MMPC-005 Solved Question Paper

Question 1: Random Variables, Probability Distributions and Managerial Decisions

Rewritten question: What do we mean by a random variable and by a probability distribution? How do probability distributions help managers take better decisions under uncertainty?

Concept of a random variable (student-friendly idea)

A random variable is a numerical way of representing the outcome of an uncertain situation. Instead of saying “demand is high or low”, we attach numbers to the outcome.

• If you toss a coin three times and count how many heads you get, that count (0, 1, 2 or 3) is a random variable.
• If a store manager tracks tomorrow’s demand for a product (50 units, 60 units, etc.), that demand is also a random variable.

In the course material, random variables arise when we map each possible outcome of an experiment (sample space) to a number.

We usually distinguish two types:

• Discrete random variable: Takes separate values such as 0, 1, 2, 3 … (e.g., number of defective items in a batch).
• Continuous random variable: Takes values on a whole interval (e.g., time taken to serve a customer, or a battery’s life in months).

Probability distribution

A probability distribution tells us how likely each possible value (or range of values) of a random variable is.

• For a discrete random variable, we use a probability mass function (PMF), which assigns a probability to each value.
• For a continuous random variable, we use a probability density function (PDF) and work with probabilities over intervals (e.g., probability that demand lies between 90 and 110 units).

Formally, if (X) is a discrete random variable taking values (x_i), the distribution satisfies:

How this helps in managerial decision-making

In real organisations, nearly every important decision involves uncertainty:

• Demand forecasting: Instead of assuming “next month’s demand is exactly 5,000 units”, we model demand as a random variable with a distribution (say normal or Poisson). This helps us plan inventory so that the probability of stockout stays within an acceptable limit.
• Quality control: The number of defective items produced in a day can be treated as a discrete random variable. Its distribution (e.g., binomial or Poisson) helps managers decide sample size and acceptable quality levels.
• Risk analysis: In capital budgeting, the net present value (NPV) of a project can be treated as a random variable. A probability distribution for NPV lets managers estimate the chance of loss and choose between risky alternatives.
• HR planning: Employee turnover can be modelled as a random variable. Its distribution helps HR estimate how many people to hire or train as backup.

So, probability distributions convert vague uncertainty into structured numbers. Once we have numbers, techniques from the MMPC-005 blocks—like expected values, variance, and probability-based decisions—can be applied systematically.

Question 2: Median (Q2) and Sixth Decile (D6) for Grouped Profit Data

Rewritten question: The profits of 100 companies (in ₹ lakhs) are grouped as follows:

20–30 (4), 30–40 (8), 40–50 (18), 50–60 (30), 60–70 (15), 70–80 (10), 80–90 (8), 90–100 (7). Using grouped-data formulas, find the median (Q2) and the 6th decile (D6), and explain what these values mean in business terms.

Step 1: Prepare the cumulative frequency table

Total number of companies (N = 100).

Step 2: Median (Q2)

For grouped data, the median is found from the class containing the median position:

Here:

So we look for the 50th value in the cumulative frequencies. The 50th value lies in the class 50–60 (the cumulative frequency first reaches 60 in this class).

• Median class: 50–60
• Lower boundary (L = 50)
• Cumulative frequency before median class (c_f = 30)
• Frequency of median class (f = 30)
• Class width (h = 10)

Grouped-data median formula:

Substituting values:

Median profit ≈ ₹56.7 lakhs.

Step 3: Sixth decile (D6)

The 6th decile corresponds to the 60th percentile. The position is:

So the 6th decile lies at the 60th observation.

The 60th value also lies in the 50–60 class.

• Decile class: 50–60
• Lower boundary (L = 50)
• Cumulative frequency before class (c_f = 30)
• Class frequency (f = 30)
• Width (h = 10)

Decile formula:

Substituting:

Sixth decile (D_6 = ₹60) lakhs.

Managerial interpretation (real-life flavour)

• The median profit of around ₹56.7 lakhs means that half of the companies earn less than this amount and half earn more. A manager benchmarking performance can see whether their company is above or below this “middle” level.
• The sixth decile of ₹60 lakhs tells us that about 60% of the companies have profits up to ₹60 lakhs, and 40% earn more than this. If you are in the top 40%, you are doing better than the majority; if not, you might need to revisit pricing, cost control or marketing strategy.

Question 3: Why Managers Rely on Sampling Instead of Studying the Whole Population

Rewritten question: Why is sampling so widely used in business and research? In many studies it is practically the only way to learn about a population. Explain the main reasons with examples.

The MMPC-005 sampling unit lists several very practical reasons why sampling is preferred over a full census.

1. Time savings

Studying every unit often takes far too long. Sampling gives quick feedback.

