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GNDU Question Paper-2023
B.A 2
nd
Semester
QUANTITATIVE TECHNIQUES
(Quantitative Techniques-II)
Time Allowed: 3 Hours Maximum Marks: 100
Note: There are Eight questions of equal marks. Candidates are required to attempt any Four
questions.
SECTION-A
1. Define statistics. Discuss functions and limitations of statistics. Elaborate scope of statistics in
economics.
2. Explain tabulation and classification of data. Highlight their importance.
SECTION-B
3. Find Mean, Median and Mode for the following data:
Size
10-19
20-29
30-39
40-49
50-59
60-69
70-79
80-89
90-99
Frequency
11
19
21
16
10
8
6
3
1
4. Calculate standard deviation and coefficient of variation from the following data:
Marks No. of students
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0-10 5
10-20 7
20-30 14
30-40 12
40-50 9
50-60 6
60-70 2
SECTION-C
5. From the following data calculate the coefficient of correlation. Given that Mean Values of X
and Y are 6 & 8 respectively.
X
6
2
10
4
8
Y
9
11
?
8
7
6. From the following data, find two regression equations and the correlation coefficient:
X
25
28
35
32
31
36
29
38
34
32
Y
43
46
49
41
36
32
31
30
33
39
SECTION-D
7. Define index numbers. Explain their uses. Explain the concepts of Laspeyre's Index number,
Paasche's Index number and Fisher's Index number.
8. Fit a linear trend to the following data by least squares method:
Year : 1990 1992 1994 1996 1998
Production in 000 Units : 18 21 23 27 16
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GNDU Answer Paper-2023
B.A 2
nd
Semester
QUANTITATIVE TECHNIQUES
(Quantitative Techniques-II)
Time Allowed: 3 Hours Maximum Marks: 100
Note: There are Eight questions of equal marks. Candidates are required to attempt any Four
questions.
SECTION-A
1. Define statistics. Discuss functions and limitations of statistics. Elaborate scope of statistics in
economics.
Ans: Definition of Statistics
Statistics is the branch of mathematics that deals with collecting, organizing, analyzing, and
interpreting numerical data. It helps in making informed decisions by identifying patterns and
trends. Simply put, statistics is a tool used to understand data in a meaningful way.
For example, if a company wants to know whether its new product is successful, it can conduct a
survey and analyze the collected data to see how many people like the product and how many do
not. This process of collecting and analyzing data is what statistics is all about.
Functions of Statistics
Statistics plays a vital role in various fields by performing several key functions. Some of the main
functions of statistics are:
1. Collection of Data
o The first step in statistics is to gather information in a systematic way. Data can be
collected from surveys, experiments, or records.
o Example: A school collects data on students' exam scores to analyze their academic
performance.
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2. Organization of Data
o Once data is collected, it needs to be arranged in a structured manner to make it
easier to study.
o Example: A company organizes sales data monthly to track business growth.
3. Presentation of Data
o Data can be presented in the form of tables, charts, and graphs to make it visually
understandable.
o Example: A weather report shows temperature variations using bar graphs or line
charts.
4. Analysis of Data
o After organizing and presenting data, it is analyzed to extract useful information.
o Example: A hospital analyzes patient records to determine the most common
illnesses during a particular season.
5. Interpretation of Data
o The final step is interpreting the data to make informed decisions.
o Example: A government analyzes unemployment data to create policies that
generate more jobs.
6. Forecasting and Predicting Trends
o Statistics helps in predicting future trends based on past data.
o Example: A business uses sales data from previous years to estimate future sales.
Limitations of Statistics
While statistics is a powerful tool, it has some limitations. These include:
1. Does Not Deal with Individual Cases
o Statistics works with groups and large sets of data rather than focusing on
individual cases.
o Example: A country’s average income does not reflect the income of every citizen.
2. Possibility of Misinterpretation
o If not used correctly, statistics can be misleading.
o Example: A politician may present selective data to show only positive aspects of
their work.
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3. Cannot Establish Causation
o Statistics can show relationships between variables but cannot prove cause and
effect.
o Example: If ice cream sales and crime rates rise together, it does not mean ice
cream causes crime; both might be influenced by hot weather.
4. Depends on Data Quality
o If data is incomplete or inaccurate, the statistical analysis will also be unreliable.
o Example: A survey with biased questions may lead to incorrect conclusions.
