MKT 317 February 12, 2010

AGENDA Quiz 1 F-Test Analysis of Variance (ANOVA)

F – Test Why do we use an F – Test? To examine the equality of variances in two populations for a business research question Two Tailed Test: H0: σ12 = σ22 H1: σ12 ≠ σ22 One Tailed Test: H0: σ12 ≤ σ22 H1: σ12 > σ22 *ALWAYS RIGHT TAILED*

Several Rules for F-Test One-tailed F-test Always write a right-tailed hypothesis Do not divide  by 2 when look for the critical value Two-tailed F-test Use the right tail in the test Divide  by 2 when look for the critical value Always: Name the population with the larger expected variance to be the population #1. Reject H0 if Fcalculated > Fcritical

F – Test (cont.) Larger Sample Variance = SL2 Smaller Sample Variance SS2 F calc = Degrees of Freedom for FCRITICAL: dfNUMERATOR: n of sample in numerator – 1 dfDENOMINATOR: n of sample in denominator – 1

Example for Comparison of Variance - 1 The graduate student union is preparing for the annual faculty versus graduate T.A. golf outing. The president of the union is worried about which team is more consistent (i.e. less variable in their scores). He has the scores from 25 out of the 50 times this challenge has happened. This is what he knows: Is there a difference in the variances of the scores? Use =0.10 Faculty n1 = 25 s1 = 9.2 strokes Graduate T.A.s n1 = 25 s1 = 4.5 strokes

Example for Comparison of Variance - 2 The marketing manager for Chrysler wants to test her hunch that women’s opinions of minivan safety varies more then men’s opinions. On a 100 point rating scale she surveyed 41 women and 31 men. The women had a standard deviation of 10.95 and the men had a standard deviation of 8.94. As her assistant you need to test this hunch and report the results. Use =0.05

ANOVA ANOVA lets you compare the means of multiple populations with a single test simultaneously. All populations must be normally distributed and have equal variances. Analysis of variance is a test for comparing means, not variances.

ANOVA Business Examples Financial Analysts would like to know the effect of the stock market (bull, bear, neutral market) on short term interest rates. Logistics people want to know the effect of the size of the shipping container (small, medium, large) on the cost of shipping , or the time it takes to ship it. Marketing people want to know the effect of store location (downtown, suburban, rural) on store sales. Accountants would like to know if the amount of accounts receivable is effected by the size of corporation (small, medium, large).

Hypotheses for ANOVA There is only one set of hypotheses for all ANOVA’s. H0: 1= 2= 3=…= r (where r is the number of populations [treatments] being compared) H1: “Not all population means are equal” OR “At least one population mean is different”

Important Notes The following hypothesis cannot be tested using ANOVA: H0: 1 2  3  …  r It takes only one mean being different from the rest to reject H0. The alternative hypothesis does not say “all means are different” OR H1: 1 2  3  …  r

ANOVA Logic Our goal is to test whether the population means of the treatments are equal to each other. ANOVA compares the between–group variance with the within-group variance. If the treatment means are different, the between-group variance should be much larger compared to the within-group variance. In fact, ANOVA simply consists of calculating the ratio of the two and measuring its significance.

Sum Squares Relationship between the three sum of squares: SST = SSTR + SSE SSTR between group variance SSE within group variance To be able to compare these variances, we need to take their averages. To take their averages, we divide each sum of squares with its respective degrees of freedom.

Mean Squares Mean Square Treatment (MSTR): Mean Square Error (MSE): r: Number of treatments. N: Total number of observations.

Conducting the Test MSTR Now that we have calculated the average variances, we are ready to calculate a test-statistic. Since it is a test of variances, it will be an F-test: MSTR F(r-1, N-r) = ———— MSE The smaller the between-group variance (MSTR), and thus the ratio (MSTR/MSE), the more likely the means are equal. The larger it is, the more likely that the means are not equal. This is always a right-tailed test. Reject when Fcalculated > Fcritical

ANOVA Table SST = SSTR + SSE N-1 = N-r + r-1

ANOVA EXAMPLE 1 The manager of a store wants to decide what kind of hand-knit sweaters to sell. The manager is considering three kinds of sweaters: Irish, Peruvian, and Shetland. The decision will depend on the results of an analysis of which kind of sweater, if any, lasts the longest before wearing out. There are 20 observations on Irish sweaters, 18 on Peruvian sweaters, and 21 on Shetland sweaters. The data are assumed to be independent random samples from the three populations of sweaters. The manager hires a statistician, who carries out an ANOVA and finds SSE = 1,240 and SSTR = 740. Construct a complete ANOVA table, and determine whether there is evidence to conclude that the three kinds of sweaters do not have equal average durability.

ANOVA EXAMPLE 2 Kevin Kelly is shopping for wellies on the Internet. He has found 400 different pairs of wellies for sale from companies in Ireland, England, China, and the U.S. (100 from each country). He is wondering if the average price for a pair of wellies is the same in each country. Here is what he knows: SSTR = 270 MSE = 9 Construct a complete ANOVA table and determine whether there is evidence to conclude that the four countries do not have equal average prices.