Statistics for Business Decision (PGCO - VII): Unit-wise Comprehensive Questions and Answers for NSOU M.Com.
Subject: Statistics for Business Decision (PGCO - VII)
Course: PGDP - NSOU M. Com.
Unit I: Probability Theory
Unit II: Random Variable and Probability Distributions
Unit III: Sampling Theory
Unit IV: Multiple Regression Analysis
Unit V: Theory of Attributes
Unit VI: Theory of Hypothesis
Unit VII: Analysis of Variance
Unit VIII: Statistical Quality Control
Subject: Statistics for Business Decision (PGCO - VII)
Course: PGDP - NSOU M. Com.
Unit I: Probability Theory
Q. 1. What is a random experiment?
Answer: A random experiment is an experiment or an act that can be repeated under identical conditions, but its result depends on chance and cannot be predicted with certainty before the experiment is performed. Although the exact outcome is uncertain, all the possible outcomes are known in advance. Examples of random experiments include drawing a card from a pack of cards, throwing a die, tossing a coin, counting the number of cars passing a road crossing during a fixed period, or counting the number of male or female students present in a class on a particular day.
Q. 2. What is an event?
Answer: An event is the outcome or a set of outcomes of a random experiment. It is any phenomenon that may occur during a random experiment. Events are generally represented by capital letters such as A, B and C. An event may be an elementary event, which cannot be divided into simpler events, or a composite event, which consists of two or more elementary events. For example, obtaining an even number when a die is thrown is a composite event because it includes the outcomes 2, 4 and 6.
Q. 3. What do mean by the term 'sample space'?
Answer: A sample space is the complete set of all elementary outcomes of a random experiment. It contains every possible result that can occur when the experiment is performed. Since one of these outcomes must occur in every trial, the sample space is also known as the sure event. For example, when a die is thrown, the sample space is {1, 2, 3, 4, 5, 6}.
Q. 4. Define mutually exclusive events.
Answer: Mutually exclusive events are events that cannot occur simultaneously in the same trial of a random experiment. The occurrence of one event automatically implies the non-occurrence of the other event. For example, when a die is thrown, the event of obtaining an even number and the event of obtaining an odd number are mutually exclusive because both cannot happen together in a single throw.
Q. 5. Define exhaustive events.
Answer: Exhaustive events are a collection of events in which at least one event must necessarily occur whenever the random experiment is performed. Together they include all possible outcomes of the experiment. For example, in the throwing of a die, the six outcomes 1, 2, 3, 4, 5 and 6 together form an exhaustive set of events because one of these outcomes must occur in every throw.
Q. 6. Define equally likely events.
Answer: Equally likely events are events that have the same chance of occurring. After considering all relevant information, no event is expected to occur in preference to another. For example, when a card is drawn randomly from a well-shuffled pack of 52 cards, each card has an equal chance of being selected, making all outcomes equally likely.
Q. 7. What do you understand by 'Union of events'?
Answer: The union of two events A and B is denoted by A ∪ B. It represents the occurrence of either event A or event B or both events together. In other words, the union includes all outcomes that belong to A, or B, or to both. For several events, their union denotes the occurrence of at least one of those events. This concept is commonly represented by a Venn diagram in which the combined area of the events is shaded.
Q. 8. What is intersection of events?
Answer: The intersection of two events A and B is denoted by A ∩ B. It represents the simultaneous occurrence of both events A and B. Only those outcomes that are common to both events are included in the intersection. For multiple events, the intersection denotes the occurrence of all the events together. In a Venn diagram, the overlapping region of the events represents their intersection.
Q. 9. What is meant by the difference of events?
Answer: The difference of events A and B is denoted by A – B. It represents the occurrence of event A together with the non-occurrence of event B. In other words, it consists of all outcomes that belong to A but do not belong to B. Mathematically, it is expressed as A – B = A – (A ∩ B). In a Venn diagram, the portion of A excluding the common part with B represents the difference of events.
