Existing Relationships between the Response Variable: Sales and Four Independent Variables - Assignment Example

Statistics Project Calculate the pair-wise correlation coefficients between sales per square meter and each of the other variables and test their statistical significance. Produce scatter plots for each pair of variables. Provide a written interpretation for each of the correlation coefficients and the related scatter plots. ANSWERS In this section we present the results on correlation between the dependent variable and the explanatory variables. Correlation refers to the relationship that exists between two variables (Lopez-Paz , et al., 2013). The relationship could either be positive or negative (Aldrich, 1995). The Pearson correlation coefficient is used to measure the strength of association and the values ranges from negative one to positive one (Nikolić, et al., 2012). A coefficient of negative one implies that the correlation between the two variables is perfectly negatively linearly correlated while a coefficient of positive one implies that the correlation between the two variables is perfectly positively linearly correlated (Mahdavi , 2013). On the other hand, a coefficient of zero means that there is no correlation between the two variables (Anscombe, 1973). The table below presents a correlation matrix. We observe that sales has a statistically significant correlation with three out four variables presented. For instance, there is a significant positive relationship between sales and number of full-timers though the power of correlation seems to be weak. There is also a positive statistically significant relationship between sales and the number of hours worked; however, just like in the case of full timers, the correlation is quite weak. Lastly, there is a statistically significant relationship between sales and Sales floor space of the store. The relationship however unlike the other two mentioned above is a weak negative relationship. It is only the number of part-timers that didn’t have a statistically significant relationship with the sales as a variable. . pwcorr sales nfull npart hoursw ssize, sig | sales nfull npart hoursw ssize -------------+--------------------------------------------- sales | 1.0000 | | nfull | 0.2372 1.0000 | 0.0000 | npart | 0.0501 0.2888 1.0000 | 0.3177 0.0000 | hoursw | 0.2630 0.5313 0.2491 1.0000 | 0.0000 0.0000 0.0000 | ssize | -0.2938 0.3497 0.3665 0.5759 1.0000 | 0.0000 0.0000 0.0000 0.0000 Figure 1: Scatterplot of sales versus number of full-timers The above figure represents a scatterplot between sales and number of full-timers; the graph shows a somewhat a positive linear relationship between the two variables (sales and number of full-timers). This is to mean, as the number of full timers increase so does the sales. Figure 2: Scatterplot of sales versus number of part-timers In figure 2, we present a scatterplot of sales versus number of part-timers; we observe some sort of negative relationship between the two variables (sales and number of part-timers). This tries to explain that as the number of part-timers increase we observe a subsequent decrease in the sales made. Figure 3: Scatterplot of sales versus number of hours worked In figure 3, we present a scatterplot of sales versus number of hours worked; we observe a positive relationship between sales and number of hours worked. That is to say, as the number of hours worked increases so does the sales. Figure 4: Scatterplot of sales versus Sales floor space of the store In figure 4 we observe a negative linear relationship between sales and Sales floor space of the store (ssize). As the size of the floor increases we would expect a decrease in the number os sales made. 2) Write down an equation representing a linear regression model in which sales per square metre depend on a constant, the total number of hours worked and floor space of the store (in square metres). Estimate the equation, report the results, and comment on the overall goodness of the model. ANSWERS In this case we expect to have an OLS model; the equation of the model (OLS) is; Where represents the coefficient of the constant/intercept represents the coefficient of number of hours worked represents the coefficient of Sales floor space of the store . reg sales hoursw ssize Source | SS df MS Number of obs = 400 -------------+---------------------------------------------- F( 2, 397) = 114.49 Model | 2.0409e+09 2 1.0204e+09 Prob > F = 0.0000 Residual | 3.5382e+09 397 8912441.34 R-squared = 0.3658 -------------+---------------------------------------------- Adj R-squared = 0.3626 Total | 5.5791e+09 399 13982690.7 Root MSE = 2985.4 ---------------------------------------------------------------------------------------------------------- sales | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+-------------------------------------------------------------------------------------------- hoursw | 37.52842 2.83722 13.23 0.000 31.95056 43.10627 ssize | -22.14457 1.625067 -13.63 0.000 -25.33938 -18.94976 _cons | 5133.59 321.6934 15.96 0.000 4501.155 5766.026 ----------------------------------------------------------------------------------------------------------- The table above represents the OLS regression results. Based on the results, the equation is given as; We check the appropriateness/fitness of the model by checking the F-statistic. From the table, the p-value for the F-statistic is given as 0.000; this is a value which is by far less than 5% significance level; we thus reject the null hypothesis and conclude that the model is significant at 5% significance level. That is to say, the model is fit/appropriate at 5% significance level. However, looking at the value of adjusted R-Squared, we observe a very a small figure of 0.3658; this means that only 36.58% of the variation in the dependent variable (ssize) is explained by the two explanatory variables in the model. The rest (63.42%) of the variation is probably explained by other factors not included in this model. 