5. Affected by Errors and Biases
o Human errors and personal biases can distort statistical results.
o Example: A company may exaggerate positive results to attract investors.
Scope of Statistics in Economics
Statistics plays a crucial role in economics by helping economists analyze and understand
economic data. Its applications in economics include:
1. Understanding Economic Trends
o Economists use statistics to study inflation, unemployment, GDP growth, and other
economic indicators.
o Example: If inflation is rising, economists use statistical data to identify reasons and
suggest solutions.
2. Formulating Economic Policies
o Governments use statistical data to create policies that improve the economy.
o Example: A government analyzing poverty statistics may introduce welfare
programs to support low-income families.
3. Market Research and Consumer Behavior
o Businesses use statistics to study consumer preferences and market trends.
o Example: A company analyzing sales data can decide which products to promote
more.
4. Production and Business Planning
o Companies use statistical methods to decide how much to produce based on
demand.
o Example: A car manufacturer predicts future demand using sales data from
previous years.
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5. Price Analysis
o Statistics helps in determining fair prices for goods and services.
o Example: By studying supply and demand trends, businesses can set competitive
prices.
6. Measuring National Income
o Economists use statistics to calculate a country’s national income, which includes
GDP (Gross Domestic Product) and GNP (Gross National Product).
o Example: If GDP is rising, it indicates economic growth.
7. Economic Forecasting
o Predicting future economic conditions based on past and current data.
o Example: If past data shows a pattern of economic recession every ten years,
economists can prepare for future recessions.
8. International Trade Analysis
o Statistics helps in studying import and export trends, trade balances, and economic
relationships between countries.
o Example: A country analyzing trade data can negotiate better deals with
international partners.
Conclusion
Statistics is an essential tool for understanding and analyzing data. It serves multiple functions
such as data collection, organization, analysis, and interpretation. However, it also has limitations,
such as the risk of misinterpretation and dependency on data quality. In economics, statistics is
widely used to understand economic trends, formulate policies, conduct market research, plan
production, and analyze trade. By using statistical methods, businesses, governments, and
researchers can make better decisions and improve economic outcomes.
2. Explain tabulation and classification of data. Highlight their importance.
Ans: Tabulation and Classification of Data: Meaning, Importance, and Examples
Introduction
Data plays a crucial role in understanding and analyzing various aspects of life, such as economics,
business, social sciences, and research. However, raw data is often unorganized and difficult to
interpret. To make sense of data, it needs to be structured and categorized systematically. This is
where classification and tabulation of data come into play.
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Classification of Data
Classification of data refers to the process of grouping similar data items together based on
common characteristics. This helps in organizing data in a meaningful way, making it easier to
analyze and draw conclusions.
Types of Classification
Classification can be done based on different criteria. The main types of classification are:
1. Qualitative Classification – When data is classified based on attributes or qualities, such as
gender, nationality, or profession, it is called qualitative classification.
o Example: A survey classifying people into categories like male and female.
2. Quantitative Classification – When data is classified based on numerical values, such as
height, weight, or income, it is called quantitative classification.
o Example: Categorizing students based on their marks (e.g., 0-50, 51-75, 76-100).
3. Geographical Classification – When data is classified based on geographical locations such
as countries, states, or cities, it is called geographical classification.
o Example: Population data classified by countries (India, USA, China, etc.).
4. Chronological Classification – When data is arranged based on time periods, such as years,
months, or days, it is called chronological classification.
o Example: The number of cars sold in 2019, 2020, and 2021.
Importance of Classification of Data
• Simplifies Large Data Sets: Classification helps in organizing large amounts of data, making
it easier to handle and analyze.
• Enhances Comparability: Grouping similar data allows for easy comparison between
different categories.
• Facilitates Decision Making: Well-classified data helps in drawing meaningful conclusions
and making informed decisions.
• Improves Data Interpretation: Structured data makes it easier to identify patterns and
trends.
Tabulation of Data
Tabulation is the process of presenting data in a systematic table format. It involves arranging
classified data in rows and columns, making it easier to read and analyze.
Types of Tables in Tabulation
There are two main types of tables used in tabulation:
1. Simple Table – A table that presents only one type of data.
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o Example: A table showing the number of students in different classes.
Class
Number of
Students
1st
30
2nd
28
3rd
35
2. Complex Table – A table that presents multiple categories of data.
o Example: A table showing the number of male and female students in different
classes.