Q. 10. Define independent events.
Answer: Independent events are events whose probabilities of occurrence are not affected by the occurrence or non-occurrence of other events. Two events A and B are said to be statistically independent if the conditional probability of A given B is equal to the unconditional probability of A, that is, P(A|B) = P(A). Similarly, P(B|A) = P(B). For independent events, the probability of their simultaneous occurrence is given by:
P(A ∩ B) = P(A) × P(B)
Thus, the occurrence of one independent event does not influence the probability of occurrence of the other event.
Q. 11. Discuss the theorem of total probability.
Answer: The Theorem of Total Probability is also known as the Addition Theorem of Probability. It is used to determine the probability of the occurrence of at least one of two or more events. The theorem states that if two events are mutually exclusive, then the probability of the occurrence of either event is equal to the sum of their individual probabilities.
If A₁ and A₂ are mutually exclusive events, then
P(A₁ ∪ A₂) = P(A₁) + P(A₂)
For n mutually exclusive events A₁, A₂, A₃, …, Aₙ,
P(A₁ ∪ A₂ ∪ … ∪ Aₙ) = P(A₁) + P(A₂) + … + P(Aₙ)
or,
P(⋃Aᵢ) = ΣP(Aᵢ)
The proof is based on the fact that mutually exclusive events cannot occur simultaneously. Therefore, the number of favourable elementary events for the union is simply the sum of the favourable elementary events for each event individually.
Some important deductions from the theorem are:
i. Complement Rule
If Aᶜ is the complement of event A, then
P(A) + P(Aᶜ) = 1
Hence,
P(Aᶜ) = 1 – P(A)
ii. Difference of Events
For any two events A₁ and A₂,
P(A₁ – A₂) = P(A₁) – P(A₁ ∩ A₂)
If A₁ and A₂ are mutually exclusive, then
P(A₁ – A₂) = P(A₁)
iii. Addition Rule for Non-Mutually Exclusive Events
If the events are not mutually exclusive, then
P(A₁ ∪ A₂) = P(A₁) + P(A₂) – P(A₁ ∩ A₂)
Similarly, for three events,
P(A₁ ∪ A₂ ∪ A₃) = P(A₁) + P(A₂) + P(A₃) – P(A₁ ∩ A₂) – P(A₁ ∩ A₃) – P(A₂ ∩ A₃) + P(A₁ ∩ A₂ ∩ A₃)
This principle can be extended to any number of events by using the inclusion-exclusion formula.
iv. Boole's Inequality
For any two events,
P(A₁ ∪ A₂) ≤ P(A₁) + P(A₂)
This follows because the probability of the intersection can never be negative.
Thus, the theorem of total probability provides a simple and useful method of calculating the probability of the occurrence of one or more events and forms the basis for many advanced probability theorems.
Q. 12. Explain the theorem of conditional probability.
Answer: The Theorem of Conditional Probability explains how the probability of an event changes when it is known that another event has already occurred. It is used when the occurrence of one event influences the likelihood of another event.
Suppose there are two events A₁ and A₂, where P(A₁) > 0. Let M(A₁) denote the number of elementary events favourable to A₁ and M(A₁ ∩ A₂) denote the number of elementary events favourable to both A₁ and A₂.
The conditional probability of A₂ given that A₁ has already occurred is defined as
P(A₂|A₁) = M(A₁ ∩ A₂) / M(A₁)
Similarly, if P(A₂) > 0,
P(A₁|A₂) = M(A₁ ∩ A₂) / M(A₂)
This means that among all the outcomes favourable to A₁, only those common to both A₁ and A₂ are considered while calculating the probability of A₂. Thus, the sample space is reduced because the occurrence of A₁ is already known.
The theorem is useful in situations where the probability of an event depends upon prior information or the occurrence of another event. It forms the foundation of several important concepts in probability, including the compound probability theorem, independent events and Bayes' theorem.
