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GNDU Question Paper-2021
Bachelor of Business Administration
BBA 3
rd
Semester
STATISTICS FOR BUSINESS
Time Allowed: Three Hours Max. Marks: 50
Note: There are EIGHT questions. Candidates are required to attempt any FIVE questions.
All questions carry equal marks.
SECTION-A
1. For the matrix show that A = [
𝟏 𝟏 𝟏
𝟏 𝟐 − 𝟑
𝟐 − 𝟏 𝟑
] show that A
3
– 6A + 11 1 = 0 Hence Find A
2. Solve following system of equation using Cramer's rule:
x
1
– x
2
+ x
3
= 6
2x
1
– x
2
+ 2x
3
= 3
3x
1
– x
2
+ x
3
= 3
find x, y, z.
SECTION-B
3. What do you mean by sampling? Discuss in detail various non-random
sampling methods.
4. From the following data of 122 persons find out the modal weight:
Weight (in lbs)
No. of Persons
Weight (in lbs)
No. of persons
100-110
4
140-150
33
110-120
6
150-160
17
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120-130
20
160-170
8
130-140
32
170-180
2
SECTION-C
5.(a) Discuss in detail the properties of regression coefficients.
(b) Calculate the rank correlation of following data:
X
12
15
18
20
16
15
Y
10
18
19
12
15
19
X
18
22
15
21
18
15
Y
17
19
16
14
13
17
6. (a) What do you mean by index numbers? Discuss its utility.
(b) Compute Laspeyres, Paasche, Fisher's, Bowley's and Marshall Edgeworth index
numbers from following data:
1980
1985
Item
Price
Quantity
Price
Quantity
A
12
100
20
120
B
4
200
4
240
C
8
120
12
150
D
20
60
24
50
SECTION-D
7. Discuss in detail properties of Poisson and normal distribution.
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8. A box contains 8 red, 3 white and 9 blue balls. If 3 balls are drawn at random find the
probability that 10
(a) all 3 are red, (b) 2 are red, (c) all 3 are white,
(d) at least 1 is white.
GNDU Answer Paper-2021
Bachelor of Business Administration
BBA 3
rd
Semester
STATISTICS FOR BUSINESS
Time Allowed: Three Hours Max. Marks: 50
Note: There are EIGHT questions. Candidates are required to attempt any FIVE questions.
All questions carry equal marks.
SECTION-A
1. For the matrix show that A = [
𝟏 𝟏 𝟏
𝟏 𝟐 − 𝟑
𝟐 − 𝟏 𝟑
] show that A
3
– 6A + 11 1 = 0 Hence Find A
Ans:
Easy2Siksha.com
Easy2Siksha.com
Easy2Siksha.com
2. Solve following system of equation using Cramer's rule:
x
1
– x
2
+ x
3
= 6
2x
1
– x
2
+ 2x
3
= 3
3x
1
– x
2
+ x
3
= 3
find x, y, z.
Ans:
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Easy2Siksha.com
Easy2Siksha.com
SECTION-B
3. What do you mean by sampling? Discuss in detail various non-random
sampling methods.
Ans: Sampling & Non-Random Sampling Methods – A Storytelling Explanation
Imagine you are the captain of a big ship. On this ship, you have thousands of passengers
traveling with you. Now, one fine evening, you want to know how the food served on your
ship is rated by passengers.
Do you think you will go to every single passenger and ask them about their food
experience? That would take hours and would be nearly impossible. Instead, you will choose
a small group of passengers and ask them. If that small group is wisely selected, their
answers will give you a good idea of what the majority of passengers think.
This process of choosing a small group of people (or items) from a larger population to
represent the whole population is what we call Sampling.
What is Sampling?
In simple words:
Sampling means selecting a part of something to understand the whole thing.
• The large set of people/items we are interested in = Population
• The smaller group we actually study = Sample
Sampling saves time, money, and effort, while still helping us make accurate conclusions.
Just like tasting a spoon of soup from a big pot helps us know if the entire soup needs more
salt, a sample helps us understand the entire population.
Types of Sampling
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There are broadly two types:
1. Random Sampling – Everyone in the population has an equal chance of being
selected. (Like picking lottery tickets.)
2. Non-Random Sampling – Selection is based on convenience, judgment, or other
non-random criteria.
Since the question is about Non-Random Sampling Methods, let’s walk through them one
by one – but instead of just listing definitions, let’s imagine real-life examples so it feels like
a story.
Non-Random Sampling Methods
1. Convenience Sampling – "The Easy Way Out"
Imagine you are standing in a shopping mall and need to know people’s opinions about a
new smartphone. Instead of roaming around the entire city, you just ask the people who are
already in the mall because they are easy to reach.
This is called Convenience Sampling – choosing people who are easiest to access.
• Pros: Quick, cheap, saves effort.
• Cons: Not always accurate because you may miss people who are not conveniently
available.
It’s like asking only your friends about your singing talent – they might say “you’re amazing”
just because they like you, not because you actually sing well.