Class
Male Students
Female Students
Total Students
1st
15
15
30
2nd
12
16
28
3rd
18
17
35
Components of a Table
A well-structured table consists of the following parts:
• Title: Clearly states what the table is about.
• Rows and Columns: Data is arranged in horizontal (rows) and vertical (columns) format.
• Headings: Each column and row should have a clear heading to specify the type of data
presented.
• Body: Contains the actual data values.
• Footnotes (if needed): Additional information about the data source or explanations.
Importance of Tabulation of Data
• Easy Understanding: Data in tables is more readable than unorganized information.
• Quick Comparison: Tabulated data allows for easy comparison between different values.
• Better Accuracy: Reduces errors and misinterpretation of data.
• Saves Time: A table presents complex data in a concise manner, reducing the time needed
for analysis.
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Example to Understand Classification and Tabulation
Imagine a school conducting a survey on students' favorite sports. The raw data consists of
students naming their favorite sport. The data is unorganized and difficult to analyze.
Step 1: Classification of Data
We classify the students based on their favorite sport:
Sport
Number of Students
Cricket
40
Football
35
Basketball
25
Tennis
20
Step 2: Tabulation of Data
We can further break down this data into categories like gender:
Sport
Male Students
Female Students
Total Students
Cricket
30
10
40
Football
25
10
35
Basketball
15
10
25
Tennis
10
10
20
Now, it is easier to see which sport is more popular among boys and girls and to draw conclusions
from the data.
Conclusion
Classification and tabulation are essential techniques in data management. Classification helps
group similar data, making it easier to analyze, while tabulation presents data in an organized
table format, enhancing clarity and comparison. Together, these methods make data analysis
more efficient, enabling better decision-making in fields such as business, research, and social
studies.
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SECTION-B
3. Find Mean, Median and Mode for the following data:
Size
10-19
20-29
30-39
40-49
50-59
60-69
70-79
80-89
90-99
Frequency
11
19
21
16
10
8
6
3
1
Ans: Understanding Mean, Median, and Mode
Before solving the problem, let’s first understand what each of these terms means in simple
language:
1. Mean – This is the average of all values. It tells us what the typical value in a data set is.
2. Median – This is the middle value when data is arranged in order. It helps us understand
the central tendency of the data.
3. Mode – This is the value that appears most frequently in the data set. It tells us which
value is the most common.
Now, let’s find these three measures for the given frequency distribution.
Given Data:
Size (Class Intervals)
Frequency (f)
10 - 19
11
20 - 29
19
30 - 39
21
40 - 49
16
50 - 59
10
60 - 69
8
70 - 79
6
80 - 89
3
90 - 99
1
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Step 1: Finding the Mean (Average)
The formula for the Mean in a grouped frequency distribution is:
where:
• f is the frequency of each class.
• X is the midpoint of each class, calculated as:
Step 1.1: Finding Midpoints
Class Interval
Midpoint (X)
Frequency (f)
fX
10 - 19
(10+19)/2 = 14.5
11
11 × 14.5 = 159.5
20 - 29
(20+29)/2 = 24.5
19
19 × 24.5 = 465.5
30 - 39
(30+39)/2 = 34.5
21
21 × 34.5 = 724.5
40 - 49
(40+49)/2 = 44.5
16
16 × 44.5 = 712
50 - 59
(50+59)/2 = 54.5
10
10 × 54.5 = 545
60 - 69
(60+69)/2 = 64.5
8
8 × 64.5 = 516
70 - 79
(70+79)/2 = 74.5
6
6 × 74.5 = 447
80 - 89
(80+89)/2 = 84.5
3
3 × 84.5 = 253.5
90 - 99
(90+99)/2 = 94.5
1
1 × 94.5 = 94.5
Step 1.2: Summing Up Values
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Step 1.3: Calculating the Mean
So, the mean (average) is 41.24.
Step 2: Finding the Median
The Median is the middle value of an ordered data set. In a grouped frequency distribution, we
use this formula:
where:
• L = Lower boundary of the median class
• N = Total frequency (95)
• CF = Cumulative frequency before the median class
• f = Frequency of the median class
• h = Class width (Difference between upper and lower boundary)
Step 2.1: Finding the Median Class
First, calculate N/2:
Now, find the class where the cumulative frequency just exceeds 47.5.
