Pair-Wise Correlation Coefficients between Sales per Square Meter and Each of the Other Variables - Statistics Project Example

Statistics project Pair-wise correlation coefficients between sales per square meter and each of the other variables The data relates to the annual sales of 400 Dutch fashion stores during 1990. There are two data types on broad basis. One is discrete type of data which has only limited categories (maximum 10) whereas the continuous data can take any value within a range. In our study, all the variables are continuous in nature. The variables dealt in the data are: Sales per square metre which is referred by sales (also the target variable), Number of full timers which is referred by nfull, Number of part timers which is referred by npart, Total number of hours worked which is referred by hoursw and Sales floor space of the store which is referred by ssize. From the above data we would be finding the correlation coefficients of all the other variables with the variable ssize. The formula which is used to find the Karl Pearson’s correlation coefficient is as follows: where the term, is the covariance term and the term  is the standard deviation of x and the term is the standard deviation of y. The range of Karl Pearson’s correlation coefficient is from -1 to +1. If the correlation coefficient is nearing to -1, it denotes negative relation between the two variables x and y (if one increases the other decreases and vice versa), and if the correlation coefficient is nearing to +1, it denotes positive relation between the two variables x and y (if one increases the other also increases and if one decreases other also decreases). To find out the existence of any type of correlation roughly, we take the help of scatter diagram. Table A: Correlation coefficient between the variables   sales nfull npart hoursw ssize sales 1 0.237 0.050 0.263 -0.294 nfull 1 0.289 0.531 0.350 npart 1 0.249 0.366 hoursw 1 0.576 ssize 1 The individual correlation coefficients for all the other variables such as nfull, npart, hoursw and ssize with sales per square metre are presented in the following table. Table B: Coefficient of correlation for all the other variables with sales per sq. mtr.   sales nfull 0.237 npart 0.05 hoursw 0.236 ssize -0.294 The above table depicts that the highest positive correlation of sales per square metre with ssize is -0.294 (highest in magnitude despite being negative), followed by nfull, hoursw and finally npart. The correlation coefficient is tested through the test of significance procedure that under null hypothesis, H0: The correlation coefficient is not significant against the alternative hypothesis H1: The correlation coefficient is significant. The formula for testing the correlation coefficient is given by Student’s t-test using the following expression: to =  with (n-2) degrees of freedom where n is the sample size. If the correlation coefficient is significant, we conclude that the relationship between the two given variables is highly significant and influential otherwise we say that the correlation coefficient is not significant. In the present study, all the other variables are having highly significant correlation coefficients with sales except npart. Figure A: Graph showing the correlation between nfull and sales per sq. mtr The above graph clearly shows a positive correlation exist between nfull and sales. Figure B: Graph showing the correlation between npart and sales per sq. mtr The above graph clearly shows a very very low correlation exist between npart and sales. Figure C: Graph showing the correlation of nfull and npart with sales per sq. mtr Blue denotes nfull and red denotes npart Figure D: Chart showing the correlation between hoursw and sales per sq. mtr. The above graph clearly shows a good correlation exist between hoursw and sales. Figure E: Chart showing the correlation between ssize and sales per sq. mtr. The above graph clearly shows a negative correlation exist between ssize and sales. Figure F: Graph showing the correlation of hoursw and ssize with sales per sq. mtr Blue denotes boursw and red denotes ssize Equation representing a linear regression model, estimates the equation, the results, and overall goodness of the model The regression equation which explains the sales per square meter on hours worked on the sales floor space of the store is y = 5133.59 + 37.53 * x1- 22.14 * x2 where x1 denotes hoursw and x2 denotes ssize. Table C: Table depicting summary output of the Regression analysis Multiple R 0.605 R Square 0.365 