2. Judgmental / Purposive Sampling – "The Expert’s Choice"
Suppose a fashion brand wants to test a new luxury clothing line. Instead of asking
everyone, they only ask fashion designers, stylists, and influencers, because these people
are believed to give more valuable feedback.
This is Judgment Sampling – where the researcher uses their own judgment to select who
should be included in the sample.
• Pros: Useful when expert opinions are needed.
• Cons: The researcher’s bias may affect the choice.
It’s like asking your cricket coach whether you should pursue cricket as a career – their
opinion may carry more weight than asking your next-door neighbor.
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3. Quota Sampling – "Divide and Choose"
Think of a journalist who wants to interview 100 people about a new government policy. But
the journalist doesn’t just talk to anyone randomly. Instead, they decide:
• 50 men and 50 women
• 25 people from rural areas and 75 from urban areas
This is Quota Sampling – the population is divided into categories, and then a fixed number
(quota) of samples are taken from each group.
• Pros: Ensures representation from different categories.
• Cons: Still not truly random, so bias may creep in.
It’s like making sure your cricket team has a balance of bowlers, batsmen, and all-rounders
before selecting players.
4. Snowball Sampling – "The Chain Reaction"
Imagine you are researching a secretive group of artists who rarely reveal their identity. You
manage to find one artist, who then introduces you to another, and that one introduces you
to more. Slowly, your network grows – like a snowball rolling down a hill, getting bigger
and bigger.
This is Snowball Sampling – existing participants help you find more participants.
• Pros: Very useful for studying hidden or hard-to-reach groups.
• Cons: The sample may not represent the population fairly since it spreads within
certain circles only.
It’s like making friends – you meet one person at a party, they introduce you to their friends,
and the circle keeps expanding.
5. Voluntary Sampling – "Whoever Wants to Join"
Imagine you put up an online survey link saying: “Share your opinion on exam stress!”
Whoever feels like it can click and respond.
This is Voluntary Sampling – where participants choose themselves to be part of the study.
• Pros: Easy and inexpensive.
• Cons: Usually attracts people with strong opinions, which can make the results
biased.
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It’s like asking, “Who wants free pizza?” – only those who love pizza will rush, while others
stay away.
Why Do We Use Non-Random Sampling?
At this point, you might wonder – if non-random sampling has so many biases, why do
researchers even use it?
The answer is simple:
• Sometimes it’s too expensive or time-consuming to do random sampling.
• Sometimes the population is hard to reach (like secretive groups, rare patients, or
specialized professionals).
• Sometimes researchers just want a quick idea rather than perfect accuracy.
So, even though it is less scientific than random sampling, non-random sampling is practical
and widely used.
Bringing It All Together
Let’s go back to our ship example. As the captain:
• If you ask any passenger you bump into, that’s Convenience Sampling.
• If you only ask the chef and food experts on board, that’s Judgment Sampling.
• If you decide to ask 10 people from first class, 20 from economy, and 5 from crew,
that’s Quota Sampling.
• If you meet one passenger who introduces you to another and so on, that’s Snowball
Sampling.
• If you announce, “Anyone who wants to give feedback, please come forward!”,
that’s Voluntary Sampling.
Each method gives you some information, though not always perfectly accurate.
Conclusion
Sampling is like tasting a small portion of food before eating the entire meal. Non-random
sampling, though not perfect, helps researchers gather insights quickly, cheaply, and in
situations where random sampling is difficult.
• Convenience Sampling = Easy access
• Judgment Sampling = Expert choice
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• Quota Sampling = Representation by groups
• Snowball Sampling = Chain referrals
• Voluntary Sampling = Self-selection
In short, non-random sampling methods are practical shortcuts – not always accurate, but
often necessary in real-world research.
4. From the following data of 122 persons find out the modal weight:
Weight (in lbs)
No. of Persons
Weight (in lbs)
No. of persons
100-110
4
140-150
33
110-120
6
150-160
17
120-130
20
160-170
8
130-140
32
170-180
2
Ans: The Story of Finding the Modal Weight
Imagine you are at a fair. There’s a big weighing machine, and people are queuing up to
check their weights. The fair organizers have noted down everyone’s weight in groups or
classes instead of writing each person’s exact weight. Why? Because handling 122 exact
numbers would be messy. So, they decided to say:
• “Okay, 4 people are between 100–110 lbs.”
• “6 people are between 110–120 lbs.”
• “20 people are between 120–130 lbs.”
• and so on.
By the end, they had the complete distribution of weights of 122 persons.
Now, one of the organizers suddenly asks you:
“Hey, can you tell me the modal weight of this crowd?”
At first, you might scratch your head. “Modal weight? What’s that?”
Well, let’s start from the basics.
Step 1: Understanding the Mode
The word mode simply means “the most common value.” In everyday life, the mode is just
the thing that appears most often.
For example:
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• If in a classroom, 10 students love pizza, 15 love burgers, and 8 love noodles, then
burgers are the mode of favorite food because they are chosen by the maximum
number of students.