Q. 13. Discuss the theorem of compound probability.
Answer: The Theorem of Compound Probability gives the probability that two or more events occur together. It combines conditional probability with the probability of the conditioning event.
If A₂ is an event such that P(A₂) > 0, then for any other event A₁,
P(A₁ ∩ A₂) = P(A₁|A₂) × P(A₂)
Since,
P(A₁|A₂) = P(A₁ ∩ A₂) / P(A₂)
Multiplying both sides by P(A₂), we obtain
P(A₁ ∩ A₂) = P(A₁|A₂) × P(A₂)
The proof is based on the classical definition of probability. If M is the total number of equally likely elementary events, then
P(A₁ ∩ A₂) = M(A₁ ∩ A₂) / M
Since
P(A₁|A₂) = M(A₁ ∩ A₂) / M(A₂)
and
P(A₂) = M(A₂) / M
their product becomes
P(A₁|A₂) × P(A₂) = M(A₁ ∩ A₂) / M = P(A₁ ∩ A₂)
Hence,
P(A₁ ∩ A₂) = P(A₁|A₂) × P(A₂)
The theorem can be extended to several events. If there are events A₁, A₂, …, Aₙ and the required conditional probabilities exist, then
P(A₁ ∩ A₂ ∩ … ∩ Aₙ)
= P(A₁) × P(A₂|A₁) × P(A₃|A₁ ∩ A₂) × … × P(Aₙ|A₁ ∩ A₂ ∩ … ∩ Aₙ₋₁)
This theorem is very important in solving problems involving sequential events, dependent events, sampling without replacement and many practical applications in statistics and probability theory. It also serves as the basis for the concept of independent events and Bayes' theorem.
Q. 14. Explain the classical definition of probability.
Answer: According to the classical approach, it is assumed that the total number of elementary events in the sample space is finite and that all elementary events are equally likely to occur. Under these assumptions, the probability of an event is defined as the ratio of the number of favourable elementary events to the total number of elementary events in the sample space.
If the total number of elementary events in the sample space is M and the number of elementary events favourable to an event A is M(A), then
P(A) = M(A) / M
This definition is applicable only when:
i. The sample space contains a finite number of elementary events.
ii. All elementary events are equally likely.
From the classical definition, the following conclusions are obtained:
i. If A is an impossible event, then M(A) = 0 and therefore P(A) = 0.
ii. If A is a sure event, then M(A) = M and therefore P(A) = 1.
iii. For every event A,
0 ≤ P(A) ≤ 1
iv. If the occurrence of event A implies the occurrence of event B, then every favourable case of A is also favourable to B. Hence,
P(A) ≤ P(B)
Thus, the classical definition provides a simple and systematic method of calculating probability whenever all outcomes are equally likely and the sample space is finite.
Q. 15. Write a brief note on Bayes' Theorem.
Answer: Bayes' Theorem is an important theorem of probability used to determine the probability of an event after obtaining additional information. It helps in revising the probability of an event based on new evidence.
Let there be n mutually exclusive and exhaustive events B₁, B₂, …, Bₙ, each having non-zero probability. Let A be another event having non-zero probability. Then, for each i,
P(Bi|A) = P(Bi) × P(A|Bi) / Σ[P(Bi) × P(A|Bi)]
The theorem is based on two important probability results:
i. Compound Probability Theorem
P(A ∩ Bi) = P(A|Bi) × P(Bi)
ii. Total Probability Theorem
P(A) = Σ P(Bi) × P(A|Bi)
Substituting the value of P(A) in the conditional probability formula gives Bayes' Theorem.
Applications of Bayes' Theorem include:
i. Revising probabilities when new information becomes available.
ii. Solving problems involving conditional probability.
iii. Decision-making under uncertainty.
iv. Quality control and reliability studies.
v. Medical diagnosis, business forecasting and risk analysis.
