Payment Time Casegrading Guide QNT561 Version 93
The Payment Time Casegrading Guideqnt561 Version 93the Payment Time C
Develop a 700-word report analyzing a new electronic billing system for a trucking company, using statistical analysis to determine if it has significantly reduced payment times. The analysis should include constructing a 95% confidence interval to assess system effectiveness, interpreting confidence intervals for mean payment time, evaluating confidence at different levels, and calculating the probability of observing a specific sample mean if the population mean payment time is 19.5 days. Incorporate appropriate statistical formulas, interpret results in context, and ensure proper APA formatting with tables, graphs, headings, and references.
Paper For Above instruction
The introduction of electronic billing systems represents a strategic advancement to enhance operational efficiency and improve cash flow management within the logistics sector. This paper examines the effectiveness of a newly implemented electronic billing system at a trucking company based in Stockton, California, focusing on whether it significantly reduces the average payment time from customers. Using statistical methodologies, including confidence intervals and probability calculations, the analysis evaluates the system’s impact on payment durations, which historically exceeded industry standards and posed cash flow challenges.
The old billing system at the trucking company recorded a mean payment time exceeding 39 days, surpassing the industry benchmark of 30 days. Management hypothesized that the new electronic system would reduce this average payment time by over 50%, aiming for a mean payment time of less than 19.5 days. To test this hypothesis, a sample of 65 invoices was randomly selected from a total of 7,823 invoices processed during the initial three months of implementation. Given the known population standard deviation of 4.2 days—based on analysis from other companies using similar systems—this sample provided the basis for statistical inference.
Constructing a 95% Confidence Interval
The primary step is to construct a 95% confidence interval (CI) for the population mean payment time (μ). The formula for a confidence interval when the population standard deviation (σ) is known is:
CI = sample mean ± z* (σ / √n)
Where z* is the z-score corresponding to the desired confidence level (1.96 for 95%), σ is 4.2 days, and n is 65. (Sample mean, denoted as x̄, is obtained from the sample data.)
Assuming the calculated sample mean is 18.1 days (derived from the Excel data), the calculation becomes:
Margin of Error = 1.96 (4.2 / √65) ≈ 1.96 0.522 ≈ 1.02 days
Constructing the interval:
Lower bound = 18.1 - 1.02 ≈ 17.08 days
Upper bound = 18.1 + 1.02 ≈ 19.12 days
Hence, the 95% confidence interval for μ is approximately (17.08, 19.12) days. The interpretation signifies that we are 95% confident that the true mean payment time lies within this range. Since the entire interval is below the threshold of 19.5 days, it suggests that the new billing system has effectively reduced the average payment time to a level consistent with the management's hypothesis.
Testing the Effectiveness: Confidence Intervals for μ ≤ 19.5 days
To assess whether we can be 95% confident that μ ≤ 19.5 days, we examine the upper bound of the 95% CI. Because the upper limit of approximately 19.12 days is less than 19.5 days, we can conclude with 95% confidence that the true mean payment time does not exceed 19.5 days, supporting the effectiveness of the new system.
Similarly, considering a 99% confidence interval, the z-score increases to approximately 2.576. Recalculating the margin of error:
Margin of Error at 99% = 2.576 (4.2 / √65) ≈ 2.576 0.522 ≈ 1.345 days
The interval becomes:
Lower bound = 18.1 - 1.345 ≈ 16.76 days
Upper bound = 18.1 + 1.345 ≈ 19.445 days
Since this interval remains entirely below 19.5 days, we can state with 99% confidence that μ ≤ 19.5 days, reinforcing the conclusion that the new billing system markedly reduces the payment duration.
Probability of Observing a Sample Mean ≤ 18.1077 Days if μ = 19.5 Days
Next, the analysis computes the probability of observing a sample mean of 18.1077 days or less, assuming the true population mean is 19.5 days. This involves calculating the z-score:
z = (x̄ - μ) / (σ / √n) = (18.1077 - 19.5) / (4.2 / √65) ≈ (-1.3923) / 0.522 ≈ -2.67
The associated probability, based on the standard normal distribution, is approximately 0.0038. This indicates only a 0.38% chance that, if the true mean is 19.5 days, a sample of this size would yield a mean as low as 18.1077 days or lower. Such a low probability provides strong evidence that the actual population mean is less than 19.5 days, supporting the effectiveness of the new billing system.
Conclusion
Through the statistical analysis, the confidence intervals and probability calculations demonstrate that the new electronic billing system significantly reduces the payment time, aligning with the management’s expectations. The 95% and 99% confidence intervals both lie below the threshold of 19.5 days, conclusively indicating system effectiveness. The exceedingly low probability of observing such a low sample mean under the assumption that μ equals 19.5 days further corroborates this conclusion. These findings suggest that the trucking company’s investment in the new billing process is justified, and the system’s successful implementation can be promoted to other firms in the logistics sector.
References
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