Class Interval
Frequency (f)
Cumulative Frequency (CF)
10 - 19
11
11
20 - 29
19
11 + 19 = 30
30 - 39
21
30 + 21 = 51
40 - 49
16
51 + 16 = 67
50 - 59
10
67 + 10 = 77
60 - 69
8
77 + 8 = 85
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Class Interval
Frequency (f)
Cumulative Frequency (CF)
70 - 79
6
85 + 6 = 91
80 - 89
3
91 + 3 = 94
90 - 99
1
94 + 1 = 95
The median class is 30 - 39 because its cumulative frequency (51) is the first to exceed 47.5.
Step 2.2: Applying the Formula
• L=29.5L (Lower boundary of 30-39 class)
• N/2=47.5
CF=30 (Cumulative frequency before 30-39 class)
f=21 (Frequency of median class)
10h=10 (Class width)
So, the median is 37.83.
Step 3: Finding the Mode
The Mode is the value that appears most frequently. In a grouped frequency table, we use this
formula:
where:
• L = Lower boundary of the modal class
• f1 = Frequency of modal class
• f0 = Frequency before modal class
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• f2 = Frequency after modal class
• h = Class width
The modal class is 30 - 39 because it has the highest frequency (21).
Using the formula:
So, the mode is 32.36.
Final Answers:
• Mean = 41.24
• Median = 37.83
• Mode = 32.36
4. Calculate standard deviation and coefficient of variation from the following data:
Marks No. of students
0-10 5
10-20 7
20-30 14
30-40 12
40-50 9
50-60 6
60-70 2
Ans: Understanding Standard Deviation and Coefficient of Variation with a Simple Explanation
When working with numbers, especially in statistics, we often want to understand how "spread
out" or "scattered" the data is. Two important measures help us with this:
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1. Standard Deviation (SD) – This tells us how much the values in a data set differ from the
average (mean). A high standard deviation means the values are spread out, while a low
standard deviation means they are close to the average.
2. Coefficient of Variation (CV) – This is used to compare variability between different data
sets, even if their units or scales are different. It tells us how much variation exists in the
data relative to the mean.
Let’s break this down step by step and calculate these for the given data.
Step 1: Understanding the Data
We have marks obtained by students in different ranges:
Marks (Class Interval)
No. of Students (Frequency)
0 – 10
5
10 – 20
7
20 – 30
14
30 – 40
12
40 – 50
9
50 – 60
6
60 – 70
2
Since we are dealing with grouped data (intervals like 0–10, 10–20, etc.), we need to use the
midpoints of these intervals for calculations.
Step 2: Finding the Midpoints
For each class interval, we find the midpoint by using the formula:
Let’s calculate midpoints:
Marks (Class Interval)
No. of Students (f)
Midpoint (x)
0 – 10
5
(0+10)/2 = 5
10 – 20
7
(10+20)/2 = 15
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Marks (Class Interval)
No. of Students (f)
Midpoint (x)
20 – 30
14
(20+30)/2 = 25
30 – 40
12
(30+40)/2 = 35
40 – 50
9
(40+50)/2 = 45
50 – 60
6
(50+60)/2 = 55
60 – 70
2
(60+70)/2 = 65
Step 3: Calculating the Mean (Average)
To find the mean, we use the formula:
Where:
• f = frequency (number of students)
• x = midpoint
• ∑f = total number of students
Now, we calculate f×xf
Marks (Class Interval)
No. of Students (f)
Midpoint (x)
F × x
0 – 10
5
5
5 × 5 = 25
10 – 20
7
15
7 × 15 = 105
20 – 30
14
25
14 × 25 = 350
30 – 40
12
35
12 × 35 = 420
40 – 50
9
45
9 × 45 = 405
50 – 60
6
55
6 × 55 = 330
60 – 70
2
65
2 × 65 = 130
Total students:
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Total f × x:
Mean:
Step 4: Finding the Standard Deviation
Standard deviation measures how much the data points deviate from the mean. The formula for
grouped data is:
Let's calculate (𝑥 − 𝑥)
2
and f(𝑥 − 𝑥)
2
Marks
(Class Interval)
No. of Students
(f)
Midpoint
(x)
𝒙 − 𝒙
(𝒙 − 𝒙
)
2
f(𝒙 − 𝒙
)
2
0 – 10
5
5
5 - 32.09 = -27.09
733.86
5 × 733.86 = 3669.30
10 – 20
7
15
15 - 32.09 = -17.09
292.06
7 × 292.06 = 2044.42
20 – 30
14
25
25 - 32.09 = -7.09
50.26
14 × 50.26 = 703.64
30 – 40
12
35
35 - 32.09 = 2.91
8.47
12 × 8.47 = 101.64
40 – 50
9
45
45 - 32.09 = 12.91
166.70
9 × 166.70 = 1500.30
50 – 60
6
55
55 - 32.09 = 22.91
524.93
6 × 524.93 = 3149.58
60 – 70
2
65
65 - 32.09 = 32.91
1083.50
2 × 1083.50 = 2167.00
Summing up:
Step 5: Calculating Coefficient of Variation (CV)
The coefficient of variation helps compare variability across different datasets. It is given by:
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Final Answer
• Mean = 32.09
• Standard Deviation = 15.57
• Coefficient of Variation = 48.52%
Conclusion
The standard deviation tells us that the marks vary significantly from the mean of 32.09. A
coefficient of variation of 48.52% indicates that there is a moderate level of dispersion in the
marks. If we were comparing this data with another dataset, CV would help determine which
dataset has more relative variability.