Thus, Bayes' Theorem is a useful statistical tool for determining posterior probabilities by combining prior probabilities with observed evidence.
Q. 16. A bag contain 5 white, 6 red and 7 green balls. Three balls are drawn at random. What is the probability that a white, a red and a green ball are drawn.
Answer:
Given:
White balls = 5
Red balls = 6
Green balls = 7
Total balls = 5 + 6 + 7 = 18
Total number of ways of drawing 3 balls
= ¹⁸C₃
= 816
Number of favourable ways
= ⁵C₁ × ⁶C₁ × ⁷C₁
= 5 × 6 × 7
= 210
Therefore,
Probability = Favourable ways / Total ways
= 210 / 816
= 35 / 136
Answer:
The probability that one white, one red and one green ball are drawn is
35/136 or approximately 0.2574.
Subject: Statistics for Business Decision (PGCO - VII)
Course: PGDP - NSOU M. Com.
Unit II: Random Variable and Probability Distributions
Q. 1. What is a random variable?
Answer: A random variable, also known as a stochastic variable, is a real-valued function defined on a sample space. It assigns a numerical value to each possible outcome of a random experiment. Every value taken by a random variable has a definite probability associated with it. For example, when a die is thrown, the numbers 1, 2, 3, 4, 5 and 6 are associated with the six possible outcomes. Thus, the number appearing on the upper face can be regarded as a random variable.
A random variable can be classified into two types:
i. Discrete Random Variable
ii. Continuous Random Variable
Q. 2. What is a discrete random variable?
Answer: A discrete random variable is a random variable that can take only a finite number or a countably infinite number of distinct values. Any finite interval of a discrete random variable contains at most a finite number of possible values. Examples include:
• Number of heads obtained when coins are tossed.
• Number of defective items in a batch.
• Number obtained when a die is thrown.
The probabilities corresponding to each value of a discrete random variable are represented by a Probability Mass Function (PMF), and the sum of all probabilities is equal to 1.
Q. 3. What is a continuous random variable?
Answer: A continuous random variable is a random variable that can assume an uncountably infinite number of values over a given interval. It can take any real value within a specified range.
If X is a random variable and (l, u) is an interval such that X can take every real value between l and u, then X is called a continuous random variable.
The probability distribution of a continuous random variable is represented by a Probability Density Function (PDF). The probability of X lying within any interval is obtained by integrating the density function over that interval. The density function satisfies the following conditions:
• f(x) ≥ 0 for every value of x.
• The total area under the density curve is equal to 1.
Examples include:
• Height of individuals.
• Weight of persons.
• Time taken to complete a task.
Q. 4. Define Binomial Distribution.
Answer: Binomial Distribution is a discrete probability distribution that arises from a fixed number of independent Bernoulli trials, where each trial has only two possible outcomes: success or failure. The probability of success remains constant for every trial.
If there are n independent trials, the probability of success is p and the probability of failure is q = 1 – p, then the probability of obtaining exactly x successes is given by
P(X = x) = ⁿCₓ pˣ qⁿ⁻ˣ
where
• x = 0, 1, 2, ..., n
• 0 < p < 1
• q = 1 – p
The Binomial Distribution is characterized by the parameters n and p. It is widely used in situations involving repeated independent trials with two possible outcomes.
Q. 5. What is Poisson Distribution?
Answer: Poisson Distribution is a discrete probability distribution used to describe the number of times an event occurs within a fixed interval of time or space when the events occur independently and at a constant average rate.
Its probability mass function is
P(X = x) = (e⁻ˡ λˣ) / x!
where
• x = 0, 1, 2, ...
• λ is a positive constant representing the mean number of occurrences.
• e ≈ 2.718.
The Poisson Distribution satisfies the following properties:
• It is a discrete distribution.
• The probabilities are always non-negative.
• The sum of all probabilities is equal to 1.
• The mean and variance of the distribution are both equal to λ.
