MMPC-005
MANAGEMENT PROGRAMME (MP)
Term-End Examination December, 2021
MMPC-005 : QUANTITATIVE ANALYSIS FOR MANAGERIAL APPLICATIONS Time : 3 hours Maximum Marks : 100 Note :
(i) Section A has six questions, each carrying 15 marks. Attempt any four questions.
(ii) Section B is compulsory and carries 40 marks. Attempt both questions.
(iii) Use of calculator is permissible.
SECTION A
Q 1. What do you understand by probability sampling methods ? Explain the various types of probability sampling.
Ans. Probability sampling methods are statistical techniques used to select a subset of individuals or items from a larger population, in such a way that each member of the population has an equal or known probability of being chosen. This ensures that the sample is representative of the population, and allows for more accurate statistical inferences to be made.
There are several types of probability sampling methods, including:
1. Simple random sampling: This method involves selecting individuals or items from the population at random, using a process that gives every member of the population an equal chance of being chosen. This can be done using a random number generator, a lottery system, or by using a table of random numbers.
2. Systematic sampling: This method involves selecting every nth individual or item from a list of the population, where n is a predetermined sampling interval. For example, if the sampling interval is 10, every 10th person on the list is selected.
3. Stratified sampling: This method involves dividing the population into strata, or subgroups, based on specific characteristics such as age, gender, income, or education level. A sample is then selected from each stratum, using either simple random sampling or another probability sampling method.
4. Cluster sampling: This method involves dividing the population into clusters, or groups, based on geographic location or some other characteristic. A sample of clusters is then selected at random, and all individuals within those clusters are included in the sample.
5. Multi-stage sampling: This method combines several sampling methods, often beginning with a large-scale cluster or area sampling, followed by a series of smaller-scale probability sampling techniques within each selected cluster or area.
Overall, probability sampling methods are useful in ensuring that a sample accurately represents a larger population, allowing for more accurate statistical analysis and inference.
Q 2. Find the coefficient of correlation from the following data :
Mean of Fertilizer used (X̄) = (15+18+20+24+30+35+40+50)/8 = 28.375
Mean of Productivity (Ȳ) = (85+93+95+105+120+130+150+160)/8 = 114.375
Step 2: Calculate the deviation of each value from the mean:
Deviation of Fertilizer used (X-X̄):
15 - 28.375 = -13.375
18 - 28.375 = -10.375
20 - 28.375 = -8.375
24 - 28.375 = -4.375
30 - 28.375 = 1.625
35 - 28.375 = 6.625
40 - 28.375 = 11.625
50 - 28.375 = 21.625
Deviation of Productivity (Y-Ȳ):
85 - 114.375 = -29.375
93 - 114.375 = -21.375
95 - 114.375 = -19.375
105 - 114.375 = -9.375
120 - 114.375 = 5.625
130 - 114.375 = 15.625
150 - 114.375 = 35.625
160 - 114.375 = 45.625
Step 3: Multiply the deviation of X and Y for each observation and find the sum of the products:
(X-X̄)(Y-Ȳ) for each observation
(-13.375) x (-29.375) = 393.359375
(-10.375) x (-21.375) = 221.015625
(-8.375) x (-19.375) = 162.640626
(-4.375) x (-9.375) = 41.015625
(1.625) x (5.625) = 9.140625
(6.625) x (15.625) = 103.515625
(11.625) x (35.625) = 414.140625
(21.625) x (45.625) = 986.015625
Sum of (X-X̄)(Y-Ȳ) = 2331.253125
Step 4: Calculate the sum of the squared deviation of X and Y
Sum of (X-X̄)^2 = 4356.859375
Sum of (Y-Ȳ)^2 = 14456.640625
Step 5: Calculate the correlation coefficient using the formula
r = (Σ(X-X̄)(Y-Ȳ)) / sqrt[ Σ(X-X̄)^2 x Σ(Y-Ȳ)^2 ]
r = 2331.253125 / sqrt(4356.859375 x 14456.640625)
r = 0.961
Therefore, the coefficient of correlation between the amount of fertilizer used and productivity is 0.961. This indicates a strong positive correlation between the two variables, which means that as the amount of fertilizer used increases, productivity.
Q. 3 Assume the mean height of soldiers to be 68·22
inches with a variance of 10·8 inches. How many soldiers in a regiment of 1000
would you expect to be over 6 feet tall ? (Given : From Z table area between 0
and 1·15 = 0·3749)
Soll:-
To begin with, we need to convert the height of 6 feet to inches:
6 feet = 6 x 12 inches = 72 inches.