Understanding these concepts helps in analyzing real-world data, such as exam scores, business
performance, or even weather patterns!
SECTION-C
5. From the following data calculate the coefficient of correlation. Given that Mean Values of X
and Y are 6 & 8 respectively.
X
6
2
10
4
8
Y
9
11
?
8
7
Ans: Understanding the Coefficient of Correlation
The coefficient of correlation (denoted as r) is a statistical measure that determines the strength
and direction of the relationship between two variables. It helps us understand how one variable
changes in relation to another. The value of r ranges from -1 to +1:
• +1 indicates a perfect positive correlation.
• -1 indicates a perfect negative correlation.
• 0 indicates no correlation at all.
Step-by-Step Calculation of the Coefficient of Correlation
We are given the following dataset:
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X
6
2
10
4
8
Y
9
11
?
8
7
Step 1: Find the Missing Value
The mean values of X and Y are given as 6 and 8, respectively. This means:
Since the mean is already provided, the sum of Y values must also satisfy the mean condition:
Calculating the total sum of Y values:
Solving for ?:
So, the missing Y value is 5. Our complete dataset is now:
X
6
2
10
4
8
Y
9
11
5
8
7
Step 2: Compute the Correlation Formula Components
The formula for Pearson’s coefficient of correlation is:
where:
• X (Mean of X) = 6
• (Mean of Y) = 8
Let’s calculate (X - X) and (Y - ):
X
6
2
10
4
8
X - 6
0
-4
4
-2
2
Y
9
11
5
8
7
Y - 8
1
3
-3
0
-1
Next, multiply (X - X) (Y - ):
X - 6
0
-4
4
-2
2
Y - 8
1
3
-3
0
-1
(X-6)(Y-8)
0
-12
-12
0
-2
Summing up (X - X
)(Y - ):
Now, we compute (X - X
)² and (Y - )²:
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X - 6
0
-4
4
-2
2
(X-6)²
0
16
16
4
4
Y - 8
1
3
-3
0
-1
(Y-8)²
1
9
9
0
1
Summing them up:
Now, calculating r:
Interpretation of the Result
The correlation coefficient is -0.92, which is very close to -1. This indicates a strong negative
correlation between X and Y. In simple terms, as the values of X increase, the values of Y tend to
decrease significantly.
Real-Life Example of Correlation
To better understand correlation, consider the relationship between time spent on social media
and academic performance. If students spend too much time on social media, their study time
decreases, leading to lower grades. If we collected data on study hours (X) and exam scores (Y), we
might find a negative correlation, just like in our dataset.
Similarly, a positive correlation example would be exercise and physical fitness. The more you
exercise, the fitter you become, meaning an increase in one variable leads to an increase in the
other.
Conclusion
The coefficient of correlation is a useful statistical tool to measure relationships between two
variables. In this problem, we found a strong negative correlation between X and Y, meaning an
increase in X leads to a decrease in Y. Understanding correlation helps in various fields, including
economics, psychology, and business, to predict trends and make informed decisions.