• Example: A bank with 30,000 employees wants to measure job satisfaction. Waiting for 30,000 questionnaires is unrealistic. Surveying, say, 500 well-chosen employees allows HR to act quickly on problems.

2. Lower cost

Contacting fewer units means fewer visits, phone calls, interviews and less data-entry work.

• Example: A market research firm testing a new cold drink flavour does not need to interview all potential customers. Talking to a representative sample in key cities keeps costs manageable while still giving reliable insights.

3. Destructive testing

In many industrial settings, the item being tested is destroyed in the process.

• Example: To check the breaking strength of electric cables, you must pull them until they snap. Testing every cable would leave nothing to sell, so only a sample is tested and the results are generalised.

4. Practical difficulties of a full census

Some populations are so large, scattered or changing that complete coverage is not realistic.

• Example: A company selling soap across India cannot track every single buyer. Samples from different regions (urban/rural, different income groups) are more feasible and still give useful information.

5. Adequate reliability from a well-designed sample

Statistics tells us that if a sample is chosen properly (e.g., via probability sampling), the estimates usually are very close to the true population values, and we can even measure the likely error (standard error, confidence intervals).

• Example: In quality control, checking 100 items from every batch often gives enough confidence to accept or reject the batch. Testing more may give only tiny improvement in accuracy, but at a much higher cost.

6. Better quality of data

With a smaller, focused sample, investigators can spend more time training enumerators, probing answers and checking entries.

• Example: For an employee engagement survey, interviewers can conduct in-depth conversations with a 5% sample, yielding richer, more honest feedback than rushing through superficial questions with everyone.

Because of these reasons, sampling is not a “shortcut” but a deliberate, scientifically justified choice in business research.

Question 4: Joint and Combined Probabilities for Job Applicants

Rewritten question: For applicants to the role of Management Accountant, the probabilities are:

• Probability that an applicant has a post-graduate degree = 0.30
• Probability that an applicant has relevant work experience as a chief financial accountant = 0.70
• Probability that an applicant has both of the above = 0.20

If there are 300 applicants, how many would be expected to have at least one of these attributes (either the degree, the experience, or both)?

Step 1: Use the addition rule of probability

Let:

• (A): applicant has a post-graduate degree
• (B): applicant has work experience as chief financial accountant

The probability that an applicant has either a degree or experience (or both) is:

Substitute the values:

So, 80% of applicants are expected to have at least one of the two qualifications.

Step 2: Convert probability into expected count

Total applicants = 300.

Expected number with either degree or experience:

Answer: Out of 300 applicants, about 240 would be expected to have either the post-graduate degree, or relevant experience, or both.

Managerial insight

From a hiring manager’s perspective, this calculation helps in designing the screening process. If around 240 candidates meet at least one key criterion, the HR team knows that the first-level shortlisting will still leave a large pool, and further filters (interviews, tests, behavioural assessments) will be needed to identify the best fit.

Question 5: Testing the Difference Between Two Proportions

Rewritten question: Suppose we want to test whether the proportion of “successes” in one group is different from that in another group. What is the usual procedure for testing a difference between two proportions? State the null and alternative hypotheses, and explain why we use a pooled estimate of the population proportion.

Typical situation

Assume we have two independent samples:

• Sample 1: size (n_1), with (x_1) “successes” (e.g., customers who like a new product).
• Sample 2: size (n_2), with (x_2) “successes”.

Sample proportions:

Step 1: Hypotheses

• Null hypothesis (H_0): the two population proportions are equal.

• Alternative hypothesis (H_1): depends on the problem.

• Two-sided:

(proportions are different)

• One-sided:

or

Step 2: Pooled estimate of the common proportion

Under (H_0), both populations share the same true proportion (P). The best estimate of this common (P) uses information from both samples:

This is called the pooled proportion.

Step 3: Standard error of the difference in proportions

Using the pooled estimate, the standard error of (p_1 – p_2) is:

Step 4: Test statistic

The test statistic (approximately standard normal for large samples) is:

We compare the calculated (Z) with the critical values from the standard normal table at the chosen significance level (e.g., 5%).

Step 5: Decision

• If (|Z|) is greater than the critical value (e.g., 1.96 for a two-sided 5% test), we reject (H_0) and conclude that the proportions differ significantly.
• Otherwise, we do not reject (H_0); any observed difference could be due to sampling variation.

Why use the pooled estimate?

• Under the null hypothesis, there is one common population proportion, not two. So combining information from both samples gives a more reliable estimate of that common value.
• This pooled (p) leads to the correct standard error formula used in classical two-proportion Z tests.
• Without pooling, you would be implicitly assuming two different proportions, which contradicts the null hypothesis being tested.

Practical example

Suppose a marketing manager wants to test if a new advertisement lifts the proportion of customers who remember the brand. Group 1 (old ad) and Group 2 (new ad) provide two proportions. This test tells the manager whether the observed improvement is statistically meaningful or likely due to chance.

