Question

Major consulting firms such as Accenture, Ernst & Young Consulting, and Deloitte & Touche Consulting employ statistical analysis to assess the effectiveness of the systems they design for their customers. In this case, a consulting firm has developed an electronic billing system for a Stockton, CA, trucking company. The system sends invoices electronically to each customer’s computer and allows customers to easily check and correct errors. It is hoped the new billing system will substantially reduce the amount of time it takes customers to make payments. Typical payment times—measured from the date on an invoice to the date payment is received—using the trucking company’s old billing system had been 39 days or more. This exceeded the industry standard payment time of 30 days.

The new billing system does not automatically compute the payment time for each invoice because there is no continuing need for this information. The management consulting firm believes the new system will reduce the mean bill payment time by more than 50 percent. The mean payment time using the old billing system was approximately equal to, but no less than, 39 days. Therefore, if µ denotes the new mean payment time, the consulting firm believes that µ will be less than 19.5 days. Therefore, to assess the system’s effectiveness (whether µ < 19.5 days), the consulting firm selects a random sample of 65 invoices from the 7,823 invoices processed during the first three months of the new system’s operation. Whereas this is the first time the consulting company has installed an electronic billing system in a trucking company, the firm has installed electronic billing systems in other types of companies.

Analysis of results from these other companies show, although the population mean payment time varies from company to company, the population standard deviation of payment times is the same for different companies and equals 4.2 days. The payment times for the 65 sample invoices are manually determined and are given in the Excel^® spreadsheet named “The Payment Time Case”. If this sample can be used to establish that new billing system substantially reduces payment times, the consulting firm plans to market the system to other trucking firms.

• 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.
• Using the 95% confidence interval, can we be 95% confident that µ ≤ 19.5 days?
• Using the 99% confidence interval, can we be 99% confident 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?

Answer 1

            Electronic billing systems were created to make life easier for companies. Technically speaking, the new billing system is supposed to significantly reduce the payment time process. For this case study, the author will use datasets provided in the assignment and apply the concepts of sampling distributions and confidence intervals of 95% and 99%. The author will be conducting this analysis to determine whether the new electronic billing system that was developed does in fact reduce the payment time. In this case, the new system was developed for a trucking company based in Stockton, CA. Upon conclusion, the author will demonstrate abilities in using datasets to apply the concepts of sampling distributions and confidence intervals to make sound management decisions.

95% Confidence Interval

            The provided standard deviation of payment times for all the companies is 4.2 day. Using this and a 95% confidence interval, the author will determine the new billing systems effectiveness at improving payment times. The author utilized the sample of 65 various payment times extracted from 7,823 invoices. The mean that was calculated from the samples is 18.11 days. The following formula will help to determine the population mean: x ± z (α/2) σ/√n. x = 18.11, σ = 4.2, n = 65 and z (α/2) is 1.96. The z value was identified as half of the α value of .05, which is .0250. Subtracting this value from the total area of half of the curve or .5000, gives a value of .4750. According to the normal Z-table, we find that the corresponding value is 1.96. Expanding upon the formula can also show the lower confidence limit with the -z value and upper confidence limit with the +z value. This formula is expressed as, x -z α/2 σ/√n<μ< x +z α/2 σ/√n. Because of the expanded formula, the mean pay time is between 17.09 days and 19.11 days, expressed mathematically as 17.09<μ<19.11. Based on these calculations, the effectiveness of the new billing system can now be determined. According to the case, the old billing system showed an average billing payment time of 39 days. Also, according to the case, the firm made a claim that prior to implementing the new system, the average payment time was estimated to be reduced to <19.5 days. The calculations show, that since the new population mean was less than the upper limit of 19.11 days, the electronic billing system was successful in reducing the mean payment time by at least 51%.

95% Confidence That µ ≤ 19.5 Days?

            Unlike the previous calculation that dealt with the average period a company took to pay its creditors; this calculation is best suited for the probability of the mean payment time. If µ ≤ 19.5 days, standard deviation is 4.2 days, Sample (n) is 65; The mean is calculated at 18.10769231. The computed standard deviation = 3.961230384, this is obtained from the computation in excel. However, the normal distribution probability, confidence intervals and the intermediate calculations give the following: Standard deviation: 3.9612304, Population standard deviation 4.2, Sample Mean 18.10769231, Standard Error of 0.5209, Z Value 1.9600, half of interval width is 1.0210, The interval lower limit 17.0867, Interval Upper Limit 19.1287. If the following equation is used, Confidence Interval = X +/-Z * α/√N, then 95% Confidence Interval results in 17.0867 and 19.1287. Based on these results, the conclusion can be drawn, that the two results that were calculated are less than the confident interval of µ ≤ 19.5 days. This means that it is evident that with 95% confidence, the billing system was effective at µ ≤ 19.5 days.

99% Confidence That µ ≤ 19.5 Days?

Formula Confidence interval :   

((18.11 – (2.58*0.52)), (18.11 + (2.58*0.52))) = (18.11 – 1.34, 18.11 + 1.34) = (16.77, 19.45)

99% Confidence Interval and Probability of ≤ 18.1077 days?

            To obtain a more accurate interval limit based on the 99% confidence, the following data must also be used just as the 95% confidence: Confidence Interval = X +/-Z * α/√N, Standard error of the mean 0.5209, Z value 2.5760, half of interval width 1.3419, Interval lower limit 16.7658, and Interval upper limit 9.4496. After computing these figures, the results indicate that the intervals give limits at 99% Confidence Interval, which are less than 19.5 days. This indicates that it is correct to assume that at 99% confidence interval, the billing system was effective that µ ≤ 19.5 days.  

            The difference in the population mean (19.5), and the sample mean (18.1077) must be calculated in order to correctly find the probability of observing a sample mean of ≤ 18.1077 days within 65 invoices. The difference is then divided by 4.2/sqrt 65. Which results in a value of -0.3315. When referencing this value to the z-table, the z-score results in a value of -0.1293. Subtracting this value from 50% leaves us with a probability of 63%. This indicates that there is a 63% probability of observing a sample mean payment ≤ 18.1077 days within 65 invoices.

Conclusion

            Based on the information provided, the calculations that were computed, the author has found that from the sample of 65 invoices, 95% Confidence Interval results in 17.0867 and 19.1287 and 99% confidence results in an average processing time between 16.7658 and 19.4496 days. These figures indicated that the findings show the new system is effective at cutting processing time down by nearly 50% of 30-day industry standard. Based on all the information and figures that were calculated, the author would suggest that the firm implement the new system. Also, to maximize profit and to receive payments in a timely manner, the new system should be marketed to other trucking companies around the country.