6. From the following data, find two regression equations and the correlation coefficient:
X
25
28
35
32
31
36
29
38
34
32
Y
43
46
49
41
36
32
31
30
33
39
Ans: Understanding Regression Equations and Correlation Coefficient
In statistics, regression analysis helps us understand the relationship between two variables, while
the correlation coefficient tells us the strength and direction of this relationship. Both concepts are
widely used in various fields to predict outcomes and interpret data patterns. Let’s break down
these concepts step by step using the data provided:
• Data for X: 25, 28, 35, 32, 31, 36, 29, 38, 34, 32
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• Data for Y: 43, 46, 49, 41, 36, 32, 31, 30, 33, 39
Step 1: Calculate Mean of X and Y
Before diving into the regression equations, we need to calculate the mean (average) of both
variables, X and Y.
The mean of a set of numbers is calculated by summing up all the values and dividing by the total
number of values.
Mean of X:
Mean of Y:
Step 2: Calculate the Slope and Intercept of the Regression Equations
There are two types of regression equations we will calculate:
• Regression of Y on X (denoted as Y = a+bX) – this equation predicts Y based on X.
• Regression of X on Y (denoted as X=c+dY) – this equation predicts X based on Y.
To calculate these, we need the slope (denoted as b for regression of Y on X and d for regression
of X on Y) and intercept (denoted as a and c, respectively).
Step 2.1: Regression of Y on X
For the regression of Y on X, we use the formula for the slope (b):
Where:
• Xi and Y
i
are individual data points,
• 𝑥and 𝑦are the means of X and Y respectively.
Let’s calculate each part step by step:
1. Calculate X
i
−𝑥 and Y
i
−𝑦 for each data point.
X
Y
Xi−𝒙
Y
i
−𝒚
(Xi−𝒙
) Y
i
− 𝒚
(X
i
− 𝑋
2
)
25
43
-11
0.1
-1.1
121
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X
Y
Xi−𝒙
Y
i
−𝒚
(Xi−𝒙
) Y
i
− 𝒚
(X
i
− 𝑋
2
)
28
46
-8
3.1
-24.8
64
35
49
-1
6.1
-6.1
1
32
41
-4
-1.9
7.6
16
31
36
-5
-6.9
34.5
25
36
32
0
-10.9
0
0
29
31
-7
-11.9
83.3
49
38
30
2
-12.9
-25.8
4
34
33
-2
-9.9
19.8
4
32
39
-4
-3.9
15.6
16
Now we can calculate the slope (b):
3. Calculate the intercept (a):
Thus, the regression equation of Y on X is:
Step 2.2: Regression of X on Y
For the regression of X on Y, we use the formula for the slope (d):
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Let’s calculate Y
i
−𝒚
and the necessary values for the formula:
X
Y
X
i
− 𝒙
Y
i
−𝒚
(X
i
−𝒙
) (Y
i
−𝒚
)
(Y
i
−𝒚)
𝟐
25
43
-11
0.1
-1.1
0.01
28
46
-8
3.1
-24.8
9.61
35
49
-1
6.1
-6.1
37.21
32
41
-4
-1.9
7.6
3.61
31
36
-5
-6.9
34.5
47.61
36
32
0
-10.9
0
118.81
29
31
-7
-11.9
83.3
141.61
38
30
2
-12.9
-25.8
166.41
34
33
-2
-9.9
19.8
98.01
32
39
-4
-3.9
15.6
15.21
Now calculate the slope (d):
Now calculate the intercept (c):
Thus, the regression equation of X on Y is:
X=35.524+0.0111Y
Step 3: Calculate the Correlation Coefficient (r)
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The correlation coefficient, denoted by r, measures the strength and direction of the linear
relationship between two variables. It is calculated as:
Substituting the values for b and d:
Conclusion
• The regression equation of Y on X is:
Y=42.048+0.0237X
• The regression equation of X on Y is:
X=35.524+0.0111Y
• The correlation coefficient is 0.872, indicating a strong positive linear relationship
between X and Y.
This means that as X increases, Y tends to increase as well, and the relationship between them is
fairly strong.
SECTION-D
7. Define index numbers. Explain their uses. Explain the concepts of Laspeyre's Index number,
Paasche's Index number and Fisher's Index number.
Ans: Index Numbers: A Simple Explanation
Index numbers are statistical tools used to measure changes in a variable or a group of variables
over time or across different locations. They help us understand the relative changes in things like
prices, quantities, production, or income. They are commonly used in economics and business to
compare data points from different periods or to track trends.
