SPSS Correlation Analysis and Multiple Regression - Case Study Example

SPSS Project Produce the Pearson correlation matrix for all variables except the of the printer. Correlations Price(£) Text Speed (seconds per page) Text Cost (pence per 10 page) Color Photo Time (min) Color Photo Cost (£) Price (£) Pearson Correlation 1 -.383 -.501 -.518* -.263 Sig. (2-tailed) .158 .057 .048 .343 N 15 15 15 15 15 Text Speed (seconds per page) Pearson Correlation -.383 1 .030 .035 .224 Sig. (2-tailed) .158 .916 .900 .422 N 15 15 15 15 15 Text Cost (pence per 10 page) Pearson Correlation -.501 .030 1 .408 .606* Sig. (2-tailed) .057 .916 .131 .017 N 15 15 15 15 15 Color Photo Time (min) Pearson Correlation -.518* .035 .408 1 .081 Sig. (2-tailed) .048 .900 .131 .773 N 15 15 15 15 15 Color Photo Cost (£) Pearson Correlation -.263 .224 .606* .081 1 Sig. (2-tailed) .343 .422 .017 .773 N 15 15 15 15 15 *. Correlation is significant at the 0.05 level (2-tailed). (a) What are the two strongest relationships shown in the Pearson correlation matrix? State in each case what the relationship is, and whether it is as one would expect. SOLUTION The two strongest relationships are; Relationship between Color Photo Cost (£) and Text Cost (pence per 10 page); the Pearson correlation coefficient is given as 0.606; it represents a positive linear relationship, that is, an increase in the variable Color Photo Cost (£) results to a subsequent increase in the variable Text Cost (pence per 10 page) Relationship between Price (£) and Color Photo Time (min); the Pearson correlation coefficient is given as -0.518; it represents a negative linear relationship, that is, an increase in the variable price (£) results to a subsequent decrease in the variable Color Photo Time (min). (b) If we are trying to model the relationship between Price and the other variables, from these correlations, which variables should be considered for inclusion in the model? SOLUTION In modelling the relationship between Price and the other variables, from these correlations, the variable on Color Photo Time (min) would be considered at 5% significance level. However, at 10% significance level, the variable Text Cost (pence per 10 page) would have to be considered in the model; therefore at 10% significance level two variables would be considered. (c) SPSS offers a variety of methods (e.g. Forwards Backwards and Stepwise) for fitting regression models involving more than one predictor variable. Explain briefly the purpose of these methods. Perform a Regression analysis to predict the price variable from all the other numeric variables, using the Enter method. SOLUTION SPSS used methods such as Enter, Stepwise, Remove, Backward and Forward. Stepwise linear regression is a method used to regress multiple variables while simultaneously removing those that arent important; forward refers to adding one variable at a time while backward refers to removing one variable at a time. The table below presents the regression analysis using the Enter method; Model Summaryb Model R R Square Adjusted R Square Std. Error of the Estimate Durbin-Watson 1 .710a .504 .305 63.783 3.066 a. Predictors: (Constant), Color Photo Cost (£), Color Photo Time (min), Text Speed (seconds per page), Text Cost (pence per 10 page) b. Dependent Variable: Price (£) ANOVAb Model Sum of Squares df Mean Square F Sig. 1 Regression 41290.309 4 10322.577 2.537 .106a Residual 40683.024 10 4068.302 Total 81973.333 14 a. Predictors: (Constant), Color Photo Cost (£), Color Photo Time (min), Text Speed (seconds per page), Text Cost (pence per 10 page) b. Dependent Variable: Price (£) Coefficientsa Model Unstandardized Coefficients Standardized Coefficients t Sig. Collinearity Statistics B Std. Error Beta Tolerance VIF 1 (Constant) 333.082 68.589 4.856 .001 Text Speed (seconds per page) -23.659 14.389 -.381 -1.644 .131 .924 1.082 Text Cost (pence per 10 page) -7.720 6.020 -.408 -1.282 .229 .490 2.041 Color Photo Time (min) -6.477 4.716 -.346 -1.373 .200 .784 1.276 Color Photo Cost (£) 17.711 54.173 .098 .327 .750 .555 1.801 a. Dependent Variable: Price (£) (d) What percentage of the variation in APR is accounted for by the four variables? SOLUTION From the regression table, the value of R-squared is given as 0.504; this implies that 50.4% of variation in the price is explained by the four variables in the model. (e) What are the null hypotheses tested by each of the sig values in your output? What conclusions should be drawn from these results? Perform another Regression analysis to predict the price variable from all the other numeric variables, using the Backward method. SOLUTION The null hypothesis are; Conclusions From the table above, it is evident that it is only the constant coefficient is significant at either 1%, 5% or 10% significance