Copyright © 2008 by Nelson, a division of Thomson Canada Limited Chapter 18 Part 5 Analysis and Interpretation of Data DIFFERENCES BETWEEN GROUPS AND RELATIONSHIPS AMONG VARIABLES

LEARNING OBJECTIVES 1.To discuss reasons to conduct test of differences 2.To understand how the type of measurement scale influences the test of difference 3.To calculate a chi-square test for a contingency table 4.To understand the analysis of variance (ANOVA) tests for differences among three or more groups 5.To give examples of marketing questions that may be answered by analyzing the associations among variables 6.To discuss the concept of the simple correlation coefficient What you will learn in this chapter Copyright © 2008 by Nelson, a division of Thomson Canada Limited 18–1

LEARNING OBJECTIVES (cont’d) 7.To understand that correlation does not mean causation 8. To explain the concept of bivariate linear regression 9.To discuss why multivariate regression is an important tool for analysis What you will learn in this chapter Copyright © 2008 by Nelson, a division of Thomson Canada Limited 18–2

Test of DifferencesTest of Differences  An investigation of a hypothesis stating that two (or more) groups differ with respect to measures on a variable Cross-Tabulation Tables: The Chi-Square Test for Goodness of FitCross-Tabulation Tables: The Chi-Square Test for Goodness of Fit  Cross-tabulation (contingency table)  A joint frequency distribution of observations on two or more sets of variables  Chi-square test for a contingency table  A test that statistically analyzes significance in a joint frequency distribution Differences Between Groups Copyright © 2008 by Nelson, a division of Thomson Canada Limited 18–3

Cross-Tabulation Tables: The Chi-Square Test for Goodness of Fit (cont’d)Cross-Tabulation Tables: The Chi-Square Test for Goodness of Fit (cont’d)  Example of chi-square test χ² = chi-square statistic O i = observed frequency in the ith cell E i = expected frequency on the ith cell R i = total observed frequency in the ith row C j = total observed frequency in the jth column n = sample size Differences Between Groups Copyright © 2008 by Nelson, a division of Thomson Canada Limited 18–4

Differences Between Two Groups When Comparing MeansDifferences Between Two Groups When Comparing Means  The null hypothesis about differences between groups is normally stated as follows:  μ 1 = μ 2 or μ 1 – μ 2 = 0 Differences Between Groups (cont’d) Copyright © 2008 by Nelson, a division of Thomson Canada Limited 18–5

Differences Between Two Groups When Comparing ProportionsDifferences Between Two Groups When Comparing Proportions  Testing the null hypothesis that the population proportion for group 1 (π 1 ) equals the population proportion for group 2 (π 2 ) is conceptually the same as the t-test or Z-test of two means  The hypothesis which is H o : π 1 = π 2 may be restated as H o : π 1 - π 2 = 0 Differences Between Groups (cont’d) Copyright © 2008 by Nelson, a division of Thomson Canada Limited 18–6

Differences Among Three or More Groups When Comparing MeansDifferences Among Three or More Groups When Comparing Means  Analysis of Variance (ANOVA)  Analysis involving the investigation of the effects of one treatment variable on an interval-scaled dependent variable; a hypothesis-testing technique to determine whether statistically significant differences on means occur among three or more groups  If we have three groups or levels of the independent variable, the null hypothesis is stated as follows:  μ 1 = μ 2 = μ 3 Differences Between Groups (cont’d) Copyright © 2008 by Nelson, a division of Thomson Canada Limited 18–7

Relationship Among Variables Measure of AssociationMeasure of Association  A general term that refers to a number of bivariate statistical techniques used to measure the strength of a relationship between two variables Correlation AnalysisCorrelation Analysis  Correlation coefficient  A statistical measure of the covariation, or association, between two variables Copyright © 2008 by Nelson, a division of Thomson Canada Limited 18–8

Correlation and CausationCorrelation and Causation  It is important to remember that correlation does not mean causation Coefficient of Determination (r 2 )Coefficient of Determination (r 2 )  A measure obtained by squaring the correlation coefficient; that proportion of the total variance of a variable that is accounted for by knowing the value of another variable Relationship Among Variables (cont’d) Copyright © 2008 by Nelson, a division of Thomson Canada Limited 18–9 r 2 =Explained variance Total variance

Correlation MatrixCorrelation Matrix  The standard form for reporting correlational results Bivariate Linear RegressionBivariate Linear Regression  A measure of linear association that investigates a straight-line relationship of the type Y = α + βX, where X is the independent variable and α and β are two constants to be estimated Relationship Among Variables (cont’d) Copyright © 2008 by Nelson, a division of Thomson Canada Limited 18–10

Relationship Among Variables (cont’d) Bivariate Linear Regression (cont’d)Bivariate Linear Regression (cont’d) Copyright © 2008 by Nelson, a division of Thomson Canada Limited 18–11

Drawing a Regression LineDrawing a Regression Line Relationship Among Variables (cont’d) Copyright © 2008 by Nelson, a division of Thomson Canada Limited 18–12 Exhibit 18.2 LEAST-SQUARES REGRESSION LINE

Drawing a Regression Line (cont’d)Drawing a Regression Line (cont’d) Relationship Among Variables (cont’d) Copyright © 2008 by Nelson, a division of Thomson Canada Limited 18–13 X Y Total Deviation Deviation explained by regression Deviation not explained by regression

Test of Statistical SignificanceTest of Statistical Significance  F-test (regression)  A procedure to determine whether more variability is explained by the regression or unexplained by the regression Relationship Among Variables (cont’d) Copyright © 2008 by Nelson, a division of Thomson Canada Limited 18–14 Total deviation = Deviation explained by the regression Deviation unexplained by the regression (Residual error) +

Test of Statistical Significance (cont’d)Test of Statistical Significance (cont’d)  Example Copyright © 2008 by Nelson, a division of Thomson Canada Limited 18–15 SS t = SS r + SS e Relationship Among Variables (cont’d)

Multiple Regression AnalysisMultiple Regression Analysis  An analysis of association in which the effects of two or more independent variables on a single, interval- scaled dependent variable are investigated simultaneously Y= a +b 1 X 1 +b 2 X 2 +b 3 X 3...+b n X n Multiple Regression Analysis Copyright © 2008 by Nelson, a division of Thomson Canada Limited 18–16

Multiple Regression Analysis (cont’d)Multiple Regression Analysis (cont’d)  F-test d.f. for the numerator = k d.f. for the denominator = n - k - 1 Copyright © 2008 by Nelson, a division of Thomson Canada Limited 18–17 Multiple Regression Analysis (cont’d)