Pair-Wise Correlation Coefficients between Sales per Square Meter - Statistics Project Example

Introduction to Statistics Introduction In organizations, both the physical infrastructure and human resources play a crucial and complimenting role. Speaking of organizational infrastructure, it has to facilitate the streamlining of operational processes and as to support as well as motivate the employees. As Townsend (2006) states “organization infrastructure links goals, activities and people through planned processes and systems”. In that direction, every aspect of organization infrastructure from rooms, cabins, hallway, and others have to be designed and operated in an effective manner. When the organizations’ infrastructure or environment is organized aptly, it will positively influence the employees. Employees are the crucial “cog” for the organizational functioning and success. This significance of employees was put forward by Mayhew (2014) who stated that the objective of any organization is profitability; and that profitability and thereby organizations success depends on the employees performance, with poor performance by the employees being detrimental to the companys success. Employees work in an organization on full-time basis as well as short-term basis. Although, full-time employees are the majority in any organization, employment of short-term employees are also on the rise. “The use of temporary workers is growing rapidly, with the number of companies using temporary workers on the increase as global competition increased and the urge to cut down on costs of undertaking businesses in order to remain competitive rises” (Wandera 2011). This role of both full-time and short-term workers brings in focus the number of hours they contribute to the organization (Simeon 2013). So, the report will focus on the data collected from 400 fashion stores located in the Netherlands thereby discussing those stores’ infrastructure, employees including full-timers and part-timers, the hours contributed by them and others. Pair-wise correlation coefficients between sales per square meter As above-mentioned, the data is regarding the study of direct annual sales of 400 Dutch fashion stores in the year 1990. The quantitative variables used are: Total Sales (tsales), Sales per square meter (sales), Number of full-times (nfull), Number of part-times (npart), Total number of hours worked (hoursw) and Sales floor space of the store in square metres (ssize). Since all of them are quantitative variables, the Karl Pearson correlation coefficient for continuous variables is calculated and tested for its significance. “Karl Pearson correlation coefficient measures quantitatively the extent to which two variables x and y are correlated” (Sharma 2012, p.454) The formula for Karl Pearson correlation coefficient is where the numerator in the above expression is called Sum product of x and y and the denominator is sum of squares of x and y respectively. As per the rules, correlation coefficient always lies between -1 and +1. Correlation coefficient nearing to 1 represents highly positive correlation and correlation coefficient nearing to -1 represents highly negative correlation between the two variables of interest. Scatter diagram is used to depict the correlation coefficient and the nature of the relationship exists between the two random variables (continuous). When the data is run through MS Excel, it gave the following results: Table 1: Table showing correlation coefficient between all the variables   tsales Sales nfull npart hoursw ssize tsales 1 sales 0.470 1 nfull 0.565 0.237 1 npart 0.391 0.050 0.289 1 hoursw 0.709 0.263 0.531 0.249 1 ssize 0.534 -0.294 0.350 0.366 0.576 1 The pair-wise correlation coefficients between sales per square meter and each of the variables is given below: Table 2: Table showing correlation coefficient between sales per sq. mtr. and other variables   tsales nfull npart hoursw ssize Sales 0.47 0.237 0.05 0.236 -0.294 Results: The correlation coefficient between sales and tsales is 0.47 The correlation coefficient between sales and nfull is 0.237 The correlation coefficient between sales and npart is 0.05 The correlation coefficient between sales and hoursw is 0.263 The correlation coefficient between sales and ssize is -0.294. To test the significance of correlation coefficient, we formulate Null Hypothesis H0: The observed correlation coefficient is not significant Vs. Alternative Hypothesis H1: The observed correlation coefficient is significant Level of significance: 5% level or α=0.05. The significance of the correlation coefficient is given by the test statistic to=------(1) which follows t distribution with n-2 d.f. Table 2A: Table showing the test statistic value and critical value for the correlations of various variables with sales per sq. mtr. done with MS Excel Data Analysis pack Variables Correlation Test statistic df Critical value tsales 0.46988817 12.0305161 398 1.966 nfull 0.23718542 5.00759816 398 1.966 npart 0.05008504 0.99918619 398 1.966 hoursw 0.26299664 5.6153534 398 1.966 ssize -0.293791 -6.3824759 398 1.966 The correlation coefficient between sales and tsales is the highest with 0.47 and it is highly significant with probability of significance 0.000. This is also justified by the test statistic value to (12.03) given by the formula (1) greater than the critical value tc=1.966 for 398 d.f. at 5% level of significance. Hence the correlation coefficient is highly significant. The correlation coefficient between sales and nfull is 0.237 and the significance of correlation is with probability of significance 0.000. Hence the correlation is highly significant. This is also justified by the test statistic value to (5.001) given