Most consulting companies just like the one mentioned in the Payment Time Case Study applies the statistical analysis on their verge of reviewing the efficiency of the systems they design for their clients. Our case study mainly deals with an electronic billing system which was developed for a Stockholm, CA trucking firm.

Our reviewed system permits for one to send invoices to the clients electronically into their personal computers. This in the end enables the clients to check for any errors in the invoices and correcting for the same in due time. The primary purpose for a billing system is to minimize the amount of time which clients take to honor their payments. The time it takes currently is 39 days or more for clients to clear the payments according to the old system. The industry standards stands at 30 days which makes 39 days to be in excess of the industry standards.

Computing the payment time on invoices is not a necessity. This arise from the fact that there is no continued need for this king of information. Those consulting firm employing such systems believes that the time taken by customers to pay can be reduced by 59%. The old systems pays in 39 days. We therefore expect that the time will be reduce by half which comes to 19.5 days. To assess the effectiveness of the new system, a random sample of 65 invoice will be selected by the consulting firm from the total of 7,000 invoices which had been processed by the new system.

It is evident that the company has already installed these kinds of systems in other businesses. Analysis from the same reveals that the standard deviation of the payment times from the population is equal to 4.2 days. The 65 invoices payment times were manually recorded in an excel file for application in this case study. The next step was basically to assess the effectiveness of the billing system after constructing a 95% confidence interval.

95% Confidence Level Calculations |
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Std Dev = | 4.2 | ||||

Mean = | 18.1077 | ||||

Sample Size = | 65 | ||||

Confidence Level = | 95% | ||||

Standard Error of Mean = | 0.49133 | ||||

Z Value = | 1.96 | Z value for 95% confidence level | |||

Standard Error = | 0.52095 | Based on 95% confidence interval 17.0866 & 19.1287 are both below 19.5 days | |||

Interval Half Width = | 1.02105 | ||||

Interval Lower Limit = | 17.0866 | ||||

Interval Upper Limit = | 19.1287 |

99% Confidence Level Calculations |
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Std Dev = | 4.2 | ||||

Mean = | 18.1077 | ||||

Sample Size = | 65 | ||||

Confidence Level = | 99% | ||||

Standard Error of Mean = | 0.49133 | ||||

Z Value = | 2.576 | Z value for 95% confidence level | |||

Standard Error = | 0.52095 | Based on 99% confidence interval 16.7657 & 19.4496 are both below 19.5 days | |||

Interval Half Width = | 1.34196 | ||||

Interval Lower Limit = | 16.7657 | ||||

Interval Upper Limit = | 19.4496 |

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

It is evident that a 95% confidence interval implies that 95% of the interval estimates includes the population parameter, as implied by the following:

95% CI = (17.0867, 19.1287) less than 19.5.

We can therefore be 95% confident that µ ≤ 19.5 days.

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

Principally, a 99% confidence interval suggests that 99% of the interval estimates includes the population parameter, as implied by the following:

99% CI =(16.7657, 19.4496) less than 19.5.

We can therefore be 99% confident that µ ≤ 19.5 days.

Taking the population mean payment time of 19.5 days, we can calculate the probability of observing sample mean time of 65 invoices which are less than or equal to 18.1077 days as follows:

Z value for 18.1077 is z = (18.1077-19.5)/0.5209 = -2.67

P (mean x <18.1077) = P (z < -2.67)

**=0.0038 **

In conclusion, this billing system has by all means revealed it is effective. From results from the sample, the payment time stands at 18.1077 days. This is some percentage less than the expected 19.5 days which was touted by the consulting company. Analysis of the data from both the 95% and 99% confidence levels reveals that all the values are below the 19.5 days. Taking results from this case study reveals a very effective system. Among the approaches which can be applied to increase the accuracy of the results include increasing the sample size.

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