The Payment Time Case

University of Phoenix

QNT/561 Applied Business Research & Statistic

The Payment Time Case

Consulting firms use many statistical analyses to assess the effectiveness of the systems they design for their customers. A consulting company is introducing a new billing system for a Stockton, CA trucking company that they hope will substantially reduce the amount of time it takes customers to make payments. The typical times from the date on the invoice to the date payment is received using the old billing system has been 39 days or more which exceeds the industry standard payment time of 30 days. This report will talk about the about the 95% confidence interval and some of its calculations. According to Black (2017), “confidence interval is a range of values within which the analyst can declare, with some confidence, the population parameter lies.” When people are using the confidence interval in estimating statistic the tends to us the 95% confidence interval. “The confidence interval formula yields a range (interval) within which we feel with some confidence that the population mean is located,” (Black, 2017). This allows researcher to obtain some kind of confidence on the samples that have been taken.

Stockton, CA is a trucking company who has decided to incorporate a new electronic billing system to help reduce the mean bill payment time by 50%. The mean payment time with the old billing system was approximately equal to, but no less than 39 days, so if the new system can reduce the bill payment time by 50% will be helpful to the company. The consulting company will take 65 random samples of invoices out of 7,823 to be analyzed. The null hypothesis will test the confidence interval of 95% and 99 % with a standard deviation of 4.2 days and the µ will be less than 19.2 days.

**Assuming that standard deviation of the payment times for all payments is 4.2 says, 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.**

To construct the 95% confidence interval estimate, the provided paytime dataset need to be used. According to Black (2017), “z value for a 95% confidence interval is always 1.96. In other words, of all the possible ̅ x x̅ values along the horizontal axis of the diagram, 95% of them should be within a z score of 1.96 from the population mean.” Using the calculation for the paytime data the calculations are as follows:

Looking at the calculations for the 95% confidence interval, we can be 95% certain that the payment time will be less than 19.5 days. The reason for this is because the confidence interval lower limit is 17.09 and the confidence interval higher limit is 19.13, which are both less than 19.5 days.

- Sample Mean – 18.108
- Sample Size – 65
- Confidence Level – 95%
- Standard Error – 4.2/√(65) = 0.52
- Z-Value – 1.96
- Confidence Interval Lower Limit – ((18.108 – (1.96*0.52)) = (18.108 – 1.02) = 17.09
- Confidence Interval Higher Limit – ((18.108 + (1.96*.52)) = (18.108 + 1.02) = 19.13

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

Using the calculation for the paytime data the calculations are as follows:

Based on the 95% confidence interval level calculations, we are 95% confidence that the µ is less than 19.5 days because of the evidence showing that the lower limits of 17.0867 and higher limit of 19.1287 are both under the 19.5 days. It is with confidence that the 95% confidence µ ≤ 19.5 days.

- Population Standard Deviation – 4.2
- Sample Mean – 18.1077
- Sample Size – 65
- Confidence Level – 95%
- Standard Error – 4.2/√(65) = 0.5209
- Z-Value – 1.9600
- Confidence Lower Limit – ((18.1077- (1.9600*0.5209)) = (18.1077 – 1.0210) = 17.0867
- Confidence Higher Limit – ((18.1077 – (1.9600*0.5209)) = (18.1077 + 1.0210) = 19.1287

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

According to Black, (2017), “for a 99% level of confidence, a z value of 2.575 is obtained.” Using the calculation for the paytime data the calculations are as follows:

It is with confidence that the new payment time system can be used because we have a 99% confidence that has a lower limit of 16.7687 and higher limit of 19.4467 which is less that the 19.5 days making the new billing system effective.

- Population Standard Deviation – 4.2
- Sample Mean – 18.1077
- Sample Size – 65
- Confidence Level – 99%
- Standard Error – 4.2/√(65) = 0.52
- Z-Value – 2.575
- Confidence Lower Limit – ((18.1077- (2.575*0.5209)) = (18.1077 – 1.339) = 16.7687
- Confidence Higher Limit – ((18.1077 – (2.575*0.5209)) = (18.1077 + 1.339) = 19.4467

**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.**

With the population mean payment of 19.5 days, there is a probability of that a sample mean of 65 invoices will be ≤ 18.1077 days is 37%.

Z value = (18.1077 – 19.5)/0.52 = -2.6775

**References**

Black, K. (2017). Business Statistics: For Contemporary Decision Making, (9th ed). Hoboken, NJ: Wiley.