by the formula (1) greater than the critical value tc=1.966 for 398 d.f. at 5% level of significance. Hence the correlation coefficient is highly significant. The correlation coefficient between sales and npart is 0.05 and the significance of correlation is with probability of significance 0.159. Hence the correlation is not significant. This is also justified by the test statistic value to (12.03) given by the formula (1) lesser than the critical value tc=1.966 for 398 d.f. at 5% level of significance. Hence the correlation coefficient is not significant. The correlation coefficient between sales and hoursw is 0.263 and the significance of correlation is with probability of significance 0.000. Hence the correlation is highly significant. This is also justified by the test statistic value to (5.615) given by the formula (1) greater than the critical value tc=1.966 for 398 d.f. at 5% level of significance. Hence the correlation coefficient is highly significant. The correlation coefficient between sales and floor space is -0.294 and the significance of correlation is with probability of significance 0.000. Hence the correlation is highly significant. This is also justified by the test statistic value to (6.382) given by the formula (1) greater than the critical value tc=1.966 for 398 d.f. at 5% level of significance. Hence the correlation coefficient is highly significant. The scatter diagrams are given below: Chart 1: Chart showing the correlation between sales per sq. mtr. and total sales From the above scatterplot, the correlation coefficient between tsales and sales is showing a positive correlation as sales per sq.ft. increases, the total sales also tends to increase. Chart 2: Chart showing the correlation between nfull and sales per sq. mtr. From the above scatterplot, the correlation coefficient between nfull and sales is showing a positive correlation as nfull increases, the total sales also tend to increase. Chart 3: Chart showing the correlation between npart and sales per sq. mtr. From the above scatterplot, the correlation coefficient between npart and sales is showing a positive correlation as npart increases, the total sales also tend to increase. Chart 4: Chart showing the correlation between hoursw and sales per sq. mtr. From the above scatterplot, the correlation coefficient between hoursw and sales is showing a positive correlation as hoursw increases, the total sales also tend to increase. Chart 5: Chart showing the correlation between ssize and sales per sq. mtr. From the above scatterplot, the correlation coefficient between ssize and sales is showing a positive correlation as ssize increases, the total sales also tend to increase. Equation representing a linear regression model, results, and overall goodness of the model The regression equation of sales per square meter on hours worked and floor space is given by the following regression analysis done in MS Excel. Table 3: Table showing Summary output of Regression Statistics Multiple R 0.60481727 R Square 0.36580394 Adjusted R Square 0.36260899 Standard Error 2985.37122 Observations 400 Table 4: Table showing ANOVA Source of variation  Df SS MS F Significance F Regression 2 2040854395 1020427198 114.4946889 5.5128E-40 Residual 397 3538239209 8912441.334 Total 399 5579093605       Table 5: Table showing the regression coefficients and their significance   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 For testing the regression, we formulate the null hypothesis H0: The regression coefficient is not significant vs. alternative hypothesis H1: The regression coefficient is significant. Level of significance: 5% level. From the above regression analysis, we observe that the intercept (constant) is highly significant with probability value less than 0.05 and also the regression coefficients of hursw and ssize are also highly significant with probability 0.000. The standard error of hoursw is 2.84 and the standard error of ssize is 1.63.The regression equation is y = 5133.59 + 37.5284 * hoursw - 22.14 * ssize ----(1) (refer to the attached Excel file) Here the constant is 5133.59 and it is highly significant (prob. 0.000), the regression coefficient of sales per sq. mtr. on hoursw is 37.5284 and it is also highly significant. The regression coefficient of sales per sq. mtr. on ssize is -22.14 and it is highly significant. It is also justified by the observed value or test statistic Fo=114.49 being greater than the critical value of Fe=1.966 at 5% level of significance. The model is well fitted as the overall F value is highly significant and the individual regression coefficients are highly significant (note down the probability is 0.000) and the standard errors are also less. The above said model given in equation (1) is highly reliable since the multiple correlation coefficient R is 0.6 and the coefficient of determination R2 is 0.366. About 36.6% of the dependent variable sales per sq. mtr are predicted through the independent variables hoursw and ssize. For the data with 400 values, this regression equation given in (1) seems to be more reliable and it suits to the prediction purposes as well. Estimated coefficients from an economic perspective and their statistical significance “Focus on the customer starts with the in-store sales personnel who assist the customers in their purchases” (Havaldar and Cavale 2006). The positive regression coefficient shows a positive correlation between the hours worked by the employees and the sales, whereas the negative regression -22.14 shows that floor space of the store is negatively correlated with sales per sq. metre. This may be because the approachability and thereby functioning of the store is not conducive when the floor spacing is