Direct Annual Sales Data of Dutch Fashion Stores - Report Example

Project Report Word Count (Excluding Graphs and Tables 1983 Introduction to Statistics-(L1025) Introduction to Statistics-(L1025) Project Report Introduction This report will analyze the annual sales data and other characteristics of 400 Dutch fashion stores in 1990. The variables inconsideration are sales per square metre, number of full-timers, number of part-timers, total number of hours worked and sales floor space of the store (in square metres). This report will use correlation and regression analysis technique for examining the annual sales data. The research question examined in this report is, “Whether all variables taken together significantly predict the sales per square metre?”. Data Analysis, Results and Discussions Scatter plots and Correlation Analysis Figure 1 to 4 shows the scatter plots between sales per square metre and each of the other variables. There appears a positive linear relationship between sales per square metre and number of full-timers (figure 1) and between sales per square metre and total number of hours worked (figure 3). There appears no linear relationship between sales per square metre and number of part-timers (figure 2). There appears a negative linear relationship between sales per square metre and sales floor space of the store (figure 4). Figure 1: Scatter plot between sales per square metre and number of full-timers Figure 2: Scatter plot between sales per square metre and number of part-timers Figure 3: Scatter plot between sales per square metre and total number of hours worked Figure 4: Scatter plot between sales per square metre and sales floor space of the store Table 1 shows the correlation matrix table. As shown in table 1, there appears a week positive linear relationship between sales per square metre and number of full-timers (r = 0.237) and between sales per square metre and total number of hours worked (r = 0.263). In addition, there appears a weak negative linear relationship between sales per square metre and sales floor space of the store (r = -0.294). However, there appears no relationship between sales per square metre and number of part-timers (r = 0.050). Table 1: Correlation matrix (n = 400)   sales nfull npart hoursw nfull 0.2372* npart 0.0501 0.2888* hoursw 0.2630* 0.5313* 0.2491* ssize -0.2938* 0.3497* 0.3665* 0.5759* p < 0.001 Statistical Significance of the Correlation Coefficient The selected level of significance, α for each of the tests is 5%. The selected test is the Pearson Correlation Test. The degrees of freedom, df for each of the test are 398 (= 400 – 2). Two-tailed critical values at the 5% level with 398 degrees of freedom are t*398 ±1.966 (Using Excel Function TINV). The decision rule for each of the correlation test will be Reject H0, if | t | > 1.966. Otherwise, do not reject H0. Sales per square metre and Number of full-timers The null and alternate hypotheses are (There is no relationship between sales and number of full-timers.) (There is a significant relationship between sales and number of full-timers.) The test statistic is Decision: Reject H0, as t = 4.871 > 1.966. Conclusion: At the 5% level, data provide enough evidence that there is a significant relationship between sales per square metre and number of full-timers. Sales per square metre and Number of part-timers The null and alternate hypotheses are (There is no relationship between sales and number of part-timers.) (There is a significant relationship between sales and number of part-timers.) The test statistic is Decision: Fail to reject H0, as t = 1.000 < 1.966. Conclusion: At the 5% level, data do not provide enough evidence that there is a significant relationship between sales per square metre and number of part-timers. Sales per square metre and Total number of hours worked The null and alternate hypotheses are (There is no relationship between sales and total number of hours worked.) (There is a significant relationship between sales and total number of hours worked.) The test statistic is Decision: Reject H0, as t = 5.438 > 1.966. Conclusion: At the 5% level, data provide enough evidence that there is a significant relationship between sales per square metre and total number of hours worked. Sales per square metre and Sales floor space of the store The null and alternate hypotheses are (There is no relationship between sales and sales floor space of the store.) (There is a significant relationship between sales and sales floor space of the store.) The test statistic is Decision: Reject H0, as | t | = 6.132 > 1.966. Conclusion: At the 5% level, data provide enough evidence that there is a significant relationship between sales per square metre and sales floor space of the store. Thus, there are significant relationships between sales per square metre and each of the other variables