• If you check which shoe size is most common in a school, that size would be the
mode of shoe sizes.
So here, the modal weight means: the weight group (or class) where the maximum number
of people fall.
But there’s a catch—our weights are not individual numbers. They are grouped into class
intervals like 100–110, 110–120, etc. That means we need a formula to find the exact modal
value instead of just picking the group.
Step 2: Observing the Data
Let’s carefully write the table again so it’s easy to see:
Weight (in lbs)
No. of Persons
100–110
4
110–120
6
120–130
20
130–140
32
140–150
33
150–160
17
160–170
8
170–180
2
Now, look at the column “No. of Persons.”
The highest number is 33, which belongs to the group 140–150.
So, the modal class is 140–150 lbs.
This is the interval where the mode lies.
Step 3: The Formula for Mode
When the data is in class intervals, we don’t just say “mode is 140–150.” Instead, we use a
special formula to calculate the exact mode:
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Where:
• L = lower boundary of the modal class
• f1 = frequency of the modal class
• f0 = frequency of the class just before the modal class
• f2 = frequency of the class just after the modal class
• h = class size (width of the interval)
Step 4: Identifying the Values
From our data:
• Modal class = 140–150
• L=140
• f1=33
• f0=32 (the frequency of the previous class, i.e., 130–140)
• f2=17 (the frequency of the next class, i.e., 150–160)
• h=10 (since each class covers 10 lbs)
Step 5: Substituting into the Formula
Now, plug the values into the formula:
Simplify step by step:
• Numerator = 33−32=133 - 32 = 133−32=1
• Denominator = 66−32−17=66−49=1766 - 32 - 17 = 66 - 49 = 1766−32−17=66−49=17
So,
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Step 6: Telling the Result Like a Story
So, the modal weight of this group of 122 people is approximately 140.6 lbs.
That means if you were to “guess” the most typical weight that people at this fair have,
you’d say it’s around 140.6 lbs. It’s not the average (mean), and it’s not the middle value
(median). Instead, it represents the weight group where the crowd is thickest.
Think of it like this:
• Imagine you are walking into a room full of these 122 people.
• You look around and ask, “Which weight range do I bump into the most?”
• Most likely, you’ll meet people around the 140–150 lbs range.
• And with the formula, we sharpened that answer to about 140.6 lbs—as if we
zoomed in with a microscope on that busy crowd.
Why This Feels Special
The mode is like the “crowd favorite.” If mean is like dividing sweets equally among
everyone, and median is like finding the exact middle person in a line, then mode is like
shouting, “Who is the star of the show? Who has the maximum fans?”
Here, the star is clearly the 140–150 lbs group, and the precise modal weight is 140.6 lbs.
Final Answer
Modal Weight≈140.6 lbs
SECTION-C
5.(a) Discuss in detail the properties of regression coefficients.
Ans: 5(a) Properties of Regression Coefficients
Imagine two friends, Height (X) and Weight (Y). They always walk together, and people
often wonder: “If one grows, how much does the other change?” This simple curiosity is the
heart of regression analysis. In statistics, regression helps us measure how one variable
moves in relation to another. And the magical tool inside regression is called the regression
coefficient.
The regression coefficient is like a translator between two friends. It tells us:
• If X changes by 1 unit, how much will Y change (on average)?
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• And vice versa.
Now, just like a good translator must follow certain rules to avoid confusion, regression
coefficients also have their own set of properties. These properties ensure that the
relationship we calculate between variables is consistent, reliable, and meaningful. Let’s
explore them one by one in a story-like journey.
1. Regression Coefficients Can Be Positive or Negative
Think of regression coefficients as the “mood” of a relationship.
• If the coefficient is positive, it means the variables are best friends – when one
increases, the other also increases. For example, height and weight usually rise
together.
• If the coefficient is negative, it’s like a push-pull relationship – when one grows, the
other shrinks. For example, as the speed of a vehicle increases, the time taken to
travel a fixed distance decreases.
This property makes regression flexible because it can capture both types of relationships in
real life: supporting (positive) or opposing (negative).
2. They Are Not Restricted Between –1 and +1
Here’s where regression coefficients differ from correlation coefficients. Correlation is
always trapped between –1 and +1, like a bird in a cage. But regression coefficients are free
birds – they can be small decimals like 0.02 or huge numbers like 250.
For example:
• If we measure the relation between weight in kilograms and height in centimeters,
the regression coefficient could be very small (say 0.3).
• But if we change height into millimeters, the coefficient could suddenly become 30.
This tells us regression coefficients depend on the scale of measurement.
3. Both Regression Coefficients Have the Same Sign
In regression, we usually calculate two lines:
• Regression of Y on X
• Regression of X on Y
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The fascinating part is: both coefficients carry the same sign. If one is positive, the other will
also be positive. If one is negative, the other will also be negative.
Why? Because they are both describing the same friendship between X and Y, just from
different sides. You can’t have one saying “we are good friends” and the other saying “we
hate each other.”