Next, we can use the normal distribution to find the probability of a soldier being taller than 6 feet, given that the mean height is 68.22 inches and the variance is 10.8 inches. We can standardize the height value of 6 feet using the z-score formula:
z = (x - μ) / σ
where x is the height value (in inches), μ is the mean height (68.22 inches), and σ is the standard deviation (the square root of the variance, which is sqrt(10.8) = 3.2863 inches).
Plugging in the values, we get:
z = (72 - 68.22) / 3.2863
z = 1.15
We are given that the area between 0 and 1.15 on the standard normal distribution table is 0.3749. This means that the probability of a soldier being taller than 6 feet is 0.3749.
To find the expected number of soldiers in a regiment of 1000 who are taller than 6 feet, we can multiply the probability by the total number of soldiers:
Expected number of soldiers = probability x total number of soldiers
Expected number of soldiers = 0.3749 x 1000
Expected number of soldiers = 374.9
Therefore, we would expect about 375 soldiers in a regiment of 1000 to be taller than 6 feet.
Q. 4 ‘‘The primary purpose of forecasting is to
provide valuable information for planning the design and operation of the
enterprise. Planning decisions may be long-term, medium-term and short-term.’’
In view of the statement, explain the application of forecasting for long-term
decisions.
Ans. Forecasting is a
critical process that involves making predictions about future events or
conditions based on past and present data. It is an essential tool for
businesses to plan for their future operations and make informed decisions.
Long-term forecasting helps organizations to identify and anticipate changes in
the market and industry trends, which can be useful for planning the design and
operation of the enterprise.
The primary purpose of long-term forecasting is to
assist in the development of strategic plans that cover a period of several
years. Such forecasts are typically used to help organizations to:
1. Identify new opportunities and emerging trends:
By analyzing past and current data, businesses can predict future trends and
identify opportunities that may arise in the future. This information can help
organizations to make strategic decisions about entering new markets,
developing new products or services, or investing in new technologies.
2. Plan for future growth: Long-term forecasting can
help businesses to plan for future growth by providing insights into future
demand for products or services. This information can be used to make decisions
about investments in new facilities, equipment, and personnel.
3. Allocate resources: Long-term forecasts can be
used to determine the level of resources that will be required to achieve
business objectives. This includes identifying the level of investment that
will be required in marketing, research and development, and production.
4. Manage risks: Long-term forecasting can help
businesses to identify potential risks that may arise in the future. This
information can be used to develop strategies to mitigate these risks, such as
developing contingency plans, diversifying product lines, or investing in new
technologies.
5. Historical data analysis: Before
making long-term forecasts, it is important to analyze historical data to
identify trends, patterns, and anomalies. This analysis can help businesses to
develop more accurate and reliable forecasts.
6. Collaboration: Long-term
forecasting often involves collaboration between different departments and
stakeholders within an organization, including marketing, finance, and
operations. It is important to involve all relevant parties to ensure that
forecasts are comprehensive and well-informed.
7. Scenario planning: Long-term
forecasting can involve uncertainty and risks, and businesses should consider
different scenarios and outcomes when developing their forecasts. Scenario
planning involves developing multiple forecasts based on different assumptions,
which can help businesses to prepare for different eventualities.
In summary, long-term forecasting is an essential
tool for businesses to plan for their future operations and make informed
decisions. By providing insights into future trends and opportunities,
long-term forecasts can help businesses to identify new opportunities, plan for
future growth, allocate resources, and manage risks.
Q.
5 What do you understand by the term ‘‘Probability theory’’ ?
Explain different approaches to Probability theory.
Ans. Probability theory is a branch of mathematics that deals with the study of random phenomena and their outcomes. It provides a framework for quantifying uncertainty and analyzing the likelihood of events happening. Probability theory is widely used in various fields, including science, engineering, finance, and statistics, among others.
There are three main approaches to probability theory: the classical approach, the frequentist approach, and the Bayesian approach.
The classical approach, also known as the a priori approach or theoretical approach, is based on the assumption of equally likely outcomes in a sample space. This approach works well for simple problems with a finite number of equally likely outcomes. The probability of an event is the ratio of the number of favorable outcomes to the total number of possible outcomes.
The frequentist approach, also known as the empirical approach, is based on the idea of repeated trials of an experiment. This approach assumes that the probability of an event is the limit of the proportion of times the event occurs in a large number of independent trials. This approach is useful when conducting experiments and collecting data.