except the number of part-timers. Regression Analysis (Model 1) A multiple regression analysis is performed for predicting sales per square metre (dependent variable) using the independent variables total number of hours worked and floor space of the store. The results of the regression analysis are presented in table 2. The regression model (estimated equation) is given by: sales = 5133.59 + 37.528(hoursw) – 22.145(ssize) The coefficient of determination, R2 value of 0.3658 indicates that this model explains about 36.6% of the variation in the sales per square metre. The goodness of fit of the model is moderately strong that is the total number of hours worked and floor space of the store has moderately strong effect on the sales per square metre. Table 2: Regression Analysis without the Inclusion of the Number of Full-timers and Part-timers SUMMARY OUTPUT Regression Statistics Multiple R 0.6048 R Square 0.3658 Adjusted R Square 0.3626 Standard Error 2985.371 Observations 400 ANOVA   df SS MS F Significance F Regression 2 2040854395.24 1020427197.62 114.495 0.0000 Residual 397 3538239209.47 8912441.33 Total 399 5579093604.71         Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept 5133.590 321.693 15.958 0.0000 4501.155 5766.026 hoursw 37.528 2.837 13.227 0.0000 31.951 43.106 ssize -22.145 1.625 -13.627 0.0000 -25.339 -18.950 Statistical Significance of the Regression Model using the F-test The null and alternate hypotheses are All the true coefficients are zero (). () At least one of the coefficients is nonzero. () The selected level of significance, α is 5%. The selected test is the F-test for overall significance (or Goodness of Fit). The degrees of freedom are df 1= 2 and df2 = 397 Critical value at 5% level with df 1= 2 and df2 = 397 degrees of freedom is 3.018 (Using Excel Function FINV). The decision rule will be Reject H0, if F > 3.018. Otherwise, do not reject H0. The test statistic is F = = 114.495 (from table 2) Decision: Reject H0, as F = 114.495 > 3.018. Conclusion: At the 5% level, the regression model is significant. Thus, the sales per square metre can be significantly predicted using the total number of hours worked and sales floor space of the store Estimated Coefficients Interpretation from an Economic Perspective Approximation of elasticities: The intercept (5133.59) is not meaningful in this context because both the number of hours worked and sales floor space of the store cannot have a value of zero. The slope coefficient value of 37.528 for the number of hours worked indicates that on average, everything else constant, an additional hour of work is associated with an increase in the sales per square metre by about 37.53. Furthermore, a 1% rise in the number of hours worked leads to a 0.72% increase in sales per square metre. The slope coefficient value of -22.145 for the sales floor space of the store indicates that on average, everything else constant, an additional square metre of sales floor space of the store is associated with a fall in the sales per square metre by about 22.14. Furthermore, a 1 % rise in sales floor space of the store leads to a 0.53% fall in the sales per square metre. Statistical Significance of the Estimated Coefficients The selected level of significance, α for each of the tests is 5%. The selected test is the t-Test for the Predictor Significance. The degrees of freedom, df for each of the test are 397. Two-tailed critical values at the 5% level with 397 degrees of freedom are t*397 ±1.966. The decision rule for each of the t-test will be Reject H0, if | t | > 1.966. Otherwise, do not reject H0. Total number of hours worked The null and alternate hypotheses are (The coefficient of the number of hours worked is zero.) (The coefficient of the number of hours worked is different from the zero.) The test statistic is = 13.227 (from table 2) Decision: Reject H0, as t = 13.227 > 1.966. Conclusion: At the 5% level, data provides enough evidence that the number of hours worked significantly predicts the sales per square metre in the regression model. Sales floor space of the store The null and alternate hypotheses are (The coefficient of the sales floor space is zero.) (The coefficient of the sales floor space is different from the zero.) The test statistic is = -13.627 (from table 2) Decision: Reject H0, as | t | = 13.627 > 1.966. Conclusion: At the 5% level, data provide enough evidence that the sales floor space of the store significantly predicts the sales per square metre in the regression model. Thus, both the variables the number of hours worked and sales floor space of the store individually significantly predicts the sales per square metre in the regression model. Regression Analysis (Model 2) Another multiple regression analysis is performed for predicting sales per square metre (dependent variable) using all the independent variables that is