12-1 Chapter Twelve McGraw-Hill/Irwin © 2006 The McGraw-Hill Companies, Inc., All Rights Reserved.

12-2 Chapter Twelve Analysis of Variance GOALS When you have completed this chapter, you will be able to: ONE List the characteristics of the F distribution. TWO Conduct a test of hypothesis to determine whether the variances of two populations are equal. THREE Discuss the general idea of analysis of variance. Goals

12-3 Chapter Twelve continued Analysis of Variance GOALS When you have completed this chapter, you will be able to: FOUR Organize data into a one-way and a two-way ANOVA table. FIVE Conduct a test of hypothesis among three or more treatment means. SIX Develop confidence intervals for the difference between treatment means. Goals

12-4 Characteristics of F- Distribution Its values range from 0 to . As F   the curve approaches the X- axis but never touches it. Characteristics of the F-Distribution There is a “family” of F Distributions. Each member of the family is determined by two parameters: the numerator degrees of freedom and the denominator degrees of freedom. F cannot be negative, and it is a continuous distribution. The F distribution is positively skewed.

12-5 Test for Equal Variances of Two Populations For the two tail test, the test statistic is given by Test for Equal Variances of Two Populations and are the sample variances for the two samples. The larger s is placed in the denominator. The degrees of freedom are n 1 -1 for the numerator and n 2 -1 for the denominator. The null hypothesis is rejected if the computed value of the test statistic is greater than the critical value.

12-6 Example 1 The mean rate of return on a sample of 8 utility stocks was 10.9 percent with a standard deviation of 3.5 percent. At the.05 significance level, can Colin conclude that there is more variation in the software stocks? Colin, a stockbroker at Critical Securities, reported that the mean rate of return on a sample of 10 internet stocks was 12.6 percent with a standard deviation of 3.9 percent.

12-7 Example 1 continued Step 3: The test statistic is the F distribution. Step 1: The hypotheses are Step 2: The significance level is.05.

12-8 Example 1 continued Step 5: The value of F is computed as follows. The p(F>1.2416) is H 0 is not rejected. There is insufficient evidence to show more variation in the internet stocks. Step 4: H 0 is rejected if F>3.68 or if p <.05. The degrees of freedom are n 1 -1 or 9 in the numerator and n 1 -1 or 7 in the denominator.

12-9 The ANOVA Test of Means The null and alternate hypotheses for four sample means is given as: H o :  1 =  2 =  3 =  4 H 1 :  1 =  2 =  3 =  4 The ANOVA Test of Means The F distribution is also used for testing whether two or more sample means came from the same or equal populations. This technique is called analysis of variance or ANOVA

12-10 The populations have equal standard deviations. ANOVA requires the following conditions Underlying assumptions for ANOVA The sampled populations follow the normal distribution. The samples are independent

12-11 F =F = Estimate of the population variance based on the differences among the sample means Estimate of the population variance based on the variation within the samples ANOVA Test of Means Degrees of freedom for the F statistic in ANOVA If there are k populations being sampled, the numerator degrees of freedom is k – 1 If there are a total of n observations the denominator degrees of freedom is n – k.

12-12 In the following table, i stands for the i th observation c stands for c th treatment group x G is the overall or grand mean k is the number of treatment groups ANOVA Test of Means Total Variation Treatment Variation Random Variation ANOVA divides the Total Variation into the variation due to the treatment, Treatment Variation, and to the error component, Random Variation.

12-13 ANOVA Table Source of Variation Sum of Squares Degrees of Freedom Mean Square F Treatments (c) SST c  n c (X c -X G ) 2 k-1SST/(k-1) =MSTMST MSE ErrorSSE i c  (X i.c -X c ) 2 n-kSSE/(n-k) =MSE TotalTSS i  (X i -X G ) 2 n-1 Anova Table Treatment variation Random variation Total variation

12-14 Rosenbaum Restaurants specialize in meals for families. Katy Polsby, President, recently developed a new meat loaf dinner. Before making it a part of the regular menu she decides to test it in several of her restaurants. Example 2 She would like to know if there is a difference in the mean number of dinners sold per day at the Anyor, Loris, and Lander restaurants. Use the.05 significance level.

