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Chapter 12: Analysis of Variance

Analysis of Variance. Introduction to Analysis of Variance (ANOVA)ANOVA: Testing for the Equality of k Population Means. Examples. Does distance it takes to stop car at 60 mph depend on tire brand?Is there a difference in the mean time that customers spend completing their transactions among fou

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Chapter 12: Analysis of Variance

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    2. Analysis of Variance Introduction to Analysis of Variance (ANOVA) ANOVA: Testing for the Equality of k Population Means

    15. Analysis of Variance: Testing for the Equality of k Population Means Between-Treatments Estimate of Population Variance Within-Treatments Estimate of Population Variance Comparing the Variance Estimates: The F Test The ANOVA Table

    19. Between-Treatments Estimate of Population Variance A between-treatment estimate of ?2 is called the mean square between and is denoted MSB. The numerator of MSB is called the sum of squares between and is denoted SSB. The denominator of MSB represents the degrees of freedom associated with SSB.

    20. Within Groups Variance Estimate

    21. The estimate of ?2 based on the variation of the sample observations within each sample is called the within mean square and is denoted by MSW. The numerator of MSW is called the within sum of squares and is denoted by SSW. The denominator of MSW represents the degrees of freedom associated with SSW. Within-Samples Estimate of Population Variance

    22. Breakdown of sums of squares

    23. Process of Hypothesis Testing

    24. Sampling Distribution

    25. What are degrees of freedom?

    26. Comparing the Variance Estimates: The F Test If the null hypothesis is true and the ANOVA assumptions are valid, the sampling distribution of MSB/MSW is an F distribution with MSB d.f. equal to k - 1 and MSW d.f. equal to N - k. If the means of the k populations are not equal, the value of MSB/MSW will be inflated. Hence, we will reject H0 if the resulting value of MSB/MSW appears to be too large to have been selected at random from the appropriate F distribution.

    34. Introduction to Analysis of Variance Analysis of Variance (ANOVA) can be used to test for the equality of three or more population means using data obtained from observational or experimental studies. We want to use the sample results to test the following hypotheses. H0: ?1?=??2?=??3?=?. . . = ?k? Ha: Not all population means are equal

    35. Test for the Equality of k Population Means Hypotheses H0: ?1?=??2?=??3?=?. . . = ?k? Ha: Not all population means are equal Test Statistic Fcal = MSB/MSW Decision Rule Reject H0 if p-value = P(F > Fcal) < a where the value of F is based on an F distribution with k - 1 numerator degrees of freedom and N - 1 denominator degrees of freedom.

    37. ANOVA Table Source of Sum of Degrees of Mean Variation Squares Freedom Squares F Between SSB k - 1 MSB MSB/MSW Within SSW N - k MSW Total SST N - 1 SST divided by its degrees of freedom N - 1 is simply the overall sample variance that would be obtained if we treated the entire nT observations as one data set.

    41. Assumptions for Analysis of Variance For each population, the response variable is normally distributed. The variance of the response variable, denoted ?2, is the same for all of the populations. The observations must be independent.

    42. Example: Ali Manufacturing Analysis of Variance Ali would like to know if the mean number of hours worked per week is the same for the department managers at her three manufacturing plants (Buffalo, Pittsburgh, and Detroit). A simple random sample of 5 managers from each of the three plants was taken and the number of hours worked by each manager for the previous week is shown on the next slide.

    43. Analysis of Variance Plant 1 Plant 2 Plant 3 Observation Buffalo Pittsburgh Detroit 1 48 73 51 2 54 63 63 3 57 66 61 4 54 64 54 5 62 74 56 Sample Mean 55 68 57 Sample Variance 26.0 26.5 24.5 Example: Ali Manufacturing

    44. Analysis of Variance Hypotheses H0: ?1?=??2?=??3? Ha: Not all the means are equal where: ????1 = mean number of hours worked per week by the managers at Plant 1 ?2 = mean number of hours worked per week by the managers at Plant 2 ????3 = mean number of hours worked per week by the managers at Plant 3 Example: Ali Manufacturing

    45.   Example: Ali Manufacturing

    47. Analysis of Variance ANOVA Table Source of Sum of Degrees of Mean Variation Squares Freedom Square F Between 490 2 245 9.55 Within 308 12 25.667 Total 798 14 Example: Ali Manufacturing

    48. Analysis of Variance F - Test If H0 is true, the ratio MSB/MSW should be near 1 since both MSB and MSW are estimating ? 2. If Ha is true, the ratio should be significantly larger than 1 since MSB tends to overestimate ?2. Example: Ali Manufacturing

    49. P-value The figure below shows the p-value associated with Fcal.

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