Copyright © 2015 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education. Analysis of Variance Chapter 12

12-2 Learning Objectives LO12-1 Apply the F distribution to test a hypothesis that two population variances are equal. LO12-2 Use ANOVA to test a hypothesis that three or more population means are equal. LO12-3 Use confidence intervals to test and interpret differences between pairs of population means. LO12-4 Use a blocking variable in a two-way ANOVA to test a hypothesis that three or more population means are equal. LO12-5 Perform a two-way ANOVA with interaction and describe the results.

12-3 Testing the Hypothesis of Two Equal Population Variances: The F Distribution  The distribution is named to honor Sir Ronald Fisher, one of the founders of modern-day statistics.  The distribution is:  Used to test a hypothesis of equal population variances.  Used to simultaneously test a hypothesis that several population means are equal. The simultaneous comparison of several population means is called analysis of variance (ANOVA). LO12-1 Apply the F distribution to test a hypothesis that two population variances are equal.

12-4 Characteristics of a F-Distribution  There is a “family” of F- distributions. A particular member of the family is determined by two parameters: the degrees of freedom in the numerator and the degrees of freedom in the denominator.  The F-distribution is continuous.  A F-value cannot be negative.  The F-distribution is positively skewed.  It is asymptotic. As F  , the curve approaches the X-axis but never touches it. LO12-1

12-5 Testing the Hypothesis of Two Equal Population Variances The F-distribution is used to test the hypothesis that the variance of one normal population equals the variance of another normal population. Examples: Two Barth shearing machines are set to produce steel bars of the same length. The bars, therefore, should have the same mean length. We want to ensure that in addition to having the same mean length they also have similar variation. The mean rate of return on two types of common stock may be the same, but there may be more variation in the rate of return in one than the other. A sample of 10 technology and 10 utility stocks shows the same mean rate of return, but there is likely more variation in the technology stocks. A study by the marketing department for a large newspaper found that men and women spend about the same amount of time per day reading the paper. However, the same report indicated there was nearly twice as much variation in time spent per day among the men than the women. LO12-1 H 0 : σ 1 2 = σ 2 2 H 1 : σ 1 2 ≠ σ 2 2

12-6 Testing the Hypothesis of Two Equal Population Variances – Example Lammers Limos offers limousine service from the city hall in Toledo, Ohio, to Metro Airport in Detroit. The president of the company, is considering two routes. One is via U.S. 25 and the other via I-75. He wants to study the time it takes to drive to the airport using each route and then compare the results. He collected the following sample data, which is reported in minutes. Using the.10 significance level, is there a difference in the variation in the driving times for the two routes? LO12-1

12-7 Testing the Hypothesis of Two Equal Population Variances – Example LO12-1 Computing the sample means and variances:

12-8 Testing the Hypothesis of Two Equal Population Variances – Example Step 1: State the null and alternate hypotheses. H 0 : σ 1 2 = σ 2 2 H 1 : σ 1 2 ≠ σ 2 2 Step 2: Select a level of significance. The requested significance level is.10. Step 3: Select the test statistic. The appropriate test statistic to test a hypothesis of equal variances is the F statistic. LO12-1

12-9 Step 4:State the decision rule. The F-statistic is computed as the ratio of two variances. The numerator is always the larger of the two variances with its corresponding degrees of freedom. Reject H 0 if F > F  /2,v1,v2 F > F.10/2,7-1,8-1 F > F.05,6,7 Testing the Hypothesis of Two Equal Population Variances – Example LO12-1

12-10 Testing the Hypothesis of Two Equal Population Variances – Example The decision is to reject the null hypothesis because the computed F value (4.23) is larger than the critical value (3.87). Step 6: Interpret the result. The data indicates that there is a difference in the variation of the travel times along the two routes. Step 5: Compute the value of F and make a decision. LO12-1

12-11 Testing the Hypothesis of Two Equal Population Variances – Example LO12-1

12-12 Testing the Hypothesis of Three or More Equal Population Means The F-distribution is also used for testing whether two or more sample means came from the same or equal populations. Assumptions:  The sampled populations follow the normal distribution.  The populations have equal standard deviations.  The samples are randomly selected and are independent. LO12-2 Use ANOVA to test a hypothesis that three or more population means are equal.