4. Relation with Correlation Coefficient
Here’s a magical property:
This means the square of the correlation coefficient is equal to the product of the two
regression coefficients. It’s like correlation is the “common link” between the two regression
lines.
For example, if the regression coefficient of Y on X is 0.8 and that of X on Y is 0.5, then:
This shows that correlation is the bridge connecting both regression coefficients.
5. Regression Coefficients Depend on Units of Measurement
Imagine measuring height in meters vs. centimeters. The correlation will remain the same
because correlation is unit-free. But regression coefficients will change dramatically.
Example:
• If 1 meter = 100 cm, then the regression coefficient will be multiplied or divided
depending on how we change the unit.
This property reminds us to always be careful about measurement units before
interpreting regression coefficients.
6. Independent of Origin, Dependent on Scale
This is one of the most exam-friendly points, but let’s simplify it with a story.
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Suppose we shift the starting point of measurement. Instead of measuring temperature
from 0°C, we suddenly start from 10°C. Will the regression change?
• No, because shifting the origin (adding or subtracting a constant) does not affect
regression coefficients.
But if we change the scale – for example, measuring temperature in Fahrenheit instead of
Celsius – then the regression coefficient will change.
So remember:
• Independent of origin (shift doesn’t matter).
• Dependent on scale (multiplication/division matters).
7. Geometric Meaning
On a graph, regression coefficients actually represent the slope of the regression line.
Imagine drawing the best-fit line through a scatter of points:
• A steep slope = large regression coefficient.
• A flat slope = small regression coefficient.
This property helps us visualize regression not just as a number, but as an actual line
explaining the relationship.
8. Both Lines Pass Through the Mean Point
Another beautiful property: both regression lines always pass through the point (𝑥
,
𝑦)
This is logical because averages are the center of the data. So no matter which regression
line we draw, it must touch the “meeting point” of both averages. It’s like a promise
regression lines make – “We will always meet at the mean.”
Conclusion
To sum it up, regression coefficients are like storytellers of relationships between variables.
They can be positive or negative, large or small, but they always obey certain rules:
• Both have the same sign.
• Their product connects to correlation.
• They change with scale but not with origin.
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• They represent slopes of regression lines.
• And they always pass through the mean point.
So next time you see a regression equation, don’t think of it as a boring formula. Think of it
as a friendship story between two variables – sometimes supporting, sometimes opposing,
but always following their own set of principles.
(b) Calculate the rank correlation of following data:
X
12
15
18
20
16
15
Y
10
18
19
12
15
19
X
18
22
15
21
18
15
Y
17
19
16
14
13
17
Ans: A Fresh Beginning: Imagine You’re a Judge at a Talent Show
Suppose you are the judge of a small singing competition in your town. Six singers perform
in front of you, and you give them scores. At the same time, another judge (your friend) also
gives scores. Now, you both are curious — do your rankings agree with each other?
If both of you rank the singers in the same order (for example, if you think Singer A is best
and your friend also thinks Singer A is best), then your rankings are in perfect agreement.
But if your opinions are completely opposite, then the rankings are in total disagreement.
This is exactly where Rank Correlation comes in — it helps measure how similar or dissimilar
two sets of rankings are.
The most common tool we use here is called Spearman’s Rank Correlation Coefficient (ρ,
pronounced as “rho”).
The formula is:
Where:
• d = difference between the ranks of each pair (X and Y)
• n = number of observations (or contestants, in our story)
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If ρ = +1 → Perfect agreement
If ρ = –1 → Perfect disagreement (completely opposite rankings)
If ρ = 0 → No correlation (your rankings are independent of each other)
So now, let’s use this storytelling lens to solve your actual problem.
Step 1: Understanding the Data
We are given two sets of X and Y values (like the two judges’ scores). Let’s write them
clearly:
First set of values:
X: 12, 15, 18, 20, 16, 15
Y: 10, 18, 19, 12, 15, 19
Second set of values:
X: 18, 22, 15, 21, 18, 15
Y: 17, 19, 16, 14, 13, 17
So, it’s like we have two groups of contestants. We’ll calculate Spearman’s Rank Correlation
for each group separately.
Step 2: First Dataset (Group 1)
(a) Ranking X values
X values = 12, 15, 18, 20, 16, 15
Let’s arrange them in ascending order and assign ranks:
• 12 → Rank 1
• 15, 15 → both are equal, so they share the average of Rank 2 and 3 → Rank 2.5 each
• 16 → Rank 4
• 18 → Rank 5
• 20 → Rank 6
So, Ranks of X:
[1, 2.5, 5, 6, 4, 2.5]
(b) Ranking Y values
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Y values = 10, 18, 19, 12, 15, 19
Arrange them:
• 10 → Rank 1
• 12 → Rank 2
• 15 → Rank 3
• 18 → Rank 4
• 19, 19 → average of Rank 5 and 6 → 5.5 each
So, Ranks of Y:
[1, 4, 5.5, 2, 3, 5.5]
(c) Finding ddd and d2d^2d2
Now, let’s calculate the difference (d) between X-rank and Y-rank, and then square it.
