication has no effect on the mean length of time it takes a pain reliever to provide relief.
Ha: Type of medication has an effect on the mean length of time it takes a pain reliever to provide relief. .

28. Two-Way ANOVA Hypotheses
Hypotheses for interaction effects:
H0: There is no interaction effect between gender and type of medication on the mean length of time it takes a pain reliever to provide relief.
Ha: There is an interaction effect between gender and type of medication on the mean length of time it takes a pain reliever to provide relief.
Perform a two-way ANOVA test, calculating the F-statistic for each hypothesis. It is possible to reject none, one, two, or all of the null hypotheses. The statistics involved with a two-way ANOVA test is beyond the scope of this course. You can use a technology tool such as MINITAB to perform the test. .

29. Chapter 10
Section 10.4 Summary
Used one-way analysis of variance to test claims involving three or more means
Introduced two-way analysis of variance .
Larson/Farber 4th ed