It is commonly used to model the number of telephone calls received, accidents occurring at a place, or printing errors on a page.
Q. 6. What is Normal Distribution?
Answer: Normal Distribution is a continuous probability distribution in which the values of a random variable are distributed symmetrically around the mean. It is one of the most important probability distributions in statistics because many natural and business phenomena approximately follow this distribution.
Its main characteristics are:
(i) It is a continuous distribution.
(ii) The distribution is perfectly symmetrical about the mean.
(iii) The mean, median and mode are equal.
(iv) The total area under the curve is equal to 1.
(v) The two tails of the curve extend indefinitely without touching the horizontal axis.
(vi) The curve is bell-shaped, with most observations concentrated around the mean.
The Normal Distribution is extensively used in statistics for data analysis, estimation, hypothesis testing and quality control.
Q. 7. What is Exponential Distribution?
Answer: Exponential Distribution is a continuous probability distribution that is commonly used to describe the waiting time between successive occurrences of independent random events that occur at a constant average rate.
It is closely related to the Poisson Distribution because when the number of events follows a Poisson Distribution, the time between consecutive events follows an Exponential Distribution.
The important characteristics of the Exponential Distribution are:
(i) It is a continuous probability distribution.
(ii) It is used to model waiting times or lifetimes.
(iii) It is suitable for events occurring independently at a constant rate.
(iv) The distribution is positively skewed, with a high probability of small values and decreasing probability for larger values.
(v) The mean and variance depend on the distribution parameter.
Typical applications include the time between customer arrivals, the lifetime of electronic components and the waiting time before the occurrence of an event.
Q. 8. What is Binomial distribution? Explain its mean and standard deviation.
Answer: A Binomial distribution is a discrete probability distribution that arises from a fixed number of independent Bernoulli trials. Each trial has only two possible outcomes, namely success and failure. The probability of success in each trial remains constant and is denoted by p, while the probability of failure is q = 1 – p. If X denotes the number of successes in n independent trials, then X follows a Binomial distribution.
The probability mass function of a Binomial distribution is:
P(X = x) = nCx px qn−x, for x = 0, 1, 2, …, n
where
n = Number of independent trials
p = Probability of success
q = 1 – p = Probability of failure
Mean of Binomial Distribution:
The mean or expected value of a Binomial distribution is
μ = E(X) = np
This indicates that the average number of successes in n independent trials is equal to the product of the number of trials and the probability of success.
Standard Deviation of Binomial Distribution:
The variance of a Binomial distribution is
σ² = npq
Therefore, the standard deviation is
σ = √(npq)
The standard deviation measures the dispersion of observations around the mean. A larger value of npq indicates greater variability in the distribution.
Q. 9. Narrate some of the moments of Binomial distribution.
Answer: The important moments of a Binomial distribution are as follows.
(a) Raw Moments:
(i) First raw moment
μ'₁ = E(X) = np
(ii) Second raw moment
μ'₂ = n(n − 1)p² + np
(iii) Third raw moment
μ'₃ = n(n − 1)(n − 2)p³ + 3n(n − 1)p² + np
(iv) Fourth raw moment
μ'₄ = n(n − 1)(n − 2)(n − 3)p⁴ + 6n(n − 1)(n − 2)p³ + 7n(n − 1)p² + np
(b) Central moments
(i) Second central moment
μ₂ = npq
(ii) Third central moment
μ₃ = npq(q − p)
(iii) Fourth central moment
μ₄ = 2n²p²q² + npq(1 − 6pq)
(c) Measure of Skewness
β₁ = (q − p)² / npq
γ₁ = (q − p) / √(npq)
When p = q = 1/2, the distribution is symmetrical and γ₁ = 0.
(d) Measure of Kurtosis
γ₂ = (1 − 6pq) / npq
If pq = 1/6, the distribution is mesokurtic.
If pq > 1/6, the distribution is platykurtic.