X
Y
Rank X
Rank Y
d = (Xrank – Yrank)
d²
12
10
1
1
0
0
15
18
2.5
4
-1.5
2.25
18
19
5
5.5
-0.5
0.25
20
12
6
2
4
16
16
15
4
3
1
1
15
19
2.5
5.5
-3
9
(d) Applying the formula
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So, the correlation is weakly positive for the first dataset.
Step 3: Second Dataset (Group 2)
(a) Ranking X values
X = 18, 22, 15, 21, 18, 15
Arrange them:
• 15, 15 → Rank (1+2)/2 = 1.5 each
• 18, 18 → Rank (3+4)/2 = 3.5 each
• 21 → Rank 5
• 22 → Rank 6
Ranks of X: [3.5, 6, 1.5, 5, 3.5, 1.5]
(b) Ranking Y values
Y = 17, 19, 16, 14, 13, 17
Arrange them:
• 13 → Rank 1
• 14 → Rank 2
• 16 → Rank 3
• 17, 17 → Rank (4+5)/2 = 4.5 each
• 19 → Rank 6
Ranks of Y: [4.5, 6, 3, 2, 1, 4.5]
(c) Finding ddd and d2d^2d2
X
Y
Rank X
Rank Y
d
d²
18
17
3.5
4.5
-1
1
22
19
6
6
0
0
15
16
1.5
3
-1.5
2.25
21
14
5
2
3
9
18
13
3.5
1
2.5
6.25
15
17
1.5
4.5
-3
9
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(d) Applying the formula
So, the second dataset also shows a weakly positive correlation.
Step 4: The Final Story Twist
Now, what do these results (0.186 and 0.214) actually tell us?
Imagine you and your friend (the two judges) mostly agree a little, but not strongly.
Sometimes you both give close ranks, but sometimes your opinions differ widely (like in the
first dataset where X=20 and Y=12 had a huge mismatch).
That’s why the correlation is low but still positive. In simple words, the rankings show that
there is a small tendency to agree, but not strongly.
Wrapping It Up in a Story Style
So, Spearman’s Rank Correlation is like checking how well two judges (or two datasets)
agree on their rankings. In our problem, we saw that:
• For the first dataset, the agreement between X and Y rankings was ρ = 0.186.
• For the second dataset, the agreement was ρ = 0.214.
Both are close to zero, meaning the rankings are weakly positively related — a little
agreement, but not too much.
This way, you can now remember that Rank Correlation is nothing but comparing how two
people (or two measures) view the same set of contestants (or data).
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And just like in a talent show, sometimes we agree with our friend’s judgment, sometimes
we don’t — and Spearman’s Rank Correlation tells us exactly how much.
Final Answer:
• For the first dataset, ρ ≈ 0.186
• For the second dataset, ρ ≈ 0.214
6. (a) What do you mean by index numbers? Discuss its utility.
Ans: 6. (a) What do you mean by index numbers? Discuss its utility.
Imagine this: You walk into a grocery store with ₹500 in your pocket. Ten years ago, that
same ₹500 could easily buy you a full bag of rice, wheat, sugar, and vegetables. But today,
with the same ₹500, you can barely fill half the bag. Why did this happen? Prices changed.
Some things became costlier, some became cheaper, and the overall value of money
changed.
Now, suppose you want to measure this change in a single number – something that tells
you whether life has become costlier or cheaper compared to the past. That magic
measuring tool is called an Index Number.
What are Index Numbers?
In the simplest sense, an Index Number is a statistical tool that shows changes in a variable
or group of variables over time.
Think of it as a thermometer, but instead of measuring temperature, it measures changes in
prices, quantities, wages, industrial output, agricultural production, stock market
performance, etc.
• If the index number goes up, it means prices or production have increased.
• If it goes down, it means they have decreased.
So, instead of saying, “The price of rice went up by 20%, sugar by 10%, wheat by 15%…”
(which sounds confusing), we summarize all of this into a single figure like: “Overall, prices
went up by 12%.”
That single number is what makes index numbers so powerful.
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Why Do We Need Index Numbers?
Let me take you into another story. Suppose you are the Finance Minister of India. Every
year you have to decide salaries, subsidies, taxes, and development budgets. But how will
you know if people’s incomes are enough to live comfortably? Or whether inflation is
hurting the common man?
You cannot go to every shop and note the price changes of hundreds of commodities.
Instead, you rely on Index Numbers like the Consumer Price Index (CPI), which tells you
whether the cost of living has increased or decreased.
Thus, index numbers act like guiding lights for governments, businesses, and individuals.
Utilities of Index Numbers
Now, let’s discuss their main uses in a clear and engaging way:
1. Measuring Changes in the Value of Money (Purchasing Power)
The value of money is not fixed. Sometimes, money buys more goods, sometimes less. Index
numbers, especially the Consumer Price Index (CPI), help measure this.
