Description
Quantitative Methods – I
Apr 2026 Examination
Q1. A call center receives calls according to a Poisson process at an average rate of 18 calls per hour. The manager wants to allocate staff in two shifts: Shift A covers 8AM–12PM and Shift B covers 12PM–6PM. (a) For Shift A, what is the probability that no more than 30 calls are received? (b) For Shift B, if at least 2 calls must be handled every 30-minute interval to avoid service failure, find the probability that the shift incurs no service failures over its entire duration. (10 Marks)
Ans 1.
Introduction
In service operations such as call centers, understanding the pattern of incoming calls is essential for effective workforce planning and service quality management. When call arrivals follow a random but stable pattern over time, probability-based models help managers estimate demand uncertainty and allocate staff efficiently. Instead of focusing only on average call volume, managers must also consider variability across different shifts and shorter time intervals. This ensures that customer service standards are maintained while avoiding overstaffing or understaffing. In this context, probabilistic reasoning based on arrival behavior allows decision-makers to assess low-demand situations as well as service failure risks. Such theoretical analysis
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Q2. A multinational electronics firm operates two quality-control lines for producing high-end processors. The processing time (in minutes) per unit on Line A follows a normal distribution with a mean of 48 minutes and a standard deviation of 6 minutes, while Line B follows a normal distribution with a mean of 45 minutes and a standard deviation of 9 minutes.
Management has set a benchmark processing time of 55 minutes, beyond which a unit is considered inefficient and incurs additional handling cost. The firm must decide which production line should be prioritized for large-volume orders to minimize inefficiency risk. Evaluate the two production lines by:
(a) Calculating the probability that a randomly selected unit from each line exceeds the benchmark processing time.
(b) Comparing the relative inefficiency risks associated with Line A and Line B using appropriate normal distribution reasoning.
(c) Recommending which line should be prioritized for large-volume production, with a clear statistical justification.
Show all calculations and clearly justify your evaluation and final recommendation. (10 Marks)
Ans 2.
Introduction
In modern manufacturing environments, managing processing time variability is as important as improving average production speed. For high-end electronic components such as processors, delays in quality-control lines can lead to higher handling costs, scheduling disruptions, and customer dissatisfaction. When management sets a benchmark processing time, the goal is not only to improve efficiency but also to minimize the risk of extreme delays. Evaluating production lines requires more than a simple comparison of average processing times. It involves understanding how consistency, variability, and probability of inefficiency influence operational performance. A theoretical assessment of processing time behavior helps managers make informed decisions about which production line should be prioritized for large-volume orders and long-term cost control.
Concept and Application
Before performing numerical calculations, it is essential to understand the conceptual reasoning
Q3(A). A pharmaceutical producer claims that less than 2% of its tablets are outside the allowable weight tolerance. As a compliance officer, you draw a stratified random sample with the following structure: Batch 1: Sample Size 70, Out-of-Tolerance Tablets 1; Batch 2: Sample Size 60, Out-of-Tolerance Tablets 0; Batch 3: Sample Size 120, Out-of-Tolerance Tablets 4. Combine the data and test the manufacturer’s claim at a 5% significance level using an appropriate hypothesis test for proportion. Calculate and interpret the corresponding p-value. Clearly state whether you should reject the claim, while detailing every intermediate step from combined sample proportion to test statistic, and then the decision. (5 Marks)
Ans 3a.
Introduction
In pharmaceutical manufacturing, product quality claims must be supported by statistical evidence to ensure regulatory compliance and consumer safety. When a company asserts that defect levels remain below a specified tolerance limit, sampling-based hypothesis testing becomes an essential evaluation tool. By combining data from multiple production batches, compliance officers can assess whether the manufacturer’s claim is statistically justified. A theoretical understanding of proportion testing helps decision-makers interpret quality risk, detect potential deviations from standards, and take corrective action before issues escalate into
Q3(B). A city’s housing board is in the process of designing a standardized maintenance cost estimation framework for future residential projects. As part of this initiative, the board decides to develop a predictive cost model based solely on the constructed area (in sq ft). They find from a random sample of 8 units the following data: Unit | Sq ft (x) | Maintenance (Rs., y) 1 | 720 | 1845 2 | 880 | 2125 3 | 1150 | 2465 4 | 900 | 2035 5 | 1040 | 2340 6 | 1300 | 2725 7 | 790 | 1920 8 | 950 | 2105 you are required to develop and present a regression-based maintenance cost model for the housing board.
(a) Construct an appropriate linear regression equation to estimate monthly maintenance cost based on constructed area.
(b) Explain how each coefficient in your model contributes to estimating maintenance cost and justify its relevance for policy use.
(c) Use the developed model to generate an estimated maintenance cost for a proposed residential unit of 1050 sq ft, and briefly discuss how this estimate can be incorporated into the board’s standardized pricing framework.
Show all necessary calculations and clearly articulate how the model supports decision-making. (5 Marks)
Ans 3b.
Introduction
In urban housing management, estimating maintenance costs accurately is essential for budgeting, pricing transparency, and long-term project planning. Regression-based models provide a structured way to predict expenses using measurable factors such as constructed area. By understanding the theoretical foundation of regression analysis, housing authorities can design standardized pricing frameworks that support consistency and financial sustainability across residential projects.
Concept and Application
Regression analysis establishes a systematic relationship between maintenance cost and built-up area.
Interpreting the Cost Relationship