If pq < 1/6, the distribution is leptokurtic.
Q. 10. What do you mean by Poisson Distribution? Explain the mean, standard deviation and some of the moments of Poisson distribution.
Answer: Poisson distribution is a discrete probability distribution that describes the probability of occurrence of a given number of events in a fixed interval of time or space when the events occur independently and at a constant average rate. It is characterized by a single parameter λ (lambda). The probability mass function is:
P(X = x) = (e−λλx)/x!, for x = 0, 1, 2, …
where λ > 0 and e ≈ 2.718.
Mean of Poisson Distribution:
The mean of a Poisson distribution is
μ = E(X) = λ
Thus, the average number of occurrences is equal to λ.
Standard Deviation of Poisson Distribution:
The variance of a Poisson distribution is
σ² = λ
Hence, the standard deviation is
σ = √λ
Therefore, in a Poisson distribution, the mean and variance are equal.
Some Moments of Poisson Distribution:
(a) Raw Moments
(i) First raw moment
μ'₁ = λ
(ii) Second raw moment
μ'₂ = λ² + λ
(iii) Third raw moment
μ'₃ = λ³ + 3λ² + λ
(iv) Fourth raw moment
μ'₄ = λ⁴ + 6λ³ + 7λ² + λ
(b) Central moments
μ₂ = λ
μ₃ = λ
μ₄ = 3λ² + λ
(c) The coefficient of skewness is
γ₁ = 1/√λ
(d) The coefficient of kurtosis is
γ₂ = 1/λ
These moments indicate that the distribution becomes more symmetrical as λ increases.
Q. 11. What is normal distribution? Explain its properties.
Answer: A Normal distribution is a continuous probability distribution that is bell-shaped, perfectly symmetrical and completely determined by two parameters, namely the mean (μ) and the standard deviation (σ). It is one of the most important probability distributions because many natural and social phenomena approximately follow this distribution.
Properties of Normal Distribution:
i. The normal distribution curve is bell-shaped and perfectly symmetrical.
ii. The mean, median and mode are equal and lie at the centre of the distribution.
iii. The total area under the normal curve is equal to 1.
iv. The curve extends indefinitely on both sides and approaches the horizontal axis asymptotically without touching it.
v. The distribution is completely determined by two parameters, the mean (μ) and the standard deviation (σ).
vi. Half of the total area lies on each side of the mean.
vii. The points of inflection occur at μ + σ and μ − σ.
viii. The distribution is unimodal, having only one peak.
ix. The normal curve is useful in statistical inference, quality control and probability calculations because of its mathematical properties.
Q. 12. What is exponential distribution? Discuss its mean and variance.
Answer: The Exponential distribution is a continuous probability distribution commonly used to describe the waiting time between successive occurrences of independent events that happen at a constant average rate. It is represented by the parameter λ, where λ > 0.
The probability density function is:
f(x) = λe−λx, for x ≥ 0
Mean of Exponential Distribution:
The mean or expected value of an exponential distribution is
E(X) = 1/λ
This represents the average waiting time between successive events.
Variance of Exponential Distribution:
The variance is
Var(X) = 1/λ²
Therefore, the standard deviation is
σ = 1/λ
The exponential distribution is widely used in reliability analysis, queuing theory, survival analysis and waiting-time problems because of its memoryless property.
Q. 13. The mean of a binomial distribution is 40 and its standard deviation is 8. Calculate its n, p and q.
Answer:
Given,
Mean (μ) = np = 40
Standard deviation (σ) = √npq = 8
Therefore,
npq = 8² = 64
Since np = 40,
40q = 64
q = 64/40 = 1.6
Since q cannot be greater than 1, the given data are inconsistent.
Hence, no valid values of n, p and q can be obtained because the information provided in the question is incorrect. For a binomial distribution, q must satisfy 0 ≤ q ≤ 1.