• If CPI goes up, it means inflation – money buys less than before.
• If CPI goes down, it means deflation – money buys more.
For example, if the CPI shows a rise of 10%, it means that what you bought for ₹100 last
year now costs ₹110.
2. Cost of Living Adjustments
Salaries, pensions, and wages are often revised based on index numbers.
For example: A government employee’s salary might automatically increase when the
Dearness Allowance (DA) is adjusted according to the Consumer Price Index. This ensures
that even if prices rise, the employee’s real income is not badly affected.
3. Economic Planning and Policy Making
For policymakers, index numbers are like a doctor’s report card about the health of the
economy.
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• Industrial production index tells whether factories are producing more or less.
• Agricultural production index shows food grain output.
• Wholesale Price Index (WPI) shows inflation trends.
These numbers guide the government in making decisions about imports, exports, subsidies,
taxation, and investment.
4. Business Forecasting
Businesses also depend on index numbers. For example, if the stock market index is rising,
it indicates investor confidence and growth. If it falls, businesses may postpone expansion
plans. Similarly, companies study production and sales indices to predict future demand.
5. Comparison Over Time and Place
Index numbers allow us to compare:
• How living costs today differ from those 10 years ago.
• How the inflation in India compares with that in the USA.
Without index numbers, such comparisons would be impossible.
6. Useful in International Trade
Governments use index numbers to fix import and export duties. For example, if the global
crude oil price index rises, India may adjust its fuel pricing or trade policies accordingly.
A Human Example
Let’s imagine a school teacher, Mr. Sharma. In 2000, his monthly salary was ₹10,000, and
that was enough for him to run his family smoothly. In 2025, his salary is ₹40,000. Now, it
looks like his income has increased four times, but when he goes shopping, he realizes that
everything – from milk to school fees – has become much more expensive.
So, even with ₹40,000, his standard of living is not as comfortable as it seems. To judge
whether he is truly better off or worse off, we need Index Numbers that measure how
much the prices of essential goods have changed.
Conclusion
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To sum up in simple words:
• Index Numbers are like economic barometers that measure changes in prices,
production, wages, and other variables over time.
• They help governments in policy-making, businesses in forecasting, workers in wage
negotiations, and individuals in understanding the cost of living.
Without index numbers, studying economics would be like sailing in the sea without a
compass. They convert scattered, confusing data into a single meaningful figure, making it
easier to analyze and act upon.
So, the next time you feel your ₹500 note doesn’t buy as much as it used to, remember –
that’s exactly what index numbers are meant to show us!
(b) Compute Laspeyres, Paasche, Fisher's, Bowley's and Marshall Edgeworth index
numbers from following data:
1980
1985
Item
Price
Quantity
Price
Quantity
A
12
100
20
120
B
4
200
4
240
C
8
120
12
150
D
20
60
24
50
Ans: Imagine you’re the shopkeeper of a small town bazaar in 1980. You sell four staples—
A, B, C, and D—to your regulars. Five years later, in 1985, the same faces come by, but
prices and shopping baskets have shifted a bit. Your question is simple: “By how much have
prices really risen?” Different index-number “lenses” answer this in slightly different ways.
Today, we’ll walk through five classic lenses—Laspeyres, Paasche, Fisher’s Ideal, Bowley’s
(Dorbish–Bowley), and Marshall–Edgeworth—turning a dry calculation into a clear little
story.
1) Meet the data (our bazaar’s receipts)
• Base year (1980): prices p0 quantities q0
• Current year (1985): prices p1, quantities q1
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Item
p0
q0
p1
q1
A
12
100
20
120
B
4
200
4
240
C
8
120
12
150
D
20
60
24
50
Let’s total the key sums we’ll need:
These four totals are the heartbeat of all five indices.
2) Laspeyres Price Index (base-year quantities)
Laspeyres asks: “If people kept buying the 1980 basket, how much would it cost at 1985
prices?”
Interpretation: Holding the old habits fixed, prices appear to be about 36.54% higher.
Why it tends to be a bit high: consumers usually adjust their basket when prices change
(substitution). Laspeyres ignores that adjustment, so it can overstate inflation.
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3) Paasche Price Index (current-year quantities)
Paasche asks: “Given what people actually bought in 1985, how does the 1985 bill compare
to pricing that same 1985 basket at 1980 prices?”
Interpretation: Pricing the current habits, we see prices about 38.26% higher.
Why it can be a bit low or high: Paasche uses current quantities (after people have already
adjusted to price changes). Because consumers may switch toward relatively cheaper items,
Paasche sometimes reads lower than Laspeyres. Here, interestingly, it lands slightly higher
than L—this can happen when quantity shifts don’t move strongly toward cheaper items
overall (item-by-item stories matter).
