Quantitative Methods – I April 2026 Solved Assignments
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 the domains of operations management and queuing theory, a comprehensive understanding of call arrival patterns is crucial for optimising staffing levels and upholding service quality within a call center environment. When call arrivals adhere to a Poisson process, probabilistic models offer a means to forecast call volumes over specified timeframes and evaluate the probability of fulfilling service obligations. Within the context of Helios Health Solutions, the managerial imperative is to ascertain the probability of call volumes remaining within acceptable thresholds during Shift A, while also ensuring that Shift B is adequately equipped to manage a minimum call volume per interval,
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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 the realm of high-precision electronics manufacturing, the regulation of processing time is paramount for upholding product quality, operational efficacy, and economic viability. A multinational corporation specialising in high-end processor production has observed divergent processing time distributions between Line A and Line B; these disparities directly impact the probability of inefficiency when benchmark times are surpassed. Through the application of normal distribution principles, man
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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 the realm of pharmaceutical production, the precise adherence of tablets to stringent weight tolerances is paramount, directly impacting product safety, therapeutic effectiveness, and regulatory adherence. Manufacturers frequently assert that the incidence of out-of-specification units is minimal, thereby suggesting robust quality control protocols. Consequently, a compliance officer must statistically validate these assertions through the examination of representative samples drawn from diverse production batches. Through the analysis of a stratified random sample, the calculation of the combined sample proportion,
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
Pinpointing maintenance costs accurately is key to successful planning, budgeting, and policy-making in residential housing. The city’s housing board is working on a standardised system. This system will connect maintenance expenses to the size of the units, offering a clear, data-backed method for future builds. By looking at existing data on unit size and the maintenance costs tied to them, a linear regression model can be created. This model will then provide a way to predict costs reliably. The resulting model will help estimate monthly maintenance for new units. It will also help with fair pricing, resource distribution,
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