The Bayesian approach, also known as the subjective approach, is based on the idea of assigning a degree of belief to the occurrence of an event. This approach involves updating prior beliefs with new information and evidence to obtain a posterior probability distribution. Bayesian probability is widely used in decision-making, forecasting, and statistical inference.
In summary, probability theory
provides a mathematical framework for analyzing and quantifying uncertainty.
The different approaches to probability theory are useful in different
situations and help provide different perspectives on probability.
Q. 6. Write short notes on any
three of the following :
(a) Mathematical Properties of
Median
(b) Normal Distribution
(c) One-Tailed Test
(d) Hypothesis Testing
Procedure
(e) Regression Analysis.
Ans.
(a)
Mathematical Properties of Median:
1. Median is not affected by extreme values or outliers in the data.
2. The median divides the data into two halves, such that half of the values are less than the median and half are greater than the median.
3. The median is unique and will always exist, unlike the mean which may not exist for certain distributions.
4. If the data is skewed, the median is a better measure of central tendency than the mean.
(b)
Normal Distribution:
The normal distribution is a probability distribution that is symmetric and bell-shaped. It is also known as the Gaussian distribution. The normal distribution is characterized by two parameters: the mean and the standard deviation. The mean determines the location of the center of the distribution, while the standard deviation determines the spread of the distribution. Many natural phenomena follow a normal distribution, and it is used widely in statistical inference and hypothesis testing.
(c)
One-Tailed Test:
In hypothesis testing, a one-tailed test is a statistical test in which the alternative hypothesis is directional. That is, the null hypothesis is tested against the possibility that the parameter being tested falls either above or below a specified value. A one-tailed test is used when the researcher has a specific prediction about the direction of the effect being studied. For example, a researcher might predict that a new drug will reduce blood pressure, and the one-tailed test will determine whether the drug reduces blood pressure by a significant amount.
(d)
Hypothesis Testing Procedure:
Hypothesis testing is a statistical method used to make decisions about a population based on a sample. The hypothesis testing procedure involves the following steps:
1. Formulate the null and alternative hypotheses.
2. Choose a level of significance (alpha) to determine the critical value or rejection region.
3. Collect data and compute the test statistic.
4. Compare the test statistic to the critical value or rejection region.
5. Make a decision to either reject or fail to reject the null hypothesis.
(e)
Regression Analysis:
Regression analysis is a
statistical method used to explore the relationship between a dependent
variable and one or more independent variables. The objective of regression
analysis is to develop a model that can be used to predict the value of the dependent
variable based on the values of the independent variables. There are two main
types of regression analysis: simple linear regression and multiple linear
regression. In simple linear regression, there is only one independent
variable, while in multiple linear regression, there are multiple independent
variables. Regression analysis is used in various fields, including economics,
finance, and social sciences.
Q.7 What do you understand by
the term ‘‘classification of data’’ ? What is the main reason of classifying
data ? Also, explain some common types of classification.
Ans. Classification of data refers to the
process of categorizing or grouping data based on common characteristics or
attributes. The main purpose of classifying data is to make it more manageable,
easier to understand, and more useful for analysis.
There are several reasons
why data is classified, some of which include:
1. Improved organization: Classification helps to organize data in a structured and meaningful way, making it easier to access and manage.
2. Better analysis: Data classification enables data analysts to identify patterns, trends, and relationships in the data more easily, which can lead to more accurate analysis and insights.
3. Effective communication: When data is classified, it can be communicated more effectively to different stakeholders, such as managers, executives, and other decision-makers.
Some common types of data
classification include:
1. Binary classification: This involves categorizing data into two groups, such as yes or no, true or false, or positive or negative.
2. Multi-class classification: This involves categorizing data into more than two groups, such as high, medium, or low.
3. Hierarchical classification: This involves organizing data into a hierarchy, with each level representing a different category or subcategory.
4. Cluster analysis: This involves grouping similar data points together based on their attributes or characteristics.
5.
Time-series classification: This involves categorizing data based on
time-related factors, such as days, weeks, months, or years.
Ans. (i) As the coin is tossed 6 times the total number of outcomes will be 2^6 = 64
And we know that the favourable outcomes of getting exactly 4 heads will be 6C4= 15
Thus, the probability of getting exactly 4 heads will be
= the favourable outcomes / the total number of outcomes
⇒ 15/64
(ii) As the coin is tossed 6 times the total number of outcomes will be 2^6= 64
And we know that the favourable outcomes of getting at least 2 heads will be 6C2 + 6C3 + 6C4 + 6C5 + 6C6 = 57
Thus, the probability of getting at least 1 head will be
= the favourable outcomes / the total number of outcomes
⇒ 57/64
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