Uses of Index Numbers
Index numbers have several important uses:
1. Measuring Inflation: One of the most common uses of index numbers is in tracking
inflation. For example, the Consumer Price Index (CPI) is an index number that measures
changes in the prices of goods and services commonly bought by households. When the
CPI rises, it indicates that the general price level is increasing, which is a sign of inflation.
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2. Comparing Economic Growth: Index numbers are also used to compare the growth of a
country's economy over time. For example, if a country’s GDP (Gross Domestic Product)
grows by a certain percentage, it is often expressed in the form of an index number to
show the growth rate.
3. Tracking Prices of Goods and Services: Index numbers are used by businesses and
governments to track changes in the prices of various goods and services, which helps in
making important economic decisions.
4. Monitoring Production Levels: Index numbers can be used to monitor the production of
goods in industries. For example, if the output of a factory increases from one year to the
next, an index number can show how much the production has grown.
5. Comparing Regional Variations: Index numbers help compare economic conditions
between different regions or countries. This is useful for businesses that operate in
multiple regions or for policymakers.
6. Adjusting for Inflation: Index numbers are used to adjust nominal data to reflect real
values. For example, wages or income can be adjusted for inflation to show how
purchasing power has changed over time.
Laspeyres' Index Number
The Laspeyres Index Number is a method of calculating an index number using the base period
quantities (or weights). In simple terms, it compares the current prices with the prices of a base
period (a reference point), but it uses the quantities of the base period to calculate the index.
Formula for Laspeyres' Index:
Where:
• P
1
is the price of a commodity in the current period
• P
0
is the price of the commodity in the base period
• Q
0
is the quantity of the commodity in the base period
Example:
Imagine you want to measure the price change of two goods, A and B, from one year to another.
In the base year, the price of A is 10 and the price of B is 20. In the current year, the price of A is 12
and the price of B is 24. If the quantities of A and B in the base year were 5 units each, the
Laspeyres index would be calculated using the base year quantities (5 units for both goods).
So, the Laspeyres Index will show how much prices have changed, keeping the quantities from the
base year constant.
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Advantages of Laspeyres' Index:
• It is simple to calculate.
• It gives a clear picture of how much the prices of goods have increased based on the base
year quantities.
Disadvantages:
• It does not account for changes in consumption patterns. For example, if people buy fewer
expensive items and more cheaper items in the current period, the index may overstate
inflation.
Paasche's Index Number
The Paasche Index Number is the opposite of the Laspeyres Index. Instead of using the base period
quantities, it uses the quantities from the current period. This means that the Paasche Index
reflects the changes in prices while considering the current consumption patterns.
Formula for Paasche's Index:
Where:
• P
1
is the price of a commodity in the current period
• P
0
is the price of the commodity in the base period
• Q
1
is the quantity of the commodity in the current period
Example:
Using the same two goods, A and B, in the example above, let’s say in the current year, you are
now consuming 6 units of A and 4 units of B. You will use these quantities to calculate the Paasche
Index, which will reflect the price change with respect to the current quantities.
Advantages of Paasche's Index:
• It better reflects current consumption patterns and prices.
• It may provide a more accurate measure of inflation when consumption changes.
Disadvantages:
• It can be more difficult to calculate as it requires knowledge of current consumption
patterns.
• It may underestimate the price increase because it assumes that people can always switch
to cheaper alternatives.
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Fisher's Index Number
The Fisher's Index Number is often considered the "ideal" index number because it combines the
Laspeyres and Paasche indices to balance the advantages and disadvantages of both. Fisher's
Index is calculated as the geometric mean of the Laspeyres and Paasche indices. This means it
takes the average of the two indices to give a more accurate representation of price changes.
Formula for Fisher's Index:
Where:
• L is the Laspeyres Index
• P is the Paasche Index
Example:
In the previous examples, you would calculate the Laspeyres Index and the Paasche Index
separately, and then take the square root of their product to get the Fisher’s Index.
Advantages of Fisher’s Index:
• It balances the strengths and weaknesses of both Laspeyres and Paasche indices.
• It provides a more accurate and reliable measure of price changes over time.
Disadvantages:
• It is more complex to calculate than Laspeyres or Paasche indexes alone.
• It requires data on both the base and current period quantities.
