Question

Case Study Description - Payment Time Case Study

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 have been taken and the sample mean has been calculated to be 18.10 days. 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.

Develop a 600-word Microsoft Word report including the following calculations and using the information to determine whether the new billing system has reduced the mean bill payment time:

• 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 18.10 days?

Answer 1

State the interpretation of 95% confidence interval and state whether or not the billing system was effective.

We have:

x = 18.10

z = 1.96

σ = 4.2

n = 65

The 95% confidence interval is:

= x ± z*(σ/√n)

= 18.10 ± 1.96*(4.2/√65)

= 17.08, 19.12

The 95% confidence interval is between 17.08 and 19.12 days.

Since the confidence interval has an upper limit lower than 19.5 days, we can say that the billing system was effective.

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

Since the confidence interval has an upper limit lower than 19.5 days, we can be 95% confident that µ ≤ 19.5 days.

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

We have:

x = 18.10

z = 1.96

σ = 4.2

n = 65

The 99% confidence interval is:

= x ± z*(σ/√n)

= 18.10 ± 2.576*(4.2/√65)

= 16.76, 19.44

The 99% confidence interval is between 16.76 and 19.44 days.

Since the confidence interval has an upper limit lower than 19.5 days, we can 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 18.10 days?

We have:

x = 18.10

µ = 19.5

σ = 4.2

n = 65

The test statistic, z = (x - µ)/σ/√n

z = (18.10 - 19.5)/4.2/√65

z = -2.69

Probability(observing a sample mean payment time of 65 invoices less than or equal to 18.10 days) = 0.0036.

The probability of observing a sample mean payment time of 65 invoices less than or equal to 18.10 days is 0.0036.