12-15 Number of Dinners Sold by Restaurant Restaurant Day AynorLorisLander Day 1 Day 2 Day 3 Day 4 Day Example 2 continued

12-16 Step One: State the null hypothesis and the alternate hypothesis. H o :  Aynor =  Loris =  Landis H 1 :  Aynor =  Loris =  Landis Step Two: Select the level of significance. This is given in the problem statement as.05. Step Three: Determine the test statistic. The test statistic follows the F distribution. Example 2 continued

12-17 Step Five: Select the sample, perform the calculations, and make a decision. Step Four: Formulate the decision rule. The numerator degrees of freedom, k-1, equal 3-1 or 2. The denominator degrees of freedom, n-k, equal 13-3 or 10. The value of F at 2 and 10 degrees of freedom is Thus, H 0 is rejected if F>4.10 or p<  of.05. Example 2 continued Using the data provided, the ANOVA calculations follow.

12-18 Anyor #sold SS(Anyor)Loris #sold SS(Loris)Lander #sold SS(Lander) ( ) 2 ( ) 2 ( ) 2 ( ) ( ) 2 ( ) 2 ( ) 2 ( ) (18-17) 2 (16-17) 2 (17-17) 2 2 XkXk SSE: = 9.75 X G : Computation of SSE i k  (X i.k -X k ) 2

12-19 Anyor #sold TSS(Anyor)Loris #sold TSS(Loris)Lander #sold TSS(Lander) (13-14) 2 (12-14) 2 (14-14) 2 (12-14) (10-14) 2 (12-14) 2 (13-14) 2 (11-14) (18-14) 2 (16-14) 2 (17-14) 2 47 TSS: = SSE: 9.75 X G : Computation of TSS i  (X i -X G ) 2 Example 2 continued Computation of TSS

12-20 Computation of SST k  n k (X k -X G ) 2 RestaurantXTXT SST Anyor Loris Lander ( ) 2 4( ) 2 5( ) Shortcut: SST = TSS – SSE = 86 – 9.75 = Example 2 continued Computation of SST

12-21 ANOVA Table Source of Variation Sum of Squares Degrees of Freedom Mean Square F Treatments = /2 = = Error = /10 =.975 Total =12 Example 2 continued

12-22 Example 2 continued The ANOVA tables on the next two slides are from the Minitab and EXCEL systems. The p(F> ) is The mean number of meals sold at the three locations is not the same. Since an F of > the critical F of 4.10, the p of < a of.05, the decision is to reject the null hypothesis and conclude that At least two of the treatment means are not the same.

12-23 Example 2 continued Analysis of Variance Source DF SS MS F P Factor Error Total Individual 95% CIs For Mean Based on Pooled StDev Level N Mean StDev Aynor (---*---) Loris (---*---) Lander (---*---) Pooled StDev =

12-24 Example 2 continued

12-25 Inferences About Treatment Means One of the simplest procedures is through the use of confidence intervals around the difference in treatment means. When I reject the null hypothesis that the means are equal, I want to know which treatment means differ.

12-26 Confidence Interval for the Difference Between Two Means If the confidence interval around the difference in treatment means includes zero, there is not a difference between the treatment means. t is obtained from the t table with degrees of freedom (n - k). MSE = [SSE/(n - k)]

12-27 EXAMPLE 3 95% confidence interval for the difference in the mean number of meat loaf dinners sold in Lander and Aynor Can Katy conclude that there is a difference between the two restaurants?

12-28 Example 3continued The mean number of meals sold in Aynor is different from Lander. Because zero is not in the interval, we conclude that this pair of means differs.