The Payment Time Directive: What You Need to Know

Introduction 

This report utilizes confidence interval calculations to check whether the new electronic billing system developed by a consulting firm for a trucking company, is effective in decreasing the mean time in bill payments. A confidence interval provides a range of values, within which there is a certain level of confidence that a given population parameter occurs (Panik, 2012). The consulting firm is convinced that the new billing system will reduce the payment time by fifty percent from 39 days to 19.5 days. Considering other billing systems in other companies with electronic billing systems, the standard deviation for payment is 4.2 days. The section below shows some calculations, which make it possible to determine whether the new billing system is working as assumed. 

Analysis and Findings 

Assuming the standard deviation of the payment times for all payments is 4.2 days, construct a 95% confidence interval estimate to determine whether the new billing system was effective. State the interpretation of 95% confidence interval and state whether or not the billing system was effective. 

Formula for calculating standard deviation at 95% interval is as given below: 

Where: 

X ̅ = the sample mean for the 65 invoices =  =  = 18.1. In this case, Ʃ X is the sum of days contained in the sample invoices, while n is the number of invoices which is 65. 

1.96 = the z- score at 95% confidence interval. 

s = the standard deviation = 4.2 days. 

n = the sample size = 65 days. 

Therefore, 

 = (18.1 ± 1.1) = (17, 19.2). 

The obtained 95% confidence interval is (17 days, 19.2 days). Therefore, there is a 95% confidence that the assumed mean of 19.5 days is not contained in the given confidence interval. From this finding, the new billing system is not statistically effective in reducing the mean bill payment time. 

The consulting firm assumes that the payment time would reduce by 50%. According to the obtained confidence interval however, the payment time would reduce by approximately 43.6% to 49.2%. Therefore, the consulting firm should make improvements on the electronic billing system in order to achieve the desired results. 

Using the 95% confidence interval, can we be 95% confident that µ ≤ 19.5 days? 

As stated above, the obtained 95% confidence interval is (17 days, 19.2 days). Therefore, there is a 95% confidence that µ ≤ 19.5 days. 

Using the 99% confidence interval, can we be 99% confident that µ ≤ 19.5 days? 

Formula for calculating standard deviation at 99% interval is as given below: 

Where: 

X ̅ = the sample mean for the 65 invoices = 18.1, as obtained in the section above. 

2.576 = the z- score at 95% confidence interval. 

s = the standard deviation = 4.2 days. 

n = the sample size = 65 days. 

Therefore, 

 = (18.1 ± 1.3) = (16.8, 19.4). 

The obtained 99% confidence interval is (16.8 days, 19.4 days). From these results, there is a 99% confidence that µ ≤ 19.5 days. 

If the population mean payment time is 19.5 days, what is the probability of observing a sample mean payment time of 65 invoices less than or equal to 18.1077 days? 

In this case, the z- value will be of use in obtaining the required probability. The formula is as given below: 

Where, 

x = sample mean = 18.1077 days. 

μ = population mean = 19.5 days. 

σ = standard deviation = 4.2 days. 

n = sample size = 65 invoices. 

Therefore, 

 =  = -2.67. 

Reading from a z- table: 

P (x ≤ 18.1077) = 0.0038 

Therefore, the probability of observing a mean payment time of the selected 65 invoices being less than 18.1077 days is 0.38%. 

Conclusions and Recommendation 

The consulting firm wanted to check whether new billing system has reduced the mean bill payment time, using a statistical approach employing confidence intervals. Data analysis involved using both 95% and 99% confidence intervals. Findings from both confidence intervals revealed that the new billing system was not effective in reducing the mean bill payment time. 

The consulting firm should therefore create mechanisms of improving the electronic billing system, in order to reduce the payment time by the required fifty percent. 

References 

Panik, M. J. (2012).  Statistical inference: A short course . Hoboken, NJ: Wiley.