Relationship between Size of Bills and Number of Days to Collect - Research Proposal Example

639450 C32 Memorandum QSCA Management QSCA Consultant Relationship between Size of Bills and Number of Days to Collect Introduction The purpose of this memo is to provide information that will enable QSCA to increase the probability of earning high profits. It will provide information that will enable the management of QSCA to obtain an understanding of whether there is a relationship between bill size and the number of days for which payment is overdue. A regression model is prepared with explanation of the variables.

Data representing the bill amount and days overdue for 96 customers, evenly split between residential and commercial will be used to determine whether a relationship exists. The data is run separately for each customer group - residential or commercial. The statistical significance of the data in the table generated is tested to determine whether the relationship is statistically significant. The Regression Model In order to determine whether a relationship exists, a regression model is required.

The equation for this model is as follows: Y = a + bX Where: a represents the point of intercept with the Y axis b is a regression coefficient which represents the net change in Y for each unit of change in X The model is dealt with separately for residential and commercial customers. Commercial Customers The result for commercial customers which is shown in Appendix 1 indicates that there is a 96.58% correlation (represented by Multiple R) between the size of the bill and the number of days overdue.

Multicollinearity does not exist as there is only one explanatory variable. Gujarati (1995) indicates that multicollinearity in its broadest sense relates to the existence of an perfect or exact linear relationship among some or all explanatory (X) variables of a regression model as well as where the X variables are inter-correlated but not perfectly. The coefficient of determination (R squared or the correlation coefficient squared) is 95.65% and indicates that 95.65% of the change in the dependent variable Y (the number of days overdue) depends on the change in the independent variable X (the bill size).

It is a measure of the goodness of fit (ie how well the sample regression line fits the data) (Gujarati 1995). Adjusted R squared which is 95.56% is the coefficient of determination (R squared) adjusted by 1 degree of freedom (df). It provides a better measure of how well the sample regression line fits the data. According to Mason and Lind (1996) the standard error of estimate in regression analysis which is 3.2205 measures the variation about the regression line. The regression equation for the commercial customers is: Y = 101.

7582 – 0.1910X This equation indicates that the intercept is 101.7582 and the slope of the coefficient of X is -0.1910. The equation indicates that as the size of the bill increases the number of days the payment becomes late decreases. This suggests an inverse relationship between bill size and days overdue. According to Madura (2006, p. 754) the slope coefficient of -0.19 suggests that every 1 percent change in the days overdue is associated with a 0.19 per cent change in the opposite direction in the bill size.

The graph below provides a better picture of this scenario. The graph shows that as the bill amount increases the number of days decrease. The result for customers which is shown in Appendix 2 indicates that there is a 96.58% correlation between bill size and the number of days overdue. The information also indicates that the 93.14% of the change in the days overdue is explained by the size of the bill and that the variation from the regression line is 3.5152. Residential Customers The equation relating to the residential customers is: Y = 2.2096 + 0.1657X This equation indicates that the regression line crosses the Y axis at 2.

2096 and that the slope of the X coefficient is a positive 0.1657. This information indicates that the days overdue is directly related to the bill size and so for every 1 percent change increase in the number of days overdue there is a 0.16 percent change in the size of the bill. The graph below shows a clearer picture of the situation. While the equations can be used to predict values for Y based on varying X values the graph can also be use to extrapolate matching figures for X and Y values.

In order to determine the statistical significance of the relationship the following hypothesis will be tested. The Null hypothesis (H0) would indicate that the relationship is statistically significant and the alternative hypothesis (H1) would indicate that the relationship is not significant. H0: ? = 0.96 H1: ? ? 0.96 The next step is to determine the level of significance - alpha (?) which is the probability of rejecting the null hypothesis when it is actually true and thus signifies a type 1 error.

Alpha is tested at the 0.1 level of significance and indicates a 90 confidence level. The p-value is compared with the critical value in order to determine whether the null hypothesis should be rejected, in which case the alternative hypothesis would be accepted. The null hypothesis is rejected if the p < 0.1 where ? = 0.1. The value of p is 5.78 in the case of commercial customers and p = 1.27 in the case of residential customers which indicates that p>1 and implies that the null hypothesis cannot be rejected.

In the case of commercial customers the 90% confidence interval lies between -0.201 and -0.1809 and b = -0.1910 lies within this region. In the case of residential customers the confidence interval lies between 0.1547 and 0.1767 and b = 0.1657 lies within it. This implies that our finding is statistically significant. References Gujarati, D.M. (1995). Basic Econometrics. 3rd ed. USA: McGraw-Hill Madura, J. (2006). Financial Markets and Institutions. 7th ed. USA: Thomson South-Western Mason, R.

D and Lind, D.A. (1996). Statistical Techniques in Business & Economics. 9th ed. USA: Irwin Appendix 1 SUMMARY OUTPUT Regression Statistics Multiple R 0.978009251 R Square 0.956502094 Adjusted R Square 0.955556487 Standard Error 3.220464879 Observations 48 ANOVA   df SS MS F Significance F Regression 1 10490.895 10490.89504 1011.522 5.78E-33 Residual 46 477.08413 10.37139404 Total 47 10967.979         Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 90.0% Upper 90.0% Intercept 101.7581844 1.1449665 88.