Question 6: Normal Distribution, Warranty Replacements and Profit

Rewritten question: A battery manufacturer claims a warranty of 24 months. In tests, battery life is normally distributed with mean 33 months and standard deviation 4 months.

1. What proportion of batteries will fail before 24 months and therefore need replacement under warranty?
2. If the firm sells 10,000 batteries a year, earning a profit of ₹50 on each sale and spending ₹100 on each warranty replacement, what is the net annual profit?

The normal distribution is covered as a key continuous probability model in the study material.

Step 1: Convert 24 months to a Z-score

Let battery life X follow N(μ = 33, σ = 4). We need P(X < 24).

The question gives the area between (X = 24) and (X = 33) (i.e., between (Z = -2.25) and 0) as 0.4878.

The area to the left of the mean (0) is 0.5. So:

So about 1.22% of batteries will fail before 24 months and be replaced under warranty.

Step 2: Expected number of replacements

Total sales = 10,000 batteries.

Expected replacements:

Step 3: Calculate net profit

Profit per battery sold (before replacements): ₹50

Total profit from sales:

Cost per replacement: ₹100

Total replacement cost:

Net profit:

Answer:

• About 1.22% of the batteries (roughly 122 out of 10,000) will require replacement under warranty.
• The firm’s net annual profit is approximately ₹4,87,800.

Managerial insight

This kind of calculation is essential when setting warranty periods. If the company extended the warranty to, say, 30 months, the proportion of replacements would increase sharply and might wipe out profits. Probability distributions help managers choose a warranty that is attractive to customers but still financially sustainable.

Question 7: Short Conceptual Notes

Rewritten question: Write short explanatory notes (student-friendly) on the following concepts:

1. Discrete and continuous data
2. Relative skewness
3. Cluster versus stratum (in sampling)
4. Type I and Type II errors
5. Least squares criterion

I will briefly explain all five so you can revise them together.

(a) Discrete and continuous data

• Discrete data: Take separate, countable values (often integers). Examples: number of sales calls made in a day, number of defective items in a lot, number of branches of a bank.
• Continuous data: Can take any value in an interval. Examples: time taken to serve a customer, monthly sales revenue, height or weight of individuals.

In the course material, discrete and continuous variables are treated differently when we build frequency distributions and probability models (e.g., binomial vs normal distributions).

(b) Relative skewness

Skewness describes how asymmetric a distribution is.

• If the right tail is longer (a few very large values), the distribution is positively skewed.
• If the left tail is longer (a few very small values), it is negatively skewed.

Relative skewness (or coefficient of skewness) is a numerical measure that shows how far and in which direction a distribution departs from symmetry. One commonly used measure is based on the mean, median and standard deviation:

In business, a positively skewed income distribution may mean that a few customers or products contribute a very large share of revenue, which affects pricing and risk decisions.

(c) Cluster vs stratum (in sampling)

• Stratum (plural: strata): A homogeneous subgroup of the population created on the basis of some characteristic such as region, income level, or department. In stratified sampling, we divide the population into strata and then draw a sample from each stratum.
• Cluster: A naturally occurring group of population units, often heterogeneous within the group. In cluster sampling, we randomly select some entire clusters and study all (or a sample of) elements within them.

Example:

• When studying bank customers, “income group” might be a stratum (high, middle, low), while “branch” might be a cluster.
• In stratified sampling, you take some customers from each income group. In cluster sampling, you might randomly choose some branches and survey all customers in those branches.

The MMPC-005 sampling unit emphasises that strata should be internally similar but different from each other, while clusters are mini-populations that are similar to each other.

(d) Type I and Type II errors

When we test a hypothesis, there are two kinds of mistakes we can make:

• Type I error: Rejecting a true null hypothesis.

Example: Concluding that a new drug is more effective than the old one when, in reality, there is no real improvement. The probability of Type I error is the significance level, usually denoted by α (for example, 5%).

• Type II error: Failing to reject a false null hypothesis.

Example: Concluding there is “no difference” between two teaching methods when in reality the new method is better. The probability of Type II error is denoted by β. Power of a test is 1 – β, the chance of correctly detecting a real effect.

Managers must balance these risks: being too strict (a low α) may increase β, and vice versa.

(e) Least squares criterion

When we fit a line or curve to data—for example, in regression analysis or trend estimation—there are many possible lines. The least squares criterion chooses the “best-fitting” line by minimising the sum of squared vertical distances between the observed values and the line.

Suppose we fit a straight line (y = a + bx) to observed points ((x_i, y_i)). The least squares method chooses (a) and (b) to minimise:

By minimising this sum of squares, we get a line that, overall, stays as close as possible to all data points. In business, this is widely used for demand forecasting, sales trend analysis and cost estimation.