# Shown below are scatter plots of each of these

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figures with the predictors of average hours worked per week and number of customers. Shown below are scatter plots of each of these predictors with sales. Note the number of customers has a slight linear shape but is more like the upper left quadrant in Tukey’s 4- quadrant approach. Hours per week produces a graph that is quite scattered. In an effort to examine each of these predictors together and with squared and interaction terms included, a “full” quadratic model was develop. The results follow. Note that none of the predictors are significant as measured by the t test for slope. Yet the R 2 value is over 90%. The stepwise regression analysis that follows reveals that there is significant interaction between the two predictors. The interaction predictor variable accounts for over 68% of the variance of sales by itself. In combination with the interaction variable, the variable, squared number of customers is also significant bringing the total R 2 to over 88% with just these two variables. This analysis demonstrates how predictors by themselves may not be significant by that the interaction of 2 or more predictors might be.

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Case Notes 43
Case Notes 44 Regression Analysis The regression equation is Sales = 41 - 2.22 Hrs/Wk + 0.67 No.Custs + 0.0316 HrsSq - 0.0196 CustSq + 0.0093 IntHrCus Predictor Coef Stdev t-ratio p Constant 41.5 117.9 0.35 0.739 Hrs/Wk -2.221 5.099 -0.44 0.681 No.Custs 0.668 3.373 0.20 0.851 HrsSq 0.03163 0.05644 0.56 0.599 CustSq -0.01965 0.02981 -0.66 0.539 IntHrCus 0.00933 0.06959 0.13 0.899 s = 1.097 R-sq = 90.1% R-sq(adj) = 80.3% Analysis of Variance SOURCE DF SS MS F p Regression 5 55.026 11.005 9.15 0.015 Error 5 6.016 1.203 Total 10 61.042 Stepwise Regression F-to-Enter: 4.00 F-to-Remove: 4.00 Response is Sales on 5 predictors, with N = 11 Step 1 2 Constant 7.089 3.139 IntHrCus 0.0115 0.0247 T-Ratio 4.41 6.27 CustSq -0.0132 T-Ratio -3.71 S 1.46 0.942 R-Sq 68.41 88.38

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Case Notes 45 3. Shown below is a MINITAB scatter plot of sales and number of employees. Notice how the graph rises and then levels out. This fits fairly closely with the upper left quadrant of Tukey’s 4-quadrant approach. From this, we attempted to use the log of number of employees as a second predictor. The result is shown below in a stepwise regression analysis. Stepwise Regression F-to-Enter: 4.00 F-to-Remove: 4.00 Response is Sales2 on 2 predictors, with N = 10 Step 1 2 Constant -94.68 -883.10 logEmpl 25.40 220.40 T-Ratio 3.58 4.79 No.Empl. -1.24 T-Ratio -4.26 S 3.89 2.19 R-Sq 61.57 89.29 Notice that log of number employees enters the analysis first and accounts for almost 62% of the variation of sales. At the second step, number employees enters the process adding another 28%. It was likely worthwhile to recode the data using logs. According to this, sales increases are associated with the log of number employees and not as much with the straight linear increase. The company should exercise caution in merely hiring more people as sales increase.
Case Notes 46 Chapter 16 DeBourgh Manufacturing Company 1. The decomposition analysis shows several things. A study of the original data shows that there is a general upward trend. This is underscored by the graph showing the trend line fit through the data. There appear to be important seasonal effects in these data. Note that for January, February, and March, the seasonal indices are less than 100.

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