Q. 14. "The mean and standard deviation of a Poisson distribution are 16 and 9 respectively." State whether the above statement is true.
Answer:
For a Poisson distribution,
Mean = λ
Standard deviation = √λ
Given,
Mean = 16
Therefore,
Standard deviation = √16 = 4
But the question states that the standard deviation is 9.
Since 9 ≠ 4, the statement is false.
Therefore, the given statement is not true.
Q. 15. In a normal distribution 25% of the items are under 100 and 5% are over 150. Find the mean and standard deviation of the distribution.
Answer:
Given,
P(X < 100) = 0.25
P(X > 150) = 0.05
Therefore,
P(X < 150) = 0.95
From the standard normal table,
For 25%,
Z = −0.674
Hence,
(100 − μ)/σ = −0.674
100 − μ = −0.674σ
μ = 100 + 0.674σ
Also,
For 95%,
Z = 1.645
Therefore,
(150 − μ)/σ = 1.645
150 − μ = 1.645σ
Substituting μ,
150 − (100 + 0.674σ) = 1.645σ
50 = 2.319σ
σ = 21.56 (approximately)
Hence,
μ = 100 + (0.674 × 21.56)
= 114.54 (approximately)
Therefore,
Mean = 114.54
Standard deviation = 21.56
Q. 16. Six unbiased coins are tossed. Find the probabilities of:
(i) getting exactly 3 heads
(ii) getting less than 3 heads
(iii) getting at least 3 heads
(iv) getting at most 3 heads
Answer:
Here,
n = 6
p = q = 1/2
(i) Probability of exactly 3 heads
P(X = 3)
= ⁶C₃ (1/2)⁶
= 20/64
= 5/16
= 0.3125
(ii) Probability of less than 3 heads
P(X < 3)
= P(0) + P(1) + P(2)
= (1 + 6 + 15)/64
= 22/64
= 11/32
= 0.34375
(iii) Probability of at least 3 heads
P(X ≥ 3)
= 1 − P(X < 3)
= 1 − 22/64
= 42/64
= 21/32
= 0.65625
(iv) Probability of at most 3 heads
P(X ≤ 3)
= P(0) + P(1) + P(2) + P(3)
= (1 + 6 + 15 + 20)/64
= 42/64
= 21/32
= 0.65625
Q. 17. The lifetime (in hours) of an electronic bulb is a random variable with the following exponential probability density function
f(x) = 0.025e^(-0.025x), x ≥ 0
Answer:
For an exponential distribution,
λ = 0.025
(i) Mean lifetime
Mean = 1/λ
= 1/0.025
= 40 hours
(ii) Probability that the bulb fails within the first 25 hours
P(X < 25)
= 1 − e^(−0.025 × 25)
= 1 − e^(−0.625)
≈ 1 − 0.5353
≈ 0.4647
Therefore, the probability is approximately 0.465.
(iii) Probability that the bulb operates for 75 hours or more
P(X ≥ 75)
= e^(−0.025 × 75)
= e^(−1.875)
≈ 0.1534
Therefore, the probability is approximately 0.153.
Q. 18. There are 10000 electric lamps in the streets of the Hooghly Chinsurah Municipality. If these lamps have an average life of 2000 burning hours with a standard deviation of 230 hours, what number of lamps may be expected to burn for:
(i) more than 2200 hours
(ii) less than 1800 hours
(iii) between 1600 hours and 2500 hours
Answer:
Given,
Mean (μ) = 2000 hours
Standard deviation (σ) = 230 hours
Total lamps = 10000
(i) More than 2200 hours
Z = (2200 − 2000)/230
= 0.87
From the normal table,
P(Z > 0.87) = 0.1922
Expected number of lamps
= 10000 × 0.1922
= 1922 lamps
(ii) Less than 1800 hours
Z = (1800 − 2000)/230
= −0.87
P(Z < −0.87) = 0.1922
Expected number
= 10000 × 0.1922
= 1922 lamps
(iii) Between 1600 hours and 2500 hours
For 1600 hours,
Z = (1600 − 2000)/230
= −1.74
For 2500 hours,
Z = (2500 − 2000)/230
= 2.17
From the standard normal table,
P(−1.74 < Z < 2.17)
= 0.9850 − 0.0409
= 0.9441
Expected number of lamps
= 10000 × 0.9441
= 9441 lamps
Therefore,
(i) More than 2200 hours = 1922 lamps
(ii) Less than 1800 hours = 1922 lamps
(iii) Between 1600 and 2500 hours = 9441 lamps.