levels. All the four variables are insignificant at either 1%, 5% or 10% significance levels. Regression analysis using the Backward method Model Summary Model R R Square Adjusted R Square Std. Error of the Estimate 1 .710a .504 .305 63.783 2 .706b .498 .362 61.139 3 .633c .401 .302 63.952 4 .518d .268 .212 67.947 a. Predictors: (Constant), Color Photo Cost (£), Color Photo Time (min), Text Speed (seconds per page), Text Cost (pence per 10 page) b. Predictors: (Constant), Color Photo Time (min), Text Speed (seconds per page), Text Cost (pence per 10 page) c. Predictors: (Constant), Color Photo Time (min), Text Speed (seconds per page) d. Predictors: (Constant), Color Photo Time (min) ANOVAe Model Sum of Squares df Mean Square F Sig. 1 Regression 41290.309 4 10322.577 2.537 .106a Residual 40683.024 10 4068.302 Total 81973.333 14 2 Regression 40855.475 3 13618.492 3.643 .048b Residual 41117.858 11 3737.987 Total 81973.333 14 3 Regression 32895.662 2 16447.831 4.022 .046c Residual 49077.672 12 4089.806 Total 81973.333 14 4 Regression 21954.333 1 21954.333 4.755 .048d Residual 60019.000 13 4616.846 Total 81973.333 14 a. Predictors: (Constant), Color Photo Cost (£), Color Photo Time (min), Text Speed (seconds per page), Text Cost (pence per 10 page) b. Predictors: (Constant), Color Photo Time (min), Text Speed (seconds per page), Text Cost (pence per 10 page) c. Predictors: (Constant), Color Photo Time (min), Text Speed (seconds per page) d. Predictors: (Constant), Color Photo Time (min) e. Dependent Variable: Price (£) Coefficientsa Model Unstandardized Coefficients Standardized Coefficients t Sig. B Std. Error Beta 1 (Constant) 333.082 68.589 4.856 .001 Text Speed (seconds per page) -23.659 14.389 -.381 -1.644 .131 Text Cost (pence per 10 page) -7.720 6.020 -.408 -1.282 .229 Color Photo Time (min) -6.477 4.716 -.346 -1.373 .200 Color Photo Cost (£) 17.711 54.173 .098 .327 .750 2 (Constant) 342.854 59.174 5.794 .000 Text Speed (seconds per page) -22.379 13.271 -.360 -1.686 .120 Text Cost (pence per 10 page) -6.456 4.424 -.341 -1.459 .172 Color Photo Time (min) -6.852 4.385 -.366 -1.563 .146 3 (Constant) 318.382 59.359 5.364 .000 Text Speed (seconds per page) -22.702 13.880 -.366 -1.636 .128 Color Photo Time (min) -9.457 4.189 -.505 -2.257 .043 4 (Constant) 239.800 37.039 6.474 .000 Color Photo Time (min) -9.700 4.448 -.518 -2.181 .048 a. Dependent Variable: Price (£) Excluded Variablesd Model Beta In t Sig. Partial Correlation Collinearity Statistics Tolerance 2 Color Photo Cost (£) .098a .327 .750 .103 .555 3 Color Photo Cost (£) -.148b -.629 .542 -.186 .944 Text Cost (pence per 10 page) -.341b -1.459 .172 -.403 .834 4 Color Photo Cost (£) -.222c -.929 .371 -.259 .993 Text Cost (pence per 10 page) -.348c -1.385 .191 -.371 .834 Text Speed (seconds per page) -.366c -1.636 .128 -.427 .999 a. Predictors in the Model: (Constant), Color Photo Time (min), Text Speed (seconds per page), Text Cost (pence per 10 page) b. Predictors in the Model: (Constant), Color Photo Time (min), Text Speed (seconds per page) c. Predictors in the Model: (Constant), Color Photo Time (min) d. Dependent Variable: Price (£) (f) Explain what has happened in the sequence of model fitting. SOLUTION In the sequence of model fitting, one variable is dropped from the preceding model. For instance, in model (1) we have the four variables included, in model (2) we have three variables included, model (3) we have two variables included and lastly in model (4) we have only one variable included. (g) Using the final model in your output, on average how much does it cost to reduce the time taken to print a text page by one second? SOLUTION From the final model; it can be observed that a total of £ 23.659 would be spent to reduce the time taken to print a text page by one second. (h) Using the final model, on average how much does it cost to reduce the time taken for a colour photograph by one minute? SOLUTION From the final model; it can be observed that a total of £ 6.477 would be spent to reduce the time taken for a colour photograph by one minute. (i) Looking at the excluded variables in the final model, are any of them almost included in the model? Explain your reasoning. SOLUTION Indeed from the final model it can be observed that one of the variables included in the model seems to be irrelevant. This can be explained by the fact that in model (2) where we had three variables included in the model had a value adjusted R-squared given as 0.362. In model (1) where all the four variables included in the model had a value of adjusted R-squared given as 0.305. In essence we expect the value of adjusted R-squared to increase by adding a relevant variable into the model; this wasn’t the case. Therefore, we can easily conclude that the variable Color Photo Cost (£) is irrelevant. Read More