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Chapter Fourteen Examining Associations Correlation and Regression Did You Know that Degree Color and Race Make a Difference in Home Refinancing A study found that broker fees for purchasers without a college degree pay 1 472 more than those with a college degree No only did a degree matter but race was also a factor African Americans on average paid 500 more than whites Hispanics 275 more than whites Regression analysis was used to determine whether various borrower characteristics had a bearing on the amount of broker fees and closing costs paid Did You Know that Disaster Area Declarations are Related to Electoral Votes A study revealed states that had been declared disaster areas are crucial to presidential elections Regression analysis revealed that states that are likely to be declared disaster areas were the states that were highest in electoral votes Did You Know that the Presence of an NFL Team Boost Rental Costs Regression analysis has revealed that in cities with an NFL team rental costs for apartment in the central city area were 8 percent higher than in cities without an NFL team Property tax receipts were also found to be higher in cities with NFL teams Overview of Techniques for Examining Associations Spearman Correlation Coefficient Technique The technique is appropriate when The degree of association between two sets of ranks pertaining to two variables is to be examined Illustrative research question s this technique can answer Is there a significant relationship between motivation levels of salespeople and the quality of their performance Assume that the data on motivation and quality of performance are in the form of ranks say 1 through 20 for 20 salespeople who were evaluated subjectively by their supervisor on each variable Overview of Techniques for Examining Associations Cont d Pearson Correlation Coefficient Technique This technique is appropriate when The degree of association between two metric scaled interval or ratio variables is to be examined Illustrative research question s this technique can answer Is there a significant relationship between customers age measured in actual years and their perceptions of our company s image measured on a scale of 1 to 7 Overview of Techniques for Examining Associations Cont d Simple Regression Analysis Technique This technique is appropriate when A mathematical function or equation linking two metric scaled interval or ratio variables is to be constructed under the assumption that values of one of the two variables is dependent on the values of the other Overview of Techniques for Examining Associations Simple Regression Analysis Cont d Illustrative Research Question s this Technique Can Answer Are sales measured in dollars significantly affected by advertising expenditures measured in dollars What proportion of the variation in sales is accounted for by variation in advertising expenditures How sensitive are sales to changes in advertising expenditures Overview of Techniques for Examining Associations Cont d Multiple Regression Analysis Technique This technique is appropriate Under the same conditions as simple regression analysis except that more than two variables are involved wherein one variable is assumed to be dependent on the others Overview of Techniques for Examining Associations Cont d Illustrative Research Question s this Technique Can Answer Are sales significantly affected by advertising expenditures and price where all three variables are measured in dollars What proportion of the variation in sales is accounted for by advertising and price How sensitive are sales to changes in advertising and price Spearman Correlation Coefficient A Spearman correlation coefficient is a measure of association between two sets of ranks n 6 d2 i i 1 rs 1 n n2 1 di the difference between the ith sample unit s ranks on the two variables n the total sample size Example Industrial Marketing Firm An industrial marketing firm has been hiring all its salespeople from among the graduates of 10 business schools in the vicinity of its headquarters The firm developed a subjective ranking of the perceived prestige levels of the 10 schools and the performance levels of the groups of graduates recruited from these schools Question What is the degree of association between the prestige levels of the schools and the sales performance levels of their graduates hired by this company Table 14 2 Association Between School Prestige and Performance of Graduates Results First step is to calculate the Spearman Correlation Coefficient rs 1 6 56 1 339 661 10 100 1 The result is rs 661 The next step is to calculate the t distrubtion Spearman Correlation Co efficient Hypotheses H0 s 0 Ha s 0 t Distribution t rs n 2 1 rs2 2 49 For 05 t for 8 degrees of freedom d f n 2 10 2 8 tc 2 31 and 2 31 Decision Rule Reject H0 if t 2 31 or if t 2 31 Since t 2 31 we reject H0 and conclude that there is a true association between the prestige of business schools and the job performance of its graduates In other words the sample correlation of 661 is unlikely to have occurred because of chance Pearson Correlation Coefficient The Pearson correlation coefficient is the degree of association between variables that are interval or ratio scaled Pearson correlation coefficient rxy between them is given by rxy n Xi X Yi Y i 1 n 1 sx sy n sample size total number of data points X and Y means Xi and Yi values for any sample unit i sx and sy standard deviations Table 14 3 Bright Detergent Data Scatter Diagram Plot in a two dimensional graph Indicates how closely and in what fashion the variables are associated Exhibit 14 1 Scatter Diagram of Sales and Advertising Data What is the relationship between dollar sales and advertising expenditure Exhibit 14 2 Scatter Diagram of Sales and Number of Competing Brands What is the relationship between dollar sales and number of competing detergents Pearson Correlation Correlation between sales and advertising is 927 Correlation between sales and number of competing brands is 910 Two Tailed Hypothesis Test For Correlations H0 0 Ha 0 For 05 19 degrees of freedom d f n 1 19 rc 433 and rc 433 Decision rule is Reject H0 if r 433 or if r 433 Reject H0 in both cases Exhibit 14 3 Scatter Diagram Showing a Nonlinear Association Between Variables National Insurance Company Computing Pearson Correlation Among Service Quality Constructs National Insurance Company was interested in the correlations between respondents overall service quality