4) Fisher’s Ideal Index (the “compromise” square root)
Fisher’s index blends both worlds—base-year and current-year quantities—by taking the
geometric mean of Laspeyres and Paasche:
Interpretation: Fisher usually sits neatly between L and P, and it often earns the nickname
“ideal” because it balances the two perspectives and passes several desirable tests (like time
reversal and factor reversal under certain conditions).
5) Bowley’s (Dorbish–Bowley) Index (the “simple average”)
Another fair-minded approach is just to average Laspeyres and Paasche arithmetically:
Interpretation: Bowley (also called Dorbish–Bowley) yields the same value to two decimals
here as Fisher—137.40—though the philosophy differs (arithmetic mean vs. geometric
mean). In many datasets they’re close but not identical.
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6) Marshall–Edgeworth Index (average quantities)
Marshall–Edgeworth uses the average of base and current quantities as weights. A
convenient (algebraically equivalent) way to compute it is:
Plug in our totals:
Interpretation: By weighting prices with a blend of old and new buying patterns, M–E gives a
middle-of-the-road view, coming out very close to Fisher and Bowley.
7) Gather the verdict (like totals on a cash register)
• Laspeyres (L): 136.54
• Paasche (P): 138.26
• Fisher’s Ideal (F): 137.40
• Bowley (Dorbish–Bowley): 137.40
• Marshall–Edgeworth: 137.44
All five indices tell a consistent story: compared to 1980, the price level in 1985 is roughly
37% higher. The small differences are the “personality” of each method—what basket is
held fixed, whether we split the difference, and whether we take an arithmetic or geometric
average.
8) Why the numbers make sense (quick intuition checks)
1. Bounded by L and P: Fisher and Marshall–Edgeworth lie between Laspeyres and
Paasche (here, 136.54≤{137.40,137.44}≤138.26) That’s a good sanity check.
2. Item B’s price didn’t change: Item B stayed at ₹4, even though its quantity rose from
200 to 240. Stable price items help “anchor” the index.
3. Some prices rose more than others:
o A: ₹12 → ₹20 (big jump) with quantity rising 100 → 120.
o C: ₹8 → ₹12 with quantity rising 120 → 150.
o D: ₹20 → ₹24 with quantity falling 60 → 50.
When more weight (quantity) sits on items with larger price increases, indices
tend to be higher.
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9) A short narrative to remember them
Think of Laspeyres as a nostalgic shopper who insists, “Let’s pretend we still buy the exact
1980 basket—what would it cost today?” Paasche is the pragmatic shopper: “We actually
buy a slightly different basket now; price that fairly.” Fisher is the diplomat who says, “Let’s
meet in the middle, respecting both viewpoints, but combine them elegantly.” Bowley is the
friendly mediator who suggests a straightforward average of the two. And Marshall–
Edgeworth is the thoughtful neighbor who weighs prices using both old and new habits
together.
In everyday policy or business, we don’t want to be fooled by only one viewpoint. If we
freeze the old basket (Laspeyres), we might exaggerate inflation if people have already
shifted toward cheaper options. If we look only at the new basket (Paasche), we might
understate the bite felt by those who didn’t or couldn’t change their habits. Fisher, Bowley,
and Marshall–Edgeworth are attempts to be fair arbiters—different recipes for combining
the two truths.
10) Final takeaway (what to tell the examiner)
Using the given data with 1980 as the base year and 1985 as the current year, we find:
• Laspeyres Index: 136.54
• Paasche Index: 138.26
• Fisher’s Ideal Index: 137.40
• Bowley’s (Dorbish–Bowley) Index: 137.40
• Marshall–Edgeworth Index: 137.44
So, prices in 1985 are about 37% higher than in 1980. The indices cluster tightly, which
strengthens our confidence in the conclusion. If you imagine the bazaar’s shelves across
those five years, the bill you’d pay today for roughly similar shopping has climbed—no
drama, just a steady rise captured through five honest lenses.
SECTION-D
7. Discuss in detail properties of Poisson and normal distribution.
Ans: Properties of Poisson and Normal Distribution
Imagine you are standing in a busy city square. Cars are honking, people are crossing roads,
and buses keep arriving at the bus stop. While looking around, you suddenly wonder:
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“Is there any way to predict random things in life — like how many buses will arrive in the
next 10 minutes, or how tall people in this crowd are likely to be?”
This curiosity takes us into the beautiful world of probability distributions. Among them,
two very famous ones are the Poisson distribution and the Normal distribution. Let’s take
them one by one, almost like meeting two interesting characters in a story.
1. Poisson Distribution – The Story of Random Events
Think of Poisson distribution as the distribution of rare, countable events. It answers
questions like:
• How many customers will enter a shop in one hour?
• How many phone calls will a call center receive in a minute?
• How many misprints might appear on a single page of a book?
Here’s the fun part: Poisson doesn’t care exactly when something happens, but rather how
many times it happens in a fixed interval (time, area, or distance).
Properties of Poisson Distribution
1. Discrete Nature:
It deals with counts — 0, 1, 2, 3, … You can’t have 2.5 customers or 3.7 printing
errors.
