Hypothetical Problem: Bank Manager Is Trying To Determine If

Hypothetical Problema Bank Manager Is Trying To Determine If He Needs

Hypothetical Problem: A bank manager is trying to determine if he needs to open another teller for the afternoon shift. At the moment, there are 2 tellers open, each providing service at a rate of 2 customers per hour, with an arrival rate of 3 customers per hour. The manager considers opening another teller, who would also serve at a rate of 2 customers per hour. Using a multiphase system with Poisson arrivals and exponential service times, analyze whether the bank manager should open another teller. Additionally, a hypothesis test analysis is required to determine if there is a statistical difference between the ages of the sexes in the group.

Paper For Above instruction

Introduction

Efficient staffing in banking operations is critical for maintaining customer satisfaction and operational effectiveness. The decision to open additional teller channels involves analyzing customer flow and service capacities, often modeled through queuing theory. In this context, the bank manager's dilemma can be addressed through analytical modeling of service systems, combined with statistical hypothesis testing to understand demographic differences that might influence staffing decisions.

Analysis of Queue System

The queuing system described is best analyzed using the M/M/m model, which is applicable in circumstances involving Poisson arrivals and exponential service times across multiple servers. This model is suitable because customer arrivals and service times follow memoryless (Markovian) properties, and the system comprises multiple servers (tellers).

In this case, initially, there are two tellers (m=2), each servicing at 2 customers per hour (μ=2), with an arrival rate of 3 customers per hour (λ=3). If the manager considers adding an additional teller, the system would shift to three servers (m=3), each with the same service rate.

The primary question is whether the current system configuration results in excessive waiting times or customer congestion, and if adding a third teller reduces wait times significantly. Using the M/M/m queue formulas, the system's utilization (ρ) can be calculated:

- For two tellers:

ρ = λ / (m μ) = 3 / (2 2) = 3 / 4 = 0.75

This indicates the system is 75% utilized, resulting in some waiting, but not excessive.

- For three tellers:

ρ = 3 / (3 * 2) = 3 /6 = 0.5

Reduced utilization means less waiting and improved customer service.

Calculating the probability that all servers are busy (the system's congestion probability) involves more detailed M/M/m formulas, including the Erlang C formula. At an utilization level of 0.75, the probability that all tellers are busy exceeds 20%, implying significant customer wait times. Reducing utilization to 0.5 by adding another teller would decrease wait probabilities and improve service quality.

Statistical analysis supports this decision-making by providing quantitative measures of system performance improvements. Based on the queuing model, opening another teller would substantially decrease wait times, justifying the additional staffing during peak hours.

Hypothesis Testing on Age Differences

The second part of the problem involves conducting a hypothesis test to evaluate if there is a significant age difference between participants of different sexes within a group. The purpose is to determine whether age distributions differ based on gender, which could influence staffing or service strategies in banking.

The null hypothesis (H0) states there is no significant difference in mean ages between males and females (μ_M = μ_F). The alternative hypothesis (H1) suggests there is a significant difference (μ_M ≠ μ_F).

Data analysis begins with calculating the sample means and standard deviations of ages for males and females in the group. Using an independent samples t-test at alpha = 0.05, the test statistic is computed:

t = (Mean_M - Mean_F) / sqrt[(SD_M^2 / n_M) + (SD_F^2 / n_F)]

The degrees of freedom are estimated via the Welch-Satterthwaite equation. The computed t-value is compared to critical t-values to decide whether to reject H0.

Suppose the sample data indicate males have an average age of 45 years with a standard deviation of 10, and females have an average age of 41 years with a standard deviation of 9, with sample sizes of 30 each. Calculations reveal that the t-value falls within the acceptance region, leading to the conclusion that no statistically significant age difference exists between sexes at alpha = 0.05.

This analysis informs the bank’s demographic considerations and staffing approaches, ensuring decisions are data-driven.

Conclusion

The application of queuing theory, specifically the M/M/m model, indicates that adding another teller during peak hours will reduce customer wait times and enhance service quality, justifying the staffing decision. Concurrently, hypothesis testing reveals no significant age differences between sexes in the group, suggesting gender-specific staffing adjustments based on age are unwarranted. Combining operational modeling with demographic statistical analysis provides a comprehensive approach to optimizing banking operations and customer experience.

References

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