Payment Time Case

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The Payment Time Case study involves evaluating whether the implementation of a new electronic billing system by a trucking company has substantially reduced the average bill payment time. The case details the methodology for assessing the effectiveness of the new system through statistical analysis, specifically using sample data to draw inferences about the population mean payment time. The original billing system had an average payment time exceeding industry standards at approximately 39 days, whereas the new system aims to reduce this time by more than 50%, targeting a mean payment time of less than 19.5 days. The firm has collected a sample of 65 invoices processed under the new system, with the population standard deviation known to be 4.2 days, based on previous analyses with similar systems in different companies. The primary objective is to determine, through confidence intervals and probability calculations, whether the new billing system significantly decreases payment times, with specific focus on whether the mean payment time is less than or equal to 19.5 days.

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Introduction

The efficiency of billing systems in the service industry, such as trucking companies, critically impacts cash flow and customer satisfaction. Payment delays can lead to significant financial strain and operational inefficiencies. As such, the implementation of electronic billing systems represents a strategic move to optimize these processes. The case at hand involves evaluating the efficacy of a newly adopted electronic billing system by measuring its impact on payment times. The core question is whether the new system can reduce average payment days from the historical average of approximately 39 days to less than the industry-standard threshold of 30 days, or more specifically, to less than 19.5 days—indicative of a 50% reduction. The significance of this analysis lies in its potential to justify further adoption of the system across other firms, with a quantitative backing based on statistical inference.

Data and Methodology

The data comprises a sample of 65 invoices processed under the new system, with observed payment times available via a spreadsheet. The case assumes a known population standard deviation of 4.2 days, derived from prior system implementations in similar contexts. This allows for the application of z-tests and confidence interval calculations for the population mean. The significance level for the primary analysis is set at 95%, with additional considerations at a 99% confidence level, to assess the robustness of the results. The overarching statistical question is whether the sample mean provides sufficient evidence to infer that the true population mean payment time is less than or equal to 19.5 days.

Constructing the 95% Confidence Interval

Given the sample size (n=65), with a known population standard deviation (σ=4.2 days), the standard error (SE) is calculated as:

SE = σ / √n = 4.2 / √65 ≈ 4.2 / 8.062 ≈ 0.5217

The z-value for a 95% confidence level is approximately 1.96. The sample mean, taken from the dataset (assuming the observed mean is 18.1077 days for this analysis), is used to estimate the population mean.

The confidence interval is computed as:

CI = x̄ ± z SE = 18.1077 ± 1.96 0.5217 ≈ 18.1077 ± 1.022

This yields a confidence interval of approximately (17.0857, 19.1297) days.

Interpretation:

This interval suggests that, with 95% confidence, the true mean payment time lies between about 17.09 and 19.13 days. Since the entire interval is below 19.5 days, this provides strong statistical evidence that the new billing system effectively reduced the payment time to less than 19.5 days.

Assessment at Different Confidence Levels

- At the 95% confidence level, since the interval (17.09, 19.13) does not include 19.5, we are 95% confident that the true average payment time is less than or equal to 19.5 days.

- At the 99% confidence level, the critical z-value increases to approximately 2.576, resulting in a wider interval:

CI = 18.1077 ± 2.576 * 0.5217 ≈ 18.1077 ± 1.345

This yields (16.7627, 19.4537). Despite the wider interval, the upper bound remains just below 19.5 days, indicating that even at the 99% confidence level, we can confidently assert that the mean payment time is less than 19.5 days.

Probability of Observing a Sample Mean Less Than or Equal to 18.1077 Days

Assuming the true population mean is 19.5 days, the sampling distribution of the sample mean is normally distributed with mean 19.5 days and standard deviation equal to the standard error (~0.5217 days). The z-score corresponding to a sample mean of 18.1077 days is:

z = (x̄ - μ) / SE = (18.1077 - 19.5) / 0.5217 ≈ -2.54

Referring to standard normal distribution tables, a z-score of -2.54 corresponds to a cumulative probability of approximately 0.0055, or 0.55%. This implies that under the assumption of a true mean of 19.5 days, there is only about a 0.55% chance of observing a sample mean as low as 18.1077 days. This very low probability reinforces evidence against the null hypothesis and supports the conclusion that the new system significantly reduces payment times.

Conclusion

The statistical analysis demonstrates that the new billing system has significantly decreased the average payment time. The constructed confidence intervals at both 95% and 99% confidence levels are below 19.5 days, providing strong evidence that the true mean payment time is less than this threshold. Furthermore, the low probability of observing such a small sample mean if the true mean were 19.5 days confirms the effectiveness of the new system. These findings justify the adoption of the electronic billing system for broader implementation within the industry, promising improved cash flow and operational efficiency.

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