12-13 The null hypothesis is when the population means are all the same. The alternative hypothesis is when at least one of the means is different. The test statistic is the F distribution. The decision rule is whether to reject the null hypothesis if F (computed) is greater than F (table) with numerator and denominator degrees of freedom. Hypothesis Setup and Decision Rule: Testing the Hypothesis of Three or More Equal Population Means H 0 : µ 1 = µ 2 =…= µ k H 1 : The means are not all equal. Reject H 0 if F > F ,k-1,n-k LO12-2

12-14 Testing the Hypothesis of Three or More Equal Population Means - Illustrated 14 Joyce Kuhlman manages a regional financial center. She wishes to compare the productivity, as measured by the number of customers served, among three employees. Four days are randomly selected and the number of customers served by each employee is recorded. LO

12-15 Testing the Hypothesis of Three or More Equal Population Means – Example Recently a group of four major carriers joined in hiring Brunner Marketing Research, Inc., to survey recent passengers regarding their level of satisfaction with a recent flight. The survey included questions on ticketing, boarding, in-flight service, baggage handling, pilot communication, and so forth. Twenty-five questions offered a range of possible answers: excellent, good, fair, or poor. A response of excellent was given a score of 4, good a 3, fair a 2, and poor a 1. These responses were then totaled, so the total score was an indication of the satisfaction with the flight. Brunner Marketing Research, Inc., randomly selected and surveyed passengers from the four airlines. Is there a difference in the mean satisfaction level among the four airlines? Use the.01 significance level. LO12-2

12-16 Testing the Hypothesis of Three or More Equal Population Means – Example LO12-2 Computing the “treatment” and grand means:

12-17 Testing the Hypothesis of Three or More Equal Population Means – Example Step 1:State the null and alternate hypotheses. H 0 : µ N = µ W = µ P = µ B H 1 : The means are not all equal. Step 2:State the level of significance. The.01 significance level is stated in the problem. Step 3:Find the appropriate test statistic. Because we are comparing means of more than two groups, use the F statistic. LO12-2

12-18 Testing the Hypothesis of Three or More Equal Population Means – Example Step 4: Formulate a decision rule. The F-statistic will be used to formulate the decision rule. The F-statistic is a ratio of two variances, each divided by their degrees of freedom. These are called mean squares. For this ANOVA, we will divide the treatment mean square by the error mean square. Therefore, we need the degrees of freedom for treatments and error to find the F value for the decision rule. The degrees of freedom in the numerator: (Number of treatments – 1) = (k - 1) = = 3 The degrees of freedom in the denominator: (Total number of observations – Number of treatments) = (n – k) = (22 - 4) = 18 LO12-2

12-19 Testing the Hypothesis of Three or More Equal Population Means – Example LO12-2 Reject H 0 if F > F ,k-1,n-k or Reject H 0 if F > 5.09 From the F-table with the.01 level of significance, the critical value of F with 3 numerator and 18 denominator degrees of freedom is Denominator degrees of freedom Numerator degrees of freedom

12-20 Testing the Hypothesis of Three or More Equal Population Means – Example Step 5: Compute the value of F and make a decision. Creating the Analysis of Variance table: LO12-2

12-21 Creating the ANOVA Table: Computing SS Total and SSE LO12-2 Computing the total sum of squares: Computing the error sum of squares:

12-22 Creating the ANOVA Table: Treatment Sum of Squares, SST, and the ANOVA Table Step 5 (continued): Compute the value of F and make a decision. The computed value of F is 8.99, which is greater than the critical value of 5.09, so the null hypothesis is rejected. LO12-2

