Description

Quantitative Methods – I

Dec 2025 Examination

 

 

Q1. A call center receives an average of 4 customer complaints per hour. Past records indicate that complaints arrive independently and follow a Poisson distribution. The center operates from 9 AM to 5 PM with 8 working hours per day.

1. What is the probability that exactly 3 complaints will be received in a randomly chosen hour?
2. What is the probability that no complaints will be received in the first hour after opening?
3. Based on your calculations, explain whether it is unusual for the call center to have zero complaints in any given hour. Use probability values to justify your answer in context (10 Marks)

Ans 1.

Introduction

In quantitative methods, probability distributions play a critical role in understanding uncertain events. Among them, the Poisson distribution is especially useful for modeling the occurrence of discrete events over a fixed time interval when the average rate of occurrence is known and the events happen independently. In the given scenario, the call center receives an average of four customer complaints every hour. This average is stable and independent, making the Poisson distribution the most suitable model to describe the situation. The problem involves finding the probability of receiving exactly three complaints in a given hour, the probability of receiving no complaints during the first hour after opening, and interpreting whether the event of having zero complaints in any given hour should be considered unusual.

Concept and Application

Before moving into the analysis, it is important to understand the theoretical foundation of the

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Q2. A manufacturing company produces ball bearings with diameters that are normally distributed, having a mean diameter of 50 mm and a standard deviation of 0.02 mm. For quality control, any ball bearing with a diameter less than 49.97 mm or greater than 50.03 mm is considered defective.

From a day’s production of 10,000 ball bearings:

1. Calculate the expected number of defective ball bearings.
2. Evaluate if the defect rate meets the company’s target of keeping defects below 2%. (10 Marks)

Ans 2.

Introduction

In the field of quality management and quantitative methods, normal distribution plays a vital role in assessing manufacturing performance. When products are produced with slight variations in measurements, the normal distribution provides a way to evaluate how many items fall within acceptable limits and how many are considered defective. In the present scenario, a manufacturing company produces ball bearings with a target diameter of 50 mm and very small deviations around the mean. Quality control standards specify that any ball bearing with a diameter less than 49.97 mm or greater than 50.03 mm will be treated as defective. The problem essentially requires estimating the expected number of defective items and assessing whether this

 

Q3 (A). A machine is designed to fill bottles with 500 ml of juice. A sample of 16 bottles has a mean fill of 495 ml and a standard deviation of 8 ml. At the 5% level of significance, apply an appropriate hypothesis test to determine whether the machine is underfilling bottles. Assume the population is normally distributed. (5 Marks)

Ans 3a.

Introduction

In statistical decision-making, hypothesis testing is a crucial method to evaluate whether a process performs according to expectations. When a machine is designed to fill bottles with a specified quantity, minor deviations may occur due to random variations, but consistent underfilling can indicate a systematic issue. In the given case, the machine is expected to fill 500 ml of juice, yet a sample study shows a lower average. A hypothesis test is therefore applied to

 

 

Q3(B). A real estate analyst wants to study the relationship between the size of a house (in square meters) and its market price (in Rs. lakhs) using simple linear regression. The following data is collected for 6 houses:

House	Size (X) in sq. m	Price (Y) in Rs. lakhs
1	140	85
2	160	95
3	170	98
4	180	102
5	200	110
6	210	115

 

Task:

1. Calculate the regression equation of Y on X.
2. Using this equation, predict the market price of a house with an area of 190 sq. m. Round all final answers to two decimal places. (5 Marks)

Ans 3b.

Introduction

In real estate analysis, understanding the relationship between property size and its market price is essential for accurate forecasting and decision-making. Statistical tools such as regression analysis provide a structured way to model this relationship based on sample data. In the given scenario, information on six houses is used to study how the size of a house, measured in square meters, influences its market price in lakhs of rupees. The goal is to develop a regression