2. Parameter (λ – lambda):
The whole distribution depends on a single number, λ (average rate of occurrence).
If a shop usually gets 10 customers per hour, λ = 10.
3. Mean and Variance:
In Poisson, something magical happens:
This makes it unique compared to many other distributions.
4. Probability Formula:
The probability of observing exactly x events is:
where e is about 2.718.
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5. Additive Property:
If one factory receives defects following Poisson(λ1), and another has Poisson(λ2),
then together they behave like Poisson(λ1 + λ2).
6. Skewness:
For small λ, the distribution is skewed (more probability mass on the left). As λ
increases, it starts looking more like a bell curve (Normal distribution).
So, Poisson is like counting how many “surprises” show up in a box of time or space.
2. Normal Distribution – The Bell-Shaped Hero
Now, let’s meet the superstar of statistics: the Normal distribution, also known as the bell
curve. Unlike Poisson, which counts rare events, Normal describes measurements that
naturally cluster around an average.
Think of heights of students in a classroom. Not everyone is exactly 165 cm, but most
students will be close to the average, with fewer students being extremely tall or extremely
short. If we plot this data, it forms that smooth, symmetric bell shape.
Properties of Normal Distribution
1. Continuous Distribution:
Unlike Poisson, which is about discrete counts, Normal is continuous. It covers all
real values, like 150.4 cm or 170.8 cm.
2. Shape:
o Perfectly symmetric about the mean (μ).
o Highest point at μ.
o Tails stretch to infinity on both sides but never touch the axis.
3. Parameters (μ and σ):
The distribution is fully described by:
o Mean (μ): the center of the curve.
o Standard deviation (σ): the spread of the curve. A small σ means data is
tightly packed around the mean; a large σ means data is spread out.
4. Mean = Median = Mode:
All three central values coincide at the middle of the curve.
5. Area Property:
The total area under the curve = 1 (since it represents probability). About:
o 68% of data lies within 1σ from the mean.
o 95% within 2σ.
o 99.7% within 3σ. (This is the famous empirical rule).
6. Standard Normal Distribution:
When μ = 0 and σ = 1, we get a special case called the standard normal distribution.
It’s often used for z-scores and statistical tests.
7. Applications Everywhere:
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o Heights, weights, exam scores, measurement errors, IQ levels — most natural
data follows normal distribution.
o That’s why it’s called the “backbone of statistics.”
Poisson vs Normal – The Connection
Now, here comes the interesting twist in the story. The two characters, Poisson and Normal,
aren’t completely strangers. In fact:
• When λ in Poisson is very large, the Poisson distribution starts resembling a Normal
distribution.
• This is why sometimes Poisson is called the “bridge” between discrete rare events
and continuous measurements.
So, think of Poisson as the younger cousin who deals with counts, and Normal as the elder
sibling who smooths everything into a beautiful bell curve.
Why Do These Distributions Matter?
• A company predicting the number of defective bulbs uses Poisson.
• A teacher analyzing exam marks uses Normal.
• Insurance companies, hospitals, banks, scientists — all rely on these to predict,
analyze, and plan.
In short, they allow us to bring order into randomness.
Conclusion
The story of probability distributions shows us that even though life looks unpredictable,
there are patterns hidden in the chaos.
• Poisson distribution is about counting random events within fixed intervals, with
mean = variance = λ.
• Normal distribution is about natural continuous measurements that form a
symmetric bell-shaped curve, ruled by μ and σ.
Together, they not only help in solving real-world problems but also highlight the beauty of
mathematics: making sense of uncertainty.
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8. A box contains 8 red, 3 white and 9 blue balls. If 3 balls are drawn at random find the
probability that 10
(a) all 3 are red, (b) 2 are red, (c) all 3 are white,
(d) at least 1 is white.
Ans: Imagine you and two friends have wandered into a tiny, whimsical carnival stall run by
a kindly mathematician. Inside the glass box on the counter there are 20 marbles: 8 ruby-red
ones, 3 pearl-white ones, and 9 ocean-blue ones. The stallkeeper smiles and says, “Pick
three marbles from the box — all at once, without peeking back in.” You close your eyes,
reach in, and wonder: what are the chances of drawing certain color combinations?
This little scene is a perfect way to understand probability with a story-like clarity. We’ll
treat the draw as “without replacement” (you don’t put marbles back after you draw them),
so the pool changes as you take marbles out — that’s crucial and what makes the
combinatorics the right tool. I’ll show two complementary ways of thinking: the clean
combinatorics (counting choices) and a sequential viewpoint (one draw after another).
Along the way I’ll keep the arithmetic explicit and friendly.
Step 1 — Count the total number of equally likely three-marble outcomes
Before any event-specific counting, find how many different sets of 3 marbles you could
possibly draw from 20. Because we only care which three marbles are chosen (not the
order), the number is the combination:
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“This paper has been carefully prepared for educational purposes. If you notice any mistakes or
have suggestions, feel free to share your feedback.”
