Probability and Statistics Case Study Sample

Executive Summary

This case study evaluates a transaction dataset of N = 250 retail store customers to optimize inventory scheduling and resource allocation. Through empirical probability analysis, the probability of customer spending exceeding $50 is calculated as P(X > 50) = 0.428. An independent two-sample t-test is formulated to determine whether weekday and weekend spending behaviors display statistical divergence. The null hypothesis of equal means is rejected (t(248) = -7.61, p < 0.001), proving that weekend transactions yield higher average values. Finally, frequency analysis of hourly transaction arrivals indicates a high-fidelity fit to a Poisson distribution (λ = 4.2 transactions/hour) over a Binomial distribution. These findings provide quantitative thresholds for operational scaling, stock buffer rates, and labor scheduling.

Introduction and Background

Operational efficiency in small-scale retail enterprises is historically constrained by supply chain unpredictability and lack of data-driven scheduling models. Applying probability models and parametric inference on the retail shop dataset demonstrates that customer purchase behavior is statistically distinct between weekdays and weekends. Quantitative mapping of consumer behavior has transitioned from qualitative estimation to rigorous statistical forecasting (Devore, 2015). By utilizing probability distributions, inventory managers can formulate mathematical thresholds that mitigate stockout risks and minimize inventory holding costs.

This study establishes key analytical frameworks to address several questions: (a) What is the probability of a customer spending more than $50? (b) Is there a significant difference between weekday and weekend spending using a t-test? (c) How well does the transaction frequency fit a Binomial or Poisson distribution? By resolving these questions, the study bridges theoretical probability and applied business logistics, establishing a deterministic basis for supply chain decision-making.

The importance of empirical verification in business analytics is underscored by the volatility of seasonal trends. Operational choices made without statistical backing often lead to capital inefficiencies. By applying foundational rules established since early probability theories (Fermat & Pascal, 1654), modern organizations can model client workflows as predictable mathematical systems rather than volatile random variables.

Methodology and Probability Models

The statistical models implemented in this study rely on classical probability foundations and parametric hypothesis testing. Probability calculations are derived from the empirical cumulative distribution function. For distribution modeling, the Poisson distribution and the Binomial distribution are compared. The Poisson Probability Mass Function (PMF) is defined as:

where λ represents the mean transaction arrival rate per hour, and y is the number of transactions observed (Hogg et al., 2018). Conversely, the Binomial distribution parameters n (total trials) and p (probability of success) are estimated via maximum likelihood estimation (MLE). Goodness-of-fit is assessed using a Chi-Square (χ²) test.

To compare customer spending patterns, a two-sample independent t-test is deployed. The test assumes that customer purchases are independent, normal distributions are approximated under the Central Limit Theorem, and variances are approximately equal (Montgomery & Runger, 2018). The t-statistic is formulated as:

where s_p is the pooled standard deviation, computed as the weighted average of the sample variances of the weekday and weekend spending. This framework ensures that any observed deviations are statistically validated before operations are adjusted. Strict adherence to mathematical formulations prevents assumptions from corrupting the integrity of the data modeling process.

Data Analysis and Statistical Inference

The sample size of the dataset is N = 250. Preliminary analysis indicates that customer spending is positively skewed. Table 1 summarizes the descriptive statistics of customer spending across the two subgroups (weekday vs. weekend).

Table 1

Descriptive Statistics of Customer Spending Subgroups

To calculate the empirical probability that a customer spends more than $50, the frequency of transactions exceeding $50 (k = 107) is divided by the total sample size (N = 250). This yields P(X > 50) = 107 / 250 = 0.428 or 42.8%. To verify if there is a statistically significant difference between weekday and weekend spending, an independent two-sample t-test is formulated under the null hypothesis H₀: μ₁ = μ₂ against the two-tailed alternative H_a: μ₁ ≠ μ₂.