Summary of Differences Between Laspeyres, Paasche, and Fisher’s Index
• Laspeyres' Index uses base period quantities to calculate the index and is more likely to
overstate inflation.
• Paasche's Index uses current period quantities and reflects changes in consumption
patterns, but it may underestimate inflation.
• Fisher's Index is the geometric mean of the Laspeyres and Paasche indices, and it is
considered the most accurate index for measuring price changes.
Conclusion
Index numbers are powerful tools in economics and business for comparing changes in data over
time or across locations. They help measure inflation, economic growth, and other key economic
indicators. While Laspeyres and Paasche indices each have their strengths and weaknesses, the
Fisher index combines both to provide a more balanced and accurate measure of price changes.
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8. Fit a linear trend to the following data by least squares method:
Year : 1990 1992 1994 1996 1998
Production in 000 Units : 18 21 23 27 16
Ans: Understanding the Problem
You are given data about production over the years, and you need to find a linear trend that best
fits the data. A linear trend means that the relationship between time (the years) and production
follows a straight line, which we can represent using the equation:
Y=a+bX
Where:
• Y is the production (in 000 units),
• X is the year (but we’ll modify it to make calculations easier),
• a is the intercept (where the line crosses the Y-axis),
• b is the slope (how much the production increases or decreases per year).
Step-by-Step Process
We have the following data:
Year
Production (in 000 Units)
1990
18
1992
21
1994
23
1996
27
1998
16
Step 1: Modify the Year Values
Since the years are large numbers (e.g., 1990, 1992, etc.), working with them directly might
complicate things. To simplify, we will subtract 1990 from each year. This way, 1990 becomes 0,
1992 becomes 2, and so on. This makes the years easier to handle in the equation. Here's how the
modified data looks:
Year (X)
Production (Y)
0
18
2
21
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Year (X)
Production (Y)
4
23
6
27
8
16
Step 2: Set Up the Least Squares Method
The goal of the least squares method is to find the best-fitting line, which minimizes the sum of
the squared differences between the actual production values and the values predicted by the
line. This is done by finding the values of aaa and bbb that minimize the error.
The formulas to find aaa (intercept) and bbb (slope) are:
1. Slope (b):
Where:
• n is the number of data points,
• ∑XY is the sum of the products of X and Y,
• ∑X and ∑Y are the sums of the values of X and Y,
• ∑X
2
is the sum of the squares of X.
2. Intercept (aaa):
Step 3: Calculate the Sums
Let’s calculate all the necessary sums. We will use the following table to organize our calculations:
Year (X)
Production (Y)
X × Y
X²
0
18
0
0
2
21
42
4
4
23
92
16
6
27
162
36
8
16
128
64
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Now, let’s calculate the sums:
• ∑X=0+2+4+6+8=20
• ∑Y=18+21+23+27+16=105
• ∑X2=0+4+16+36+64=120
• n=5 (since there are 5 data points)
Step 4: Calculate the Slope b
Now, we can plug these sums into the formula for b:
So, the slope of the line (b) is 0.1.
Step 5: Calculate the Intercept a
Next, we can find the intercept (a) using the formula:
So, the intercept (a) is 20.6.
Step 6: Write the Equation of the Trend Line
Now that we have both the slope (b=0.1b = 0.1b=0.1) and the intercept (a=20.6a = 20.6a=20.6),
we can write the equation of the trend line:
Y=20.6+0.1X
This is the linear trend equation that fits the data.
Step 7: Interpretation of the Trend
The equation Y=20.6+0.1X tells us that the production increases by 0.1 units (in thousands) per
year. The value 20.6 represents the estimated production when X=0X = 0X=0, or the year 1990.
Step 8: Make Predictions (Optional)
You can now use this trend line to predict the production for any year. For example, if you want to
predict the production for the year 2000 (i.e., X=10X = 10X=10):
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Y=20.6+0.1×10=20.6+1=21.6
So, the predicted production for the year 2000 is 21.6 thousand units.
Conclusion
By using the least squares method, we were able to fit a linear trend line to the data. This method
is useful for identifying patterns in data and making predictions. The steps we followed included
simplifying the data, applying the formulas for the slope and intercept, and then interpreting the
results. The trend we found suggests that, on average, production increases by 0.1 units each year.
This method can be applied to various types of data where a linear relationship is expected.
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