Power FifteenAnalysis of VarianceOne-Way ANOVAAdvertising Strategies & Weekly Sales for 20 WeeksSlide 5Slide 6Apple Juice Concentrate ANOVASlide 8One-Way ANOVA and RegressionSlide 10Slide 11Slide 12Slide 13Slide 14Slide 15Anova and Regression: One-Way InterpretationSlide 17Anova and Regression: One-Way Alternative Specification: Drop PriceSlide 19Slide 20Anova and Regression: One-Way Alternative SpecificationAnova and Regression: One-Way Alternative Specification, Drop QualitySlide 23Two-Way ANOVASlide 25Mean Weekly Sales By Strategy and MediumSlide 27Slide 28Is There Any Difference In Mean Sales Among the Six Cities?Slide 30Slide 31Slide 32Slide 33Slide 34Slide 35Slide 36F-Distribution TestsTwo-Way ANOVA and RegressionSlide 39Slide 40Slide 41ANOVA and Regression: Two-Way Series of Regressions; Compare to Table 11, Lecture 15Slide 43Slide 44Slide 45Slide 46ANOVA By DifferenceANOVA and Regression: Two-Way Series of RegressionsSlide 49Slide 501Power FifteenAnalysis of Variance (ANOVA)2Analysis of VarianceOne-Way ANOVA•Tabular•RegressionTwo-Way ANOVA•Tabular •Regression3One-Way ANOVAApple Juice Concentrate Example, Data File xm 15-01New productTry 3 different advertising strategies, one in each of three cities•City 1: convenience of use•City 2: quality of product•City 3: priceRecord Weekly Sales4Advertising Strategies & Weekly Sales for 20 WeeksConvenience Quality Price529 804 672658 630 531793 774 443- - -614 624 532Mean: 577.5 Mean: 653.0 Mean: 608.655Figure 1: Mean Apple Juice Sales By Advertising Strategy520540560580600620640660convenience quality priceA dvertising StrategyIs There a Significant Difference in Average Sales?Null Hypothesis, H0 : 2Alternative Hypothesis:323121,,6Table : -Way ANOVA of Apple Juice Sales By Advertising StrategySource of VariationSum of Squares Degrees ofFreedomMeanSquareExplained(betweentreatments)ESS = kj 1nj (xj-x)2k- ESS/(k-)Unexplained(withintreatments)USS = kj )(jni(xij - xj)2n-kUSS/(n-k)TotalTSS = kj )(jni(xij - x)2n-Fk-1, n-k = [ESS/(k-1)]/[USS/(n-k)]7Apple Juice Concentrate ANOVASource ofVariationSum ofSquaresDegrees ofFreedomMeanSquareExplained(BetweenTreatments)ESS=57,512.23k-1 = 2 ESS/(k-1)=28,756.12Unexplained(WithinTreatments)USS=506,984n-k = 57 USS/(n-k)=8894.45Total TSS=564,496n-1 = 59F2, 57 = 28,756.12/8894.45 = 3.2380.00.20.40.60.81.00 2 4 6 8 10F VariableDENSITYFigure 2: F-D istribution Density For 2 DOF, 57 DOFF-Distribution Test of the Null Hypothesis of No Difference in Mean Sales with Advertising StrategyF2, 60 (critical) @ 5% =3.159One-Way ANOVA and Regression10y(1)y(2)y(3)1 0 00 1 00 0 1Regression Set-Up: y(1) is column of 20 sales observationsFor city 1, 1 is a column of 20 ones, 0 is a column of 20 Zeros. Regression of a quantitative variable on three dummiesY = C(1)*Dummy(city 1) + C(2)*Dummy(city 2) + C(3)*Dummy(city 3) + e11Table 5: One-Way ANOVA Estimated Using RegressionDependent Variable: SALESAJMethod: Least SquaresSample: 1 60Included observations: 60Variable Coefficient Std. Error t-Statistic Prob.CONVENIENCE 577.5500 21.08844 27.38704 0.0000QUALITY 653.0000 21.08844 30.96483 0.0000PRICE 608.6500 21.08844 28.86178 0.0000R-squared 