87437457 4.08E-53 99.45349 104.0629 99.83617 103.6802 BILL -0.190961488 0.0060042 -31.80443638 5.78E-33 -0.20305 -0.17888 -0.20104 -0.18088 RESIDUAL OUTPUT Observation Predicted DAYS Residuals Standard Residuals 1 62.61107927 -2.6110793 -0.819542542 2 86.67222678 -0.6722268 -0.210992617 3 83.23492 -2.23492 -0.701476986 4 64.13877117 -4.1387712 -1.299041009 5 46.76127574 0.2387243 0.074928665 6 71.58626921 -0.5862692 -0.184013013 7 83.04395851 -0.0439585 -0.013797309 8 58.7918495 -3.7918495 -1.

190152294 9 73.11396112 -4.1139611 -1.291253848 10 92.21010994 -2.2101099 -0.693689825 11 92.9739559 1.0260441 0.322045678 12 83.61684297 -0.616843 -0.193609234 13 82.66203553 1.3379645 0.419948492 14 75.023576 3.976424 1.248084908 15 44.66069937 2.3393006 0.734239057 16 67.38511647 1.6148835 0.506865405 17 42.560123 -3.560123 -1.117420023 18 62.61107927 0.3889207 0.122071011 19 87.43607274 -2.4360727 -0.764613035 20 83.61684297 -0.616843 -0.193609234 21 55.92742718 -2.9274272 -0.918835035 22 42.

36916152 4.6308385 1.453486758 23 70.82242326 -0.8224233 -0.258134963 24 60.70146439 -1.7014644 -0.534040641 25 72.35011517 -2.3501152 -0.73763343 26 83.23492 -0.23492 -0.073734618 27 54.0178123 -5.0178123 -1.574946687 28 67.57607796 3.423922 1.074670464 29 73.11396112 0.8860389 0.278102072 30 63.37492522 3.6250748 1.137806513 31 49.62569807 3.3743019 1.059096143 32 59.74665694 -2.7466569 -0.862096467 33 80.75242065 -0.7524206 -0.23616316 34 61.65627183 -1.6562718 -0.519855999 35 44.08781491 5.9121851 1.855664535 36 66.

04838606 1.9516139 0.612555379 37 44.2787764 -0.2787764 -0.087499878 38 46.57031426 0.4296857 0.134865973 39 63.7568482 3.2431518 1.017931896 40 73.30492261 -0.3049226 -0.09570642 41 88.39088018 2.6091198 0.818927528 42 84.57165041 -2.5716504 -0.807166961 43 61.46531034 1.5346897 0.481694861 44 72.54107666 1.4589233 0.457913998 45 86.4812653 5.5187347 1.732171796 46 73.87780707 -8.8778071 -2.786487818 47 90.30049506 8.6995049 2.730523916 48 51.34435146 -0.3443515 -0.108082001 Appendix 2 SUMMARY OUTPUT Regression Statistics Multiple R 0.

965846 R Square 0.932858 Adjusted R Square 0.931398 Standard Error 3.515189 Observations 48 ANOVA   df SS MS F Significance F Regression 1 7897.265 7897.265 639.1154 1.27E-28 Residual 46 568.4016 12.35656 Total 47 8465.667         Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 90.0% Upper 90.0% Intercept 2.209624 1.249749 1.768053 0.083684 -0.30599 4.725239 0.111719 4.307528587 BILL 0.165683 0.006554 25.28073 1.27E-28 0.152491 0.178875 0.154681 0.176684452 RESIDUAL OUTPUT Observation Predicted DAYS Residuals Standard Residuals 1 37.83146 3.168537 0.

911129 2 35.5119 1.488099 0.427911 3 52.24588 -0.24588 -0.0707 4 27.06207 -1.06207 -0.3054 5 47.44108 0.558925 0.160722 6 26.39934 -1.39934 -0.40239 7 33.19234 -0.19234 -0.05531 8 45.94993 1.050072 0.301954 9 18.28087 0.719128 0.206789 10 31.86688 4.133124 1.188502 11 27.7248 2.275199 0.654245 12 20.43475 -3.43475 -0.98768 13 18.77792 2.222079 0.63897 14 52.0802 -3.0802 -0.88573 15 14.63585 -1.63585 -0.4704 16 15.29858 0.701422 0.201697 17 34.84917 5.150831 1.481148 18 51.74883 -3.74883 -1.078 19 41.97354 1.026463 0.

295165 20 28.38753 2.612467 0.751228 21 26.89639 3.103613 0.89246 22 32.03256 1.967441 0.565748 23 36.17463 1.825367 0.524894 24 38.65988 3.340122 0.96047 25 29.05027 -0.05027 -0.01445 26 53.73703 -3.73703 -1.0746 27 27.55912 -2.55912 -0.73589 28 15.46426 0.535739 0.154054 29 39.48829 3.511708 1.00981 30 53.57135 -2.57135 -0.7394 31 18.28087 3.719128 1.069455 32 17.12109 -12.1211 -3.48548 33 10.49377 -0.49377 -0.14199 34 50.092 -3.092 -0.88912 35 13.80743 1.192568 0.342929 36 12.1506 -1.1506 -0.

33086 37 37.00305 4.996952 1.4369 38 36.17463 -0.17463 -0.05022 39 17.94951 4.050494 1.164741 40 9.83104 1.16896 0.33614 41 18.44655 0.553445 0.159146 42 27.06207 -3.06207 -0.88051 43 49.92632 -2.92632 -0.84148 44 37.16873 1.831269 0.526591 45 25.40524 1.59476 0.458582 46 43.63037 0.369633 0.10629 47 35.18054 -0.18054 -0.05191 48 17.94951 -11.9495 -3.43614