Adjusted R Square 0.363 Standard Error 2985.37 Observations 400 Table D: Table depicting ANOVA of the regression analysis Source df Sum of Squares Mean Sum of Squares F ratio Significance F Regression 2 2040854395 1020427198 114.49 0.000 Residual 397 3538239209 8912441.334 Total 399 5579093605 Table E: Table depicting the regression coefficients and their significance Variables  Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept 5133.59 321.69 15.96 0.000 4501.15 5766.03 hoursw 37.5284 2.84 13.23 0.000 31.95 43.11 ssize -22.14 1.63 -13.63 0.000 -25.34 -18.95 From the table E which are the regression coefficients of sales per sq. mtr on hoursw (total hours worked) and ssize (sales floor space of the store), the prediction for the sales based on hoursw and ssize is y = 5133.59 + 37.53 * x1- 22.14 * x2 ----(a) where x1 refers to hoursw and x2 refers to ssize (refer the attached Excel file). The regression model given in the equation (a) is highly consistent and can be used to predict the sales based on the two independent variables hoursw and ssize since the coefficient of determination is 0.366 which emphasizes that 37% of the sales per sq. mtr is predicted through hoursw and ssize. Estimated coefficients from an economic perspective and their statistical significance The regression coefficient of the explained variable sales (refers to sales per sq. mtr.) on the explanatory variables (i) hoursw (refers to total hours worked) is 37.53 which says that there is a positive impact of sales on hoursw (total hours worked) and a (ii) ssize is -22.94 which says that there is a negative impact of sales on ssize (sales floor space of the store). The intercept on y axis is 5133.59 which is also more precise and can be used to predict sales per sq. mtr. This positive regression predicts that whenever the hoursw is high, the sales is also high and whenever the hoursw is low, the sales is also low, but whenever the ssize is high, the sales is low and whenever the ssize is low the sales is high (see the regression coefficient is negative). These regression relationship clearly emphasizes that whenever the total hours put in by the employees is high, the sales is also going to be high which is quite naturally acceptable, and when ssize is low the sales is going to be more and more. Full timers and part timers Inclusion of two more independent variables number of full timers which is referred by nfull and number of part timers which is referred by npart in the regression model: The summary output is given below: Table F: Table used to estimate sales based on the four variables Multiple R 0.634 R Square 0.402 Adjusted R Square 0.396 Standard Error 2905.82 Observations 400 Table G: Table of ANOVA using the regression analysis Source of variation df Sum of Squares Mean Sum of Squares F ratio Significance F Regression 4 2243801304 560950326.1 66.43 0.000 Residual 395 3335292300 8443777.98 Total 399 5579093605 Table H: Table depicting the regression coefficients of the regression equation Variables Coefficients Standard Error t Stat P-value Lower 95% Upper 95% intercept 3751.31 427.7 8.77 0.000 2910.46 4592.16 hoursw 32.76 3.06 10.70 0.000 26.75 38.78 ssize -23.91 1.65 -14.50 0.000 -27.15 -20.67 nfull 557.31 172.25 3.24 0.00131 218.67 895.94 npart 685 225.44 3.04 0.0025 241.78 1128.22 From the table H which is used to predict sales per sq. mtr based on four explanatory variables hoursw (total no. of hours worked), ssize (sales floor space of the store), nfull (no. of full timers) and npart (no. of part timers), the following regression equation is arrived which is highly precise and used to predict the sales per sq. mtr. y = 3751.31 + 32.76 * x1 – 23.91 * x2 + 557.31 * x3 + 685 * x4----(b) where x1 refers to hoursw, x2 refers to ssize, x3 refers to nfull and x4 refers to npart. (please see the attached Microsoft Excel file) The regression equation given in the equation (b) is highly precise due to the fact that the multiple correlation coefficient R is 0.634 (there is an increase in multiple correlation coefficient after the inclusion of the two new explanatory variables nfull and npart) and the coefficient of determination R2 is 0.402. At the outset, 40% of the explained variable viz. sales per sq. mtr is estimated by the explanatory variables hoursw (hours worked), ssize (sales floor space of the store), nfull (no. of full timers) and npart (no. of part timers). The sample size is large enough to have the regression coefficients significant despite it being less influential in case of small samples. Read More