12-23 Testing the Hypothesis of Three or More Equal Population Means – Example Step 6: Interpret the result. The population means are not all equal. The mean scores are not the same for the four airlines; at this point we can only conclude there is a difference in at least one pair of treatment means. We cannot determine which of the airlines’ satisfaction mean scores differ. LO12-2

12-24 Testing the Hypothesis of Three or More Equal Population Means – Excel LO12-2

12-25 Testing for Differences Between Pairs of Population Means When we reject the null hypothesis that the means are equal, we may want to know which treatment means differ. One of the simplest procedures is to use confidence intervals for the difference between two means. LO12-3 Use confidence intervals to test and interpret differences between pairs of population means.

12-26 Testing for Differences Between Pairs of Population Means – Example From the previous example, develop a 95% confidence interval for the difference in the mean customer satisfaction between Northern and Branson airlines. Can we conclude that there is a difference between the two airlines’ ratings? LO12-3

12-27 Testing for Differences Between Pairs of Population Means – Minitab Output The differences in mean satisfaction ratings between each pair of airlines can be obtained directly from Minitab using the one-way ANOVA analysis and Fisher’s method to compare means. The Minitab output follows. If zero is in the interval, the means are not different. LO12-3

12-28 Two-way ANOVA: Testing the Hypothesis of Three or More Equal Population Means Using a Blocking Variable For the two-factor ANOVA, we test whether there is a significant difference between the treatment effect and whether there is a difference in the blocking effect. The two-way ANOVA table now includes a “blocks” source of variation in addition to treatment, error, and total. Notice there are “b” blocks. LO12-4 Use a blocking variable in a two-way ANOVA to test a hypothesis that three or more population means are equal.

12-29 Two-Way ANOVA: Testing the Hypothesis of Three or More Equal Population Means using a Blocking Variable WARTA, the Warren Area Regional Transit Authority, is expanding bus service from the suburb of Starbrick into the central business district of Warren. There are four routes being considered from Starbrick to downtown Warren: (1) via U.S. 6, (2) via the West End, (3) via the Hickory Street Bridge, and (4) via Route 59. WARTA conducted several tests to determine whether there was a difference in the mean travel times along the four routes. Because there will be many different drivers, the test was set up so each driver drove along each of the four routes. The next slide shows the travel time, in minutes, for each driver-route combination. At the.05 significance level, is there a difference in the mean travel time along the four routes? If we remove the effect of the drivers, is there a difference in the mean travel time? This is a two-way ANOVA. The routes are the treatments and the drivers are the blocks. LO12-2

12-30 Two-Way ANOVA: Testing the Hypothesis of Three or More Equal Population Means Using a Blocking Variable – Example Observed Sample Data: LO12-4

12-31 Two-Way ANOVA: Testing the Hypothesis of Three or More Equal Population Means Using a Blocking Variable – Example Step 1: State the null and alternate hypotheses. H 0 : µ u = µ w = µ h = µ r H 1 : Not all treatment means are the same. Step 2: State the level of significance. The.05 significance level is stated in the problem. Step 3: Find the appropriate test statistic. Because we are comparing means of more than two groups, use the F-statistic. LO12-4

12-32 Two-Way ANOVA: Testing the Hypothesis of Three or More Equal Population Means Using a Blocking Variable – Example Step 4: Formulate a decision rule. The F-statistic will be used to formulate the decision rule. The F-statistic is a ratio of two variances, each divided by their degrees of freedom. These are called mean squares. For the two-way ANOVA, we will divide the treatment mean square by the error mean square. Therefore, we need the degrees of freedom for treatments and error to find the F value for the decision rule. The degrees of freedom in the numerator: (Number of treatments – 1) = (k – 1) = 4 – 1 = 3 The degrees of freedom in the denominator: (Number of treatments – 1)(Number of blocks – 1) = (k – 1)(b - 1) = (4 – 1)(5 – 1) = (3)(4) = 12 LO12-4