more. “Space productivity is defined as net sales divided by the total square feet of retail floor space” (Dunne, ‎Lusch and Carver 2013, p.53). In that direction, when the floor space increases, the sales per sq. metre decreases and on the contrary when the floor space decreases, the sales per sq. metre increases. This is because of the point that the construction of floor space is not conducive for the effective maneuverability of the sales personnel. When that happens, the customers may not be served well and that may be also the cause for this negative regression coefficient. “Sales personnel provide their customers with the amount of service prescribed in the retail strategy of the store. Retail salespeople serve another important selling function: They persuade shoppers to buy” (Lamb, Hair and McDaniel 2011, p.457). However, the regression coefficient is highly significant and that implies that the floor space has an impact on sales per sq. mtr. This can not be denied at all. Inclusion of number of full timers and part timers The summary output is given below: Table 6: Table showing the summary output of Regression Statistics Multiple R 0.63417683 R Square 0.40218026 Adjusted R Square 0.39612639 Standard Error 2905.81795 Observations 400 Table 7: Table showing ANOVA Source of variation  df SS MS F Significance F Regression 4 2243801304 560950326.1 66.4335713 5.9985E-43 Residual 395 3335292300 8443777.975 Total 399 5579093605 Table 8: Table showing Regression coefficients   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 above regression analysis, we observe that the intercept 3751.31 (constant) is highly significant with probability value less than 0.05 and also the regression coefficients of hursw (32.76), ssize (-23.91), nfull (557.31) and npart (685) are also highly significant with probability less than 0.01. It is also justified by the observed value or test statistic Fo=66.434 being greater than the critical value of Fe=1.966 at 5% level of significance. The standard error of hoursw is 3.06, the standard error of ssize is 1.65, the standard error of nfull is 172.25 and the standard error of npart is 225.44.The regression equation is y = 3751.31 + 32.76 * hoursw - 23.91 * ssize +557.31 * nfull + 685 *npart ---- (2) (refer attached Excel file) Here the constant is 3751.31 and it is highly significant (prob. 0.000), the regression coefficient of sales per sq. mtr. on hoursw is 32.76 and it is also highly significant. The regression coefficient of sales per sq. mtr. on ssize is -23.91 and it is highly significant. The regression coefficient of sales per sq. mtr. on nfull is 557.31 and it is highly significant. The regression coefficient of sales per sq. mtr. on npart is 685 and it is highly significant. The model is well fitted as the overall F value is highly significant and the individual regression coefficients are highly significant (note down the probability is less than 0.01) and the standard errors are also less. The above said model given in equation (2) is highly reliable since the multiple correlation coefficient R is 0.634 and the coefficient of determination R2 is 0.4. About 40% of the dependent variable sales per sq. mtr are predicted through the independent variables hoursw, ssize, nfull and npart. This indicates that nfull and npart add to the prediction of sales per sq. ft. to the extent of 4% more rather than the other two variables already dealt in the second part. This clearly emphasizes that the nfull and npart are also deciding factors in sales per sq. ft. It is quite natural that the customers should be attended with utmost care by the sales persons in the stores. Also in queuing models, it is emphasized that the service rate should be considerably greater than the arrival rate so that the traffic intensity is always less than 1. For the data with 400 values, this regression equation given in (2) seems to be more reliable and it suits to the prediction purposes as well. Conclusion Although all the independent variables predict the sales per sq. ft., the most influencing variable is ssize (negative) followed by hoursw, nfull and finally npart. The number of part timers seems to influence the least in the prediction of sales per sq. mtr. The sales floor space of the store adversely (negatively) influence the sales per sq. mtr. In that sense, if the sales floor space of the store increases, sales per sq. mtr decreases and if the sales floor space of the store decreases, sales per sq. mtr increases. The other variables hoursw (number of hours worked) and nfull (Number of full timers) have a positive impact on sales per sq. mtr. References Dunne, P. , Lusch, R and Carver, J., 2013. Retailing. Cengage Learning Havaldar, KK and Cavale, VM., 2006. Sales and Distribution Management: Text and Cases. Tata McGraw-Hill Education Lamb, C., Hair, J and McDaniel, C., 2011. Essentials of Marketing. Cengage Learning Mayhew, R., 2014. Importance of Employee Performance in Business Organizations. Available from http://smallbusiness.chron.com/importance-employee-performance-business- organizations-1967.html(accessed on March 17, 2015) Sharma, JK., 2012. Business Statistics. Pearson Education. Simeon, R., 2013. Working in the Global Economy: How to Develop and Manage Your Career across Borders. Routledge. Townsend, M., 2006. Priorities for building organizational infrastructure. Available from: http://www.bizjournals.com/boston/blog/mass-high-tech/2006/05/priorities-for-building- organizational.html?page=all(accessed on March 17, 2015) Wandera, HT., 2011. “The Effects of short-term employment contract on an organization: A case of Kenya Forest Service,” International Journal of Humanities and Social Science, vol. 1, no. 21, pp. 184-204. Read More