3) Interpret the estimated coefficients from an economic perspective and comment on their statistical significance. ANSWERS Looking at the regression table given above, we see the coefficient for the number of hours (hoursw) to be given as 37.52842; the value is positive, this shows for any unit increase in the explanatory variable (hoursw) we would expect an increase in the dependent variable (sales) this indicates that for any unit change in hoursw, the dependent variable (sales) as well; and that if we increase the explanatory variable by one unit then the dependent variable will increase by 37.52842. This is true for decreasing the explanatory variable as well. Lastly, considering the coefficient of the ssize which is given as -22.14457; this shows for any unit increase in the explanatory variable (ssize) we would expect an decrease in the dependent variable (sales) this indicates that for any unit change in ssize, the dependent variable (sales) as well; and that if we increase the explanatory variable by one unit then the dependent variable will decrease by 22.14457. The vice versa is true. 4) Does the inclusion of the number of full-timers and part-timers significantly improve the model? ANSWERS To find out this, we need to include the two explanatory variables (full-timers and part-timers) into the model. The new equation model becomes; To get this we estimate the model just like in (2) above. The results are shown in the table below; . reg sales hoursw ssize nfull npart Source | SS df MS Number of obs = 400 -------------+--------------------------------------------- F( 4, 395) = 66.43 Model | 2.2438e+09 4 560950319 Prob > F = 0.0000 Residual | 3.3353e+09 395 8443777.96 R-squared = 0.4022 -------------+--------------------------------------------- Adj R-squared = 0.3961 Total | 5.5791e+09 399 13982690.7 Root MSE = 2905.8 ----------------------------------------------------------------------------------------------------------- Sales | Coef. Std. Err. t P>|t| [95% Conf. Interval] -------------+-------------------------------------------------------------------------------------------- hoursw | 32.76485 3.061793 10.70 0.000 26.74541 38.7843 ssize | -23.9079 1.648595 -14.50 0.000 -27.14901 -20.66678 nfull | 557.3077 172.2475 3.24 0.001 218.6712 895.9442 npart | 684.9984 225.4428 3.04 0.003 241.7805 1128.216 _cons | 3751.311 427.696 8.77 0.000 2910.466 4592.156 ------------------------------------------------------------------------------------------------------------ Looking at the regression table presented above, we see the value of adjusted R-squared is given as 0.3961 this is when the number of full-timers and part-timers have been included into the model while the same value of R-squared is given as 0.3626 when the number of full-timers and part-timers have not been included; this shows that inclusion number of full-timers and part-timers increases the value of adjusted R-squared from 0.3626 to 0.3961 by a factor of 0.0335. From this values we can therefore conclude that inclusion of the number of full-timers and part-timers significantly improves the model otherwise the value of adjusted R-Squared would have remained the same or would have decreased if the variables were irrelevant. Conclusion The aim of this paper was to explore and investigate the existing relationships between the response variable named sales and other four independent variables (that is, Number of full-timers, Number of part-timers, Total number of hours worked and Sales floor space of the store (in square metres) ). That is, we looked at the correlations and the multiple regression analysis. The results showed that the dependent variable (sales) has relationship with all the independent variables given in the model. However, three explanatory variables (Number of full-timers, total number of hours worked and Sales floor space of the store (in square metres)) had a statistically significant relationship with the dependent variable (sales). There was no statistically significant relationship between sales and number of part timers. Sales floor space of the store (in square metres) had a statistically negative relationship with the response variable (sales). The other two independent variables had a positive linear relationship with the dependent variable. In the regression table, we see that the model is fit (that is, it is appropriate for modelling) at 5% significance level. However, it seems a number of relevant variables are left out of the model since it is clear that only 36.58% of the change in the response variable (sales) is explained by the independent variables; the remaining 63.42% is possibly explained by the error term. In the first regression model, we only included two explanatory variables and the value of adjusted R-squared is given as 0.3626. In the second regression model, we include the two explanatory variables (full-timers and part-timers) into the model and the value of adjusted R-squared becomes 0.3961; this shows an increase in the value of adjusted R-squared hence showing that the two variables added in the model were relevant and ought not to have been left out in the model. References Aldrich, J., 1995. Correlations Genuine and Spurious in Pearson and Yule. Statistical Science, 10(4), pp. 364-376. Anscombe, F. J., 1973. Graphs in statistical analysis. The American Statistician, 27(5), pp. 17-21. Lopez-Paz , D., Hennig, P. & Schölkopf , B., 2013. The Randomized Dependence Coefficient. Conference on Neural Information Processing Systems, 5(4), pp. 189-197. Mahdavi , D. B., 2013. The Non-Misleading Value of Inferred Correlation: An Introduction to the Cointelation Model. Wilmott Magazine, 7(3), pp. 45-52. Nikolić, D., Muresan, R. C., Feng, W. & Singer, W., 2012. Scaled correlation analysis: a better way to compute a cross-correlogram. European Journal of Neuroscience, 5(1), pp. 1-21. Read More