0.101882 Mean dependent var 613.0667Adjusted R-squared0.070370 S.D. dependent var 97.81474S.E. of regression 94.31038 Akaike info criterion 11.97977Sum squaredresid506983.5 Schwarz criterion 12.08448Log likelihood -356.3930 F-statistic 3.233041Durbin-Watsonstat1.525930 Prob(F-statistic) 0.046773One-Way ANOVA and RegressionRegression Coefficients are the City Means; F statisticDependent Variable: SALESAJMethod: Least Squares Sample: 1 60Included observations: 60Variable Coefficient Std. Error t-Statistic Prob. CONVENIENCE 577.5500 21.08844 27.38704 0.0000QUALITY 653.0000 21.08844 30.96483 0.0000PRICE 608.6500 21.08844 28.86178 0.0000R-squared 0.101882 Mean dependent var 613.0667 Adjusted R-squared 0.070370 S.D. dependent var 97.81474 S.E. of regression 94.31038 Akaike info criterion 11.97977 Sum squared resid 506983.5 Schwarz criterion 12.08448 Log likelihood-356.3930 Durbin-Watson stat 1.525930Regression Coefficients are the City Means; F statistic (?)1415Table 6: Test of the Null Hypothesis: All Treatment Means Are EqualWald Test:Equation: UntitledNullHypothesis:C(1)=C(3)C(2)=C(3)F-statistic 3.233041 Probability 0.046773Chi-square 6.466083 Probability 0.03943716Anova and Regression: One-WayInterpretationSalesaj = c(1)*convenience+c(2)*quality+c(3)*price+ eE[salesaj/(convenience=1, quality=0, price=0)] =c(1) = mean for city(1)•c(1) = mean for city(1) (convenience)•c(2) = mean for city(2) (quality)•c(3) = mean for city(3) (price)•Test the null hypothesis that the means are equal using a Wald test: c(1) = c(2) = c(3)Table 5: One-Way ANOVA Estimated Using RegressionDependent Variable: SALESAJMethod: Least SquaresSample: 1 60Included observations: 60Variable Coefficient Std. Error t-Statistic Prob.CONVENIENCE 577.5500 21.08844 27.38704 0.0000QUALITY 653.0000 21.08844 30.96483 0.0000PRICE 608.6500 21.08844 28.86178 0.0000R-squared 0.101882 Mean dependent var 613.0667Adjusted R-squared0.070370 S.D. dependent var 97.81474S.E. of regression 94.31038 Akaike info criterion 11.97977Sum squaredresid506983.5 Schwarz criterion 12.08448Log likelihood -356.3930 F-statistic 3.233041Durbin-Watsonstat1.525930 Prob(F-statistic) 0.046773One-Way ANOVA and RegressionRegression Coefficients are the City Means; F statistic18Anova and Regression: One-WayAlternative Specification: Drop PriceSalesaj = c(1) + c(2)*convenience+c(3)*quality+eE[Salesaj/(convenience=0, quality=0)] = c(1) = mean for city(3) (price, the omitted one)E[Salesaj/(convenience=1, quality=0)] = c(1) + c(2) = mean for city(1) (convenience)•so mean for city(1) = c(1) + c(2)•so mean for city(1) = mean for city(3) + c(2)•and so c(2) = mean for city(1) - mean for city(3)1920Anova and Regression: One-WayAlternative Specification: Drop PriceSalesaj = c(1) + c(2)*convenience+c(3)*quality+eE[Salesaj/(convenience=0, quality=0)] = c(1) = mean for city(3) (price, the omitted one)E[Salesaj/(convenience=1, quality=0)] = c(1) + c(2) = mean for city(1) (convenience)•so mean for city(1) = c(1) + c(2)•so mean for city(1) = mean for city(3) + c(2)•and so c(2) = mean for city(1) - mean for city(3)21Anova and Regression:
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