1. Section 8.1-1 Copyright © 2014, 2012, 2010 Pearson Education, Inc. Lecture Slides Elementary Statistics Twelfth Edition and the Triola Statistics Series by Mario F. Triola

2. Section 8.1-2 Copyright © 2014, 2012, 2010 Pearson Education, Inc. Review In chapter 9, we introduced methods for comparing the means from two independent samples. Now we look to test the equality of three or more means by using the method of one-way analysis of variance (ANOVA).

3. Section 8.1-3 Copyright © 2014, 2012, 2010 Pearson Education, Inc. ANOVA Methods Require the F Distribution 1. The F distribution is not symmetric; it is skewed to the right. 2. The values of F cannot be negative. 3. The exact shape of the F distribution depends on the two different degrees of freedom.

4. Section 8.1-4 Copyright © 2014, 2012, 2010 Pearson Education, Inc. F Distribution

5. Section 8.1-5 Copyright © 2014, 2012, 2010 Pearson Education, Inc. 9-2 Key Concept This section introduces the method of one-way analysis of variance, which is used for tests of hypotheses that three or more population means are all equal. Because the calculations are very complicated, we emphasize the interpretation of results obtained by using technology.

6. Section 8.1-6 Copyright © 2014, 2012, 2010 Pearson Education, Inc. Key Concept 1.Understand that a small P-value (such as 0.05 or less) leads to rejection of the null hypothesis of equal means. With a large P-value (such as greater than 0.05), fail to reject the null hypothesis of equal means. 2.Develop an understanding of the underlying rationale by studying the examples in this section.

7. Section 8.1-7 Copyright © 2014, 2012, 2010 Pearson Education, Inc. Definition One-way analysis of variance (ANOVA) is a method of testing the equality of three or more population means by analyzing sample variances. One-way analysis of variance is used with data categorized with one factor (or treatment), which is a characteristic that allows us to distinguish the different populations from one another.

8. Section 8.1-8 Copyright © 2014, 2012, 2010 Pearson Education, Inc. One-Way ANOVA Requirements 1. The populations have approximately normal distributions. 2. The populations have the same variance σ 2 (or standard deviation σ). 3. The samples are simple random samples of quantitative data. 4. The samples are independent of each other. 5. The different samples are from populations that are categorized in only one way.

9. Section 8.1-9 Copyright © 2014, 2012, 2010 Pearson Education, Inc. Procedure 1. Use STATDISK, Minitab, Excel, StatCrunch, a TI-83/84 calculator, or any other technology to obtain results. 2. Identify the P-value from the display. 3.Form a conclusion based on these criteria: If the P-value ≤ α, reject the null hypothesis of equal means. Conclude at least one mean is different from the others. If the P-value > α, fail to reject the null hypothesis of equal means.

10. Section 8.1-10 Copyright © 2014, 2012, 2010 Pearson Education, Inc. Caution When we conclude that there is sufficient evidence to reject the claim of equal population means, we cannot conclude from ANOVA that any particular mean is different from the others.

11. Section 8.1-11 Copyright © 2014, 2012, 2010 Pearson Education, Inc. Example Use the performance IQ scores listed in Table 12-1 and a significance level of α = 0.05 to test the claim that the three samples come from populations with means that are all equal.

12. Section 8.1-12 Copyright © 2014, 2012, 2010 Pearson Education, Inc. Example - Continued Here are summary statistics from the collected data:

13. Section 8.1-13 Copyright © 2014, 2012, 2010 Pearson Education, Inc. Example - Continued Requirement Check: 1.The three samples appear to come from populations that are approximately normal (normal quantile plots OK). 2.The three samples have standard deviations that are not dramatically different. 3.We can treat the samples as simple random samples. 4.The samples are independent of each other and the IQ scores are not matched in any way. 5.The three samples are categorized according to a single factor: low lead, medium lead, and high lead.

14. Section 8.1-14 Copyright © 2014, 2012, 2010 Pearson Education, Inc. Example - Continued The hypotheses are: The significance level is α = 0.05. Technology results are presented on the next slides.

15. Section 8.1-15 Copyright © 2014, 2012, 2010 Pearson Education, Inc. Example - Continued

16. Section 8.1-16 Copyright © 2014, 2012, 2010 Pearson Education, Inc. Example - Continued The displays all show that the P-value is 0.020 when rounded. Because the P-value is less than the significance level of α = 0.05, we can reject the null hypothesis. There is sufficient evidence that the three samples come from populations with means that are different. We cannot conclude formally that any particular mean is different from the others, but it appears that greater blood lead levels are associated with lower performance IQ scores.

17. Section 8.1-17 Copyright © 2014, 2012, 2010 Pearson Education, Inc. P-Value and Test Statistic Larger values of the test statistic result in smaller P-values, so the ANOVA test is right-tailed. The figure on the next slide shows the relationship between the F test statistic and the P-value. Assuming that the populations have the same variance σ 2 (as required for the test), the F test statistic is the ratio of these two estimates of σ 2 : (1) variation between samples (based on variation among sample means) (2) variation within samples (based on the sample variances).

18. Section 8.1-18 Copyright © 2014, 2012, 2010 Pearson Education, Inc. Relationship Between F Test Statistic and P-Value

19. Section 8.1-19 Copyright © 2014, 2012, 2010 Pearson Education, Inc. Test Statistic for One-Way ANOVA

20. Section 8.1-20 Copyright © 2014, 2012, 2010 Pearson Education, Inc. Caution When testing for equality of three or more populations, use analysis of variance. Do not use multiple hypothesis tests with two samples at a time.