Using the pooled standard deviation s_p = √[((149 * 12.40²) + (99 * 15.65²)) / 248] = 13.80. The t-statistic is calculated as t = (41.25 - 54.80) / (13.80 * √(1/150 + 1/100)) = -13.55 / 1.78 = -7.61. At α = 0.05 and df = 248, the critical value for a two-tailed test is approximately 1.97. Because the calculated |t| = 7.61 > 1.97, the null hypothesis is rejected. The corresponding p-value is p < 0.001, indicating high statistical significance (Wasserstein & Lazar, 2016). Additionally, a 95% confidence interval for the difference between the two population means is constructed: (μ₁ - μ₂) ∈ [-17.06, -10.04].

Next, the transaction frequency distribution is evaluated. For hourly transaction arrivals, the observed frequency distribution is fitted to both a Binomial distribution (n = 10, p = 0.42) and a Poisson distribution (λ = 4.2). A Chi-Square goodness-of-fit test is performed. The Poisson distribution yields a Chi-Square statistic of χ² = 3.24 with df = 4 (p = 0.518), indicating that the observed data does not significantly deviate from a Poisson distribution. The Binomial distribution yields χ² = 14.85 with df = 4 (p = 0.005), indicating a poor fit. Hence, transaction frequency follows a Poisson arrival process.

Results and Interpretation

The statistical test outcomes confirm that weekend customers spend on average $13.55 more than weekday customers. This difference is statistically significant (p < 0.001). The rejection of the null hypothesis demonstrates that weekday and weekend transaction behaviors cannot be managed under a singular operational model. The 95% confidence interval indicates that the difference in average spend is estimated to be between $10.04 and $17.06, ensuring that the increase on weekends is substantial and not a product of random sampling variation.

The goodness-of-fit analysis for transaction arrivals indicates that the Poisson model is highly representative of consumer entry rates. A Poisson distribution with λ = 4.2 transactions per hour suggests that arrivals are independent and occur at a constant average rate, which aligns with standard queuing theory formulations (Ross, 2014). In contrast, the binomial model fails to capture the arrival variance due to its fixed trial assumption. This statistical confirmation of a Poisson arrival process allows managers to apply standard queuing models to predict customer wait times, queue lengths, and register requirements.

This validation provides a baseline for predictive analysis. When arrival rates follow a Poisson distribution, the inter-arrival times conform to an exponential distribution. This enables the calculation of service latency probabilities. In comparison with historic benchmarks, this mathematically sound methodology avoids common estimation pitfalls.

Conclusion and Recommendations

The quantitative analysis of the transaction dataset provides empirical validation for several strategic adjustments. First, the probability that a transaction exceeds $50 is 42.8%, reflecting a highly active consumer base. Second, because weekend customer spend is significantly higher (p < 0.001, mean difference $13.55), marketing and inventory resources should be disproportionately directed toward weekend operations. Weekend product mixes should feature high-margin inventory items to capture the higher average transaction capacity.

Third, because customer arrivals follow a Poisson distribution with an average rate of λ = 4.2 transactions per hour, the store can optimize scheduling using queue-theory models (Wackerly et al., 2008). Specifically, the probability of experiencing more than 6 transactions in any given hour is P(Y > 6) = 1 - P(Y ≤ 6) = 0.133. To maintain a service level where customer wait times are minimal, staffing should be scaled dynamically to match the Poisson arrival probabilities. These data-driven policies reduce operational overhead while maximizing weekend revenue capture.

References

• Devore, J. L. (2015). Probability and Statistics for Engineering and the Sciences. Cengage Learning.
• Fisher, R. A. (1925). Statistical Methods for Research Workers. Oliver and Boyd.
• Hogg, R. V., McKean, J., & Craig, A. T. (2018). Introduction to Mathematical Statistics. Pearson.
• Kolmogorov, A. N. (1933). Foundations of the Theory of Probability. Julius Springer.
• Montgomery, D. C., & Runger, G. C. (2018). Applied Statistics and Probability for Engineers. John Wiley & Sons.
• Ross, S. M. (2014). Introduction to Probability and Statistics for Engineers and Scientists. Academic Press.
• Wackerly, D., Mendenhall, W., & Scheaffer, R. L. (2008). Mathematical Statistics with Applications. Cengage Learning.
• Wasserstein, R. L., & Lazar, N. A. (2016). The ASA's Statement on p-Values: Context, Process, and Purpose. The American Statistician, 70(2), 129-133. https://doi.org/10.1080/00031305.2016.1154108