12-33 Two-Way ANOVA: Testing the Hypothesis of Three or More Equal Population Means Using a Blocking Variable – Example LO12-4 Step 4: State the decision rule. Reject H 0 if F > F ,v1,v2 F > F.05,k-1,(k – 1)(b – 1) F > F.05,4-1,(4 – 1)(5 – 1) F > F.05,3,12 F > 3.49 From the F-table with the.05 level of significance, the critical F value with 3 numerator and 12 denominator degrees of freedom is Denominator degrees of freedom Numerator degrees of freedom

12-34 Two-Way ANOVA: Testing the Hypothesis of Three or More Equal Population Means Using a Blocking Variable LO12-4 Calculating the Block Sum of Squares

12-35 Two-Way ANOVA: Testing the Hypothesis of Three or More Equal Population Means – Example Using a Blocking Variable LO12-4 Step 5: Compute the value of F and make a decision. The computed value of F is 7.93, which is greater than the critical value of 3.49, so the null hypothesis is rejected. Step 6: Interpret the Result. At least one pair of mean route times are different.

12-36 Using Excel to perform the calculations, we conclude: (1)The mean time is not the same for all drivers. (2)The mean times for the routes are not all the same. Two-Way ANOVA: Testing the Hypothesis of Three or More Equal Population Means Using a Blocking Variable – Excel Example LO12-4

12-37 Two-Way ANOVA with Interaction In the previous example, we studied the separate or independent effects of two variables, routes into the city and drivers, on mean travel time. There is another effect that may influence travel time. This is called an interaction or combined effect of route and driver on travel time. For example, is it possible that one of the drivers is especially good driving one or more of the routes? To measure interaction effects, it is necessary to have at least two observations in each cell. LO12-5 Perform a two-way ANOVA with interaction and describe the results. INTERACTION The effect of one factor on a response variable differs depending on the value of another factor.

12-38 Two-Way ANOVA with Interaction When we use a two-way ANOVA to study interaction, we call the two variables factors instead of treatments and blocks. Interaction occurs if the combination of two factors has some combined effect on the variable under study, in addition to each factor alone. The variable being studied is referred to as the response variable. One way to study interaction is by plotting factor means in a graph called an interaction plot. LO12-5

12-39 Graphical Observation of Interaction Between Driver and Route Our graphical observations show us that interaction effects are possible. For example, Deans (green line) is not the fastest driver on all the routes. So, the travel times depend on both the driver and the route driven. The next step is to conduct statistical tests of hypothesis to further investigate the possible interaction effects. In summary, our study of travel times has several questions: Is there a significant interaction between routes and drivers? Are the travel times for the drivers the same? Are the travel times for the routes the same? Of the three questions, we are most interested in the test for interactions. To put it another way, does a particular route/driver combination result in significantly faster (or slower) driving times? Also, the results of the hypothesis test for interaction affect the way we analyze the route and driver questions. LO12-5

12-40 Two-way ANOVA with Replication – Example Suppose the WARTA blocking experiment discussed earlier is repeated by measuring two more travel times for each driver and route combination with the data shown in the worksheet. LO12-5

12-41 The two-way ANOVA with interaction now has three sets of hypotheses to test: 1. H 0 : There is no interaction between drivers and routes. H 1 : There is interaction between drivers and routes. 2. H 0 : The driver means are the same. H 1 : The driver means are not the same. 3. H 0 : The route means are the same. H 1 : The route means are not the same. LO12-5

12-42 Two-way ANOVA Table with Interaction LO12-5 The interaction of driver and route is significant, the p-value is less than.05.

12-43 Analysis of Interaction: One-way ANOVAs for Each Route H 0 : For each route, the mean driver times are equal. H 1 : For each route, at least one pair of driver times are not equal. LO12-5