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Copyright © 2009 Cengage Learning Chapter 14 ANOVA
Copyright © 2009 Cengage Learning 14.2 Analysis of Variance Allows us to compare ≥2 populations of interval data. ANOVA is:. a procedure which determines whether differences exist between population means. a procedure which analyzes sample variance. Referred to as treatments Not necessarily equal
Copyright © 2009 Cengage Learning 14.3 One Way ANOVA New Terminology: Population classification criterion is called a factor Each population is a factor level Example 14.1 (Stock Ownership) (1) The analyst was interested in determining whether the ownership of stocks varied by age. (Xm14-01)Xm14-01 The age categories are: Young (Under 35) Early middle-age (35 to 49) Late middle-age (50 to 65) Senior (Over 65)
Copyright © 2009 Cengage Learning 14.4 Question: Does the stock ownership between the four age groups differs? n=366; % of financial assets invested in equity Factor: Age Category “One Way” ANOVA since only 1 factor 4 factor levels: (1)Young; (2)Early middle age; (3)Late middle age; & (4)Senior Hypothesis: H 0 :µ 1 = µ 2 = µ 3 = µ 4 i.e. there are no differences between population means H 1 : at least two means differ Example 14.1 (Stock Ownership) (2)
Copyright © 2009 Cengage Learning Example 14.1 (Stock Ownership) (3) Rule….F-stat macam chapter 13 (ada paham?) Rejection Region…. F > F,k–1, n–k F-stat
Copyright © 2009 Cengage Learning 14.6 SST gave us the between-treatments variation SSE (Sum of Squares for Error) measures the within-treatments variation Sample Statistics & Grand Mean Example 14.1 (Stock Ownership) (4)
Copyright © 2009 Cengage Learning 14.7 Hence, the between-treatments variation SST, is Sample variances: Then SSE: 161,871 Example 14.1 (Stock Ownership) (5)
Copyright © 2009 Cengage Learning 14.8 Example 14.1 (Stock Ownership) (6) F >F, (k–1),(n–k) = F 5%, (4 –1), (366–4) = 2.61 H 0 :µ 1 = µ 2 = µ 3 = µ 4 H 1 : at least two means differ 2.79 > 2.61 thus reject H 0 B-12
Copyright © 2009 Cengage Learning 14.9 Using Excel: ClickData, Data Analysis, Anova: Single Factor Example 14.1 (Stock Ownership) (7)
Copyright © 2009 Cengage Learning p-value = < 0.05 Reject the null hypothesis (H 0 :µ 1 = µ 2 = µ 3 = µ 4 ) in favor of the alternative hypothesis (H 1 : at least two population means differ). Conclusion: there is enough evidence to infer that the mean percentages of assets invested in the stock market differ between the four age categories. Example 14.1 (Stock Ownership) (8)
Copyright © 2009 Cengage Learning Multiple Comparisons If the conclusion is “at least two treatment means differ” i.e. reject the H 0 : We often need to know which treatment means are responsible for these differences 3 statistical inference procedures highlights it: Fisher’s least significant difference (LSD) method Tukey’s multiple comparison method
Copyright © 2009 Cengage Learning Fisher’s Least Significant Difference A better estimator of the pooled variances = MSE Substitute MSE in place of s p 2 Compares the difference between means to the Least Significant Difference LSD, given by: LSD will be the same for all pairs of means if all k sample sizes are equal. If some sample sizes differ, LSD must be calculated for each combination. Differ if absolute value of difference between means > LSD
Copyright © 2009 Cengage Learning Example 14.2 ( Car bumper Quality ) (1) North American automobile manufacturers have become more concerned with quality because of foreign competition. One aspect of quality is the cost of repairing damage caused by accidents. A manufacturer is considering several new types of bumpers. To test how well they react to low-speed collisions, 10 bumpers of each of 4 different types were installed on mid-size cars, which were then driven into a wall at 5 miles per hour.
Copyright © 2009 Cengage Learning The cost of repairing the damage in each case was assessed. Xm14-02Xm14-02 Questions… Is there sufficient evidence to infer that the bumpers differ in their reactions to low-speed collisions? If differences exist, which bumpers differ? Objective: to compare 4 populations Data: interval Samples: independent Statistical method: One-way ANOVA. Example 14.2 ( Car bumper Quality ) (2)
Copyright © 2009 Cengage Learning F = 4.06, p-value = If 5% SL, then….Reject H o There is enough evidence to infer that a difference exists between the 4 bumpers The question now is…….which bumpers differ? Example 14.2 ( Car bumper Quality ) (3)
Copyright © 2009 Cengage Learning The sample means are…. We calculate the absolute value of the differences between means and compare them. Example 14.2 ( Car bumper Quality ) (4)
Copyright © 2009 Cengage Learning Click Add-Ins > Data Analysis Plus > Multiple Comparisons Example 14.2 ( Car bumper Quality ) (5)
Copyright © 2009 Cengage Learning Hence, µ 1 and µ 2, µ 1 and µ 3, µ 2 and µ 4, and µ 3 and µ 4 differ. The other two pairs µ 1 and µ 4, and µ 2 and µ 3 do not differ. Example 14.2 ( Car bumper Quality ) (6)
Copyright © 2009 Cengage Learning Tukey’s Multiple Comparison Method As before, we are looking for a critical number to compare the differences of the sample means against. In this case: Note: is a lower case Omega, not a “w” Critical value of the Studentized range with n–k degrees of freedom Table 7 - Appendix B harmonic mean of the sample sizes Different if the pair means diff >
Copyright © 2009 Cengage Learning Example 14.2 k = 4 N 1 = n 2 = n 3 = n 4 = n g = 10 Ν = 40 – 4 = 36 MSE = 12,399 Thus,
Copyright © 2009 Cengage Learning Example 14.1 (Tukey’s Method) Using Tukey’s method µ 2 and µ 4, and µ 3 and µ 4 differ.
Chapter 15 ANOVA. Comparing Means for Several Populations When we wish to test for differences in means for only 1 or 2 populations, we use one- or two-sample.
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