Characteristics of Dutch Fashion Stores - Examining the Sales Model - Statistics Project Example

CHARACTERISTICS OF DUTCH FASHION STORES (1990 EXAMINING THE SALES’ MODEL By Presented to Introduction This report provides an analysis of data collected from 400 Dutch fashion stores in 1990. The data consists of cloth material sales made (in square meters), the number of full-time employees, number of part-time employees, total number of hours input by employees, and the size of the sales’ store. The aim of the analysis is to investigate how each of the independent variables (the number of full-time employees, number of part-time employees, total number of hours input by employees, and the size of the sales’ store) affects the response (sales made). The report provides an introduction (this section), an analysis, and a conclusion based on the findings. Analysis The first point of investigation is the correlation between sales and each of the independent variables outlined in the introduction. To test the relationships, the hypothesis: H0: Sales size is not significantly correlated with number of full-timers, part-timers, hours worked and size of store; against H1: Sales size is significantly correlated with number of full-timers, part-timers, hours worked and size of store. Table 1 below shows the pairwise correlation coefficients and their respective p-values. Table 1. Correlation coefficients and respective p-values. Independent variable Full-time Part-time Hours worked Store size Correlation with sales 0.237 0.050 0.263 -0.294 p-value 0.000 0.318 0.000 0.000 Referring to Table 1 above, the sales size was significantly, positively correlated with the number of full-time employees (ρ = 0.237, p < 0.001) and total number of hours worked (ρ = 0.263, p < 0.0010). The sales size was significantly, negatively correlated with the store size (ρ = -0.294, p < 0.001). The sales size was, however, not significantly correlated with the number of part-time employees in a firm (ρ = 0.050, p = 0.318). The significant correlation coefficients indicate that the respective relationships between the pairs of variables are statistically true. For instance, the rate of increase in the number of full-timers corresponds with a proportional increase in the rate of change in the size of sales. The number of hours input by employees also positively corresponds with an increase in the sales size. However, the relationship between the sales size and store size (in square meters) is inverse. This shows that an increase in the store size corresponds with a decrease in the sales size. From the onset, it is clear that the relationship between the number of part-timers and sales size does not follow a linear pattern. This is because of the earlier noted correlation coefficient that is not statistically significant. The bivariate relationships between the sales size and each of the independent variables are shown in Figures 1-4. The scatter plots indicate whether the nature of the relationship between two variables can be depicted either in a linear or non-linear manner. A linear relationship indicates that the rate of change in one variable results in a proportional change in the other (Bryan and Heagerty 2014). If the relationship is non-linear, it is impossible to arrive at the same conclusion as above. From Figure 1, the rate of increase in the size of sales (in square meters) does not appear to correspond to the increase in the number of full-timers in a store. While the correlation analysis shows that there is a significant linear relationship between the sales size and the number of full-timers, the scatter plot does not depict a similar relationship. Consequently, the accuracy of the two tests is not is questionable. Similarly, the size of sales and the number of part-timers does not appear to depict a linear relationship. This is also the case in the correlation analysis results for the relationship between the two variables. Consequently, it is noted that the level of accuracy in both the correlation output and scatter plot diagram is relatively high since the two results produced highly similar results. Figure 3 is the first clear proof of corresponding linear increase in both variables. From the plot, the rate of change in the total number of hours worked matches with a corresponding increase in the sales size. This relationship is similar to that depicted by the correlation analysis. On its part, Figure 4 represents a classic inverse relationship between two linearly dependent variables. Smaller sales’ stores correspond with higher sales sizes and the vice versa. This nature of relationship is also evident from the correlation results. Figure 1. Scatter plot of sales size against number of full-timers. Figure 2. Scatter plot of sales size against number of part-timers. Figure 3. Scatter plot of sales size against number of hours worked. Figure 4. Scatter plot of sales size against the size of sales store. The probable regression equation representing the linear relationship between the sales size and both the hours worked and the size of sales’ stores can be written as: Sales Size = β0 (Constant) + β1*(total hours worked) + β2*(size of sales store) where βi represents the coefficient corresponding to each particular independent variable. To test the significance of the resultant model, the hypothesis: H0: The regression coefficients are equal to zero (0); against H1: The regression coefficients are not equal to zero. Using the Excel function for regression calculation, the actual equation obtained is: Sales Size = 5133.59 + 37.53*(total hours worked) - 22.14*(size of sales store). The adjusted R square is the measure of the goodness of fit. This statistic informs on the percentage of change observed on the response variable that is attributable to the independent variables. For the above model, the adjusted R square is 0.3626; implying that the two independent variables selected only account for 36.26% of the change in the sales size. This shows that the model is relatively poor, with the independent variables accounting for less than half the change in the dependent. Certainly, there are more, stronger variables that are attributable to higher levels of change and which were not included in the present model. The model’s coefficients are shown as 5133.59 for the intercept, 37.53 for the total number of hours worked, and -22.14 for the store size. The intercept’s coefficient is quite large, and it denotes the point at which the trend line cuts off at the y-axis. That is, at the very point of beginning production, the minimal sales size expected at zero input level is 5133.59 units. Any further production is subject to the combined effects of variations in the number of hours and the store size. For every one further unitary change in the sales size, 37.53 hours are required. At the same time, an increase per square meter of sales size is possible when the size of the sales store is reduced by 22.14 square meters. The p-values for the three coefficients are less than 0.05 (all less than 0.001). Considering that the test was actually carried out at the 5% level of significance, all the coefficients are significant since their p-values are less than the level of significance of the test. In essence, this has the statistical implication that the coefficients are not equal to zero. The resultant curve is consequently bound to grow (either positively or negatively) by some magnitude. Finally, the model was reconstructed using all four independent variables available at the initial stage. The model developed is: Sales Size = β0 + β1*(total hours worked) + β2*(size of sales store) + β3*(number of full-time employees) + β4*(number of part-time employees). Sales Size = 3751.31 + 32.76*(total hours worked) – 23.91*(size of sales store) + 557.31*(number of full-time employees) + 685.00*(number of part-time employees). This new model has an adjusted R square value of 0.3961. This implies that the independent variables account for 39.61% of the change in the response. This represents an increase in the percentage of change accounted for from the previous model with only two independent variables. The actual increase is (39.61 – 36.26) = 3.35%. The model with all four independent variables is, therefore, a better fit than the one with two independent variables. Consequently, it is worth noting that the inclusion of the numbers of full-timers and part-timers positively improved the goodness of fit, though just slightly. Conclusion The analysis has revealed some important characteristics of the data. The correlation analysis proved that two of the variables; number of full-timers and total number of hours worked are positively correlated with the sales size. For both variables, an increase corresponded to an increase in the sales size. The number of part-timers was not significantly correlated with the sales size. Consequently, it is inappropriate to employ linear methods to predict values of the sales size using the number of part-timers across the stores. On the other hand, the store size was negatively correlated with the sales size. Decreasing store size led to marginal increase in the sales size. The regression model with hours worked and store size as the only predictors of sales size was a poor fit, with the independent variables accounting for 36.26% of the variation in the sales size. Adding the number of part-timers and full-timers marginally improved the model, although it still remained a poor fit. The regression model involving hours worked and store size as the only predictors had significant positive and negative coefficients respectively. These were in line with the correlation findings on each of these variables. The overall model involving all four predictors had all non-zero coefficients, indicating that each independent variable was a significant predictor of the sales size. References Bryan, M. and Heagerty, P. J. 2014. Direct regression models for longitudinal rates of change. Statistics in Medicine. 33(12): 2115-2136. Read More