two new independent variables the number of full-timers and part-timers are added to the regression model. The results of the regression analysis are presented in table 3. The regression model is given by: sales = 3751.31 + 557.308(nfull) + 684.998(npart) + 32.765(hoursw) – 23.908(ssize) The coefficient of determination, R2 value of 0.4022 indicates that this model explains about 40.22% of the variation in the sales per square metre as compared to 36.58% for the earlier model. The goodness of fit of the model is moderately strong and better as compared to the earlier model. Furthermore, the adjusted R2 value for this model is 0.3961 and is higher as compared to the earlier model of 0.3626 indicating a better goodness of fit of the model, when two new variables are added. Table 3: Regression Analysis with the Inclusion of the Number of Full-timers and Part-timers SUMMARY OUTPUT Regression Statistics Multiple R 0.6342 R Square 0.4022 Adjusted R Square 0.3961 Standard Error 2905.82 Observations 400 ANOVA   df SS MS F Significance F Regression 4 2243801304.50 560950326.12 66.434 0.0000 Residual 395 3335292300.21 8443777.98 Total 399 5579093604.71         Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept 3751.311 427.696 8.771 0.0000 2910.466 4592.156 nfull 557.308 172.248 3.236 0.0013 218.671 895.944 npart 684.998 225.443 3.038 0.0025 241.781 1128.216 hoursw 32.765 3.062 10.701 0.0000 26.745 38.784 ssize -23.908 1.649 -14.502 0.0000 -27.149 -20.667 Approximation of elasticities: The slope coefficient value of 557.308 for the number of full-timers indicates that on average, everything else constant, an additional full-timer is associated with an increase in the sales per square metre by about 557.31. Furthermore, a 1% rise in the number of full-timers leads to a 0.18% increase in sales per square metre. The slope coefficient value of 684.998 for the number of part-timers indicates that on average, everything else constant, an additional part-timer is associated with an increase in the sales per square metre by about 685.00. Furthermore, a 1 % rise in number of part-timers leads to a 0.17% increase in the sales per square metre. Statistical Significance of the Regression Model using the F-test The null and alternate hypotheses are Critical value at 5% level with df1= 4 and df2 = 395 degrees of freedom is 2.395. The decision rule will be Reject H0, if F > 2.395. Otherwise, do not reject H0. The test statistic is F = = 66.434 (from table 3) Decision: Reject H0, as F = 114.49 > 2.395. Conclusion: At the 5% level, the regression model is significant. Statistical Significance of the Added Coefficients The degrees of freedom, df for each of the test are 395. Two-tailed critical values at the 5% level with 395 degrees of freedom are t*397 ±1.966. The decision rule for each of the t-test will be Reject H0, if | t | > 1.966. Otherwise, do not reject H0. Number of Full-timers The null and alternate hypotheses are The test statistic is = 3.236 Decision: Reject H0, as t = 3.236 > 1.966. Conclusion: At the 5% level, data provide enough evidence that the number of full-timers significantly predicts the sales per square metre in the new regression model. Number of Part-timers The null and alternate hypotheses are The test statistic is = 3038 Decision: Reject H0, as t = 3.038 > 1.966. Conclusion: At the 5% level, data provide enough evidence that the number of part-timers significantly predicts the sales per square metre in the new regression model. From the above analysis, it can be said that the inclusion of the number of full-timers and part-timers significantly improve the model. Conclusion There are significant relationships between sales per square metre and each of the other variables except the number of part-timers. The sales per square metre can be significantly predicted using a regression model using the total number of hours worked and sales floor space of the store. The inclusion of the number of full-timers and part-timers significantly improve the model. Bibliography Anderson, D. R., Sweeney, D. J. & Williams, T. A., 2012. Essentials of Statistics for Business & Economics. 6th ed. United States: South-Western Cengage Learning. Anderson, D. R., Sweeney, D. J. & Williams, T. A., 2012. Modern Business Statistics with Microsoft Office Excel. Fourth ed. United States: Cengage Learning. Anderson, D. R., Sweeney, D. J. & Williams, T. A., 2012. Statistics for Business and Economics, Revised. 11th ed. United States of America: South Western, Cengage Learning. Doane, D. P. & Seward, L. E., 2007. Applied Statistics in Business and Economics. London: McGraw-Hill Irwin. Siegel, A. F., 2011. Practical Business Statistics. 6 ed. London: Mcgraw-Hill/Irwin. Triola, M. F., 2010. Elementary Statistics. 11 ed. New York: Addison-Wesley. Weiss, N. A., 2012. Introductory Statistics. 9 ed. New York: Addison-Wesley. Read More