Subject: Statistics for Business Decision (PGCO - VII)
Course: PGDP - NSOU M. Com.
Unit III: Sampling Theory
Q. 1. What is sampling?
Answer: Sampling is the process of selecting a part of an aggregate statistical material (population) with the objective of obtaining information about the whole population. Since it is often impractical to collect data from every member of a population because of limitations of time, cost, and resources, only a representative part of the population, known as the sample, is examined. The aggregate statistical material is called the population, while the selected part is called the sample. Good sampling should satisfy the principles of validity and optimisation.
Q. 2. What is sampling error?
Answer: Sampling error is the difference or discrepancy between the value of a sample statistic and the corresponding population parameter arising due to the process of selecting only a sample instead of the entire population. Even when appropriate sampling procedures are followed, the results obtained from different samples may not exactly coincide with the characteristics of the population because of chance variation. Sampling errors mainly arise due to sampling bias and chance factors.
Q. 3. What is non-sampling error?
Answer: Non-sampling errors are errors that arise not because of the selection of a sample but due to procedural and operational defects during data collection and processing. These include response errors, observational errors, errors arising from non-response, and interviewer’s errors. Such errors may occur because respondents provide inaccurate information, observers make incorrect judgments, respondents refuse to cooperate, or interviewers are influenced by personal bias or prejudice.
Q. 4. What is simple random sampling?
Answer: Simple random sampling is a probability sampling method in which every unit of the population has an equal probability of being selected, and every possible sample of a given size has the same probability of selection. In finite populations, selection may be made through methods such as the lottery method or random number tables. In infinite populations, successive selections are independent, and each item continues to have an equal probability of selection.
Q. 5. What do you mean by the term 'deliberate sampling'?
Answer: Deliberate sampling, also known as purposive or judgment sampling, is a non-probability sampling method in which the researcher deliberately selects particular units of the population according to personal judgment or the purpose of the study. The choice of sample depends on the investigator's experience, knowledge, or convenience rather than on random selection. This method is commonly used when only specific types of respondents are required for the investigation.
Q. 6. What is systematic sampling?
Answer: Systematic sampling is a probability sampling method in which the first unit is selected randomly, and thereafter every kth unit is selected from the population. The sampling interval k is determined in advance. For example, every 20th person in a list or every 15th house in a locality may be selected after choosing a random starting point. This method combines randomness with simplicity and is easy to implement.
Q. 7. What is multi-stage sampling?
Answer: Multi-stage sampling is an extension of cluster sampling in which sampling is carried out in several successive stages. Large primary sampling units are selected first, such as states, followed by districts, then towns or villages, and finally households or individuals. When random sampling is applied at each stage, the procedure is called multi-stage random sampling. This method is particularly useful when the population is widely scattered over a large geographical area.
Q. 8. What is cluster sampling?
Answer: Cluster sampling is a probability sampling method in which the population is first divided into several groups or clusters, each of which generally contains heterogeneous units representing the population. Instead of selecting individual elements, entire clusters are selected randomly for inclusion in the sample. Area sampling is a special form of cluster sampling in which geographical areas serve as clusters. This method is especially useful when a complete list of population units is not available.
Q. 9. What is sequential sampling?
Answer: Sequential sampling is a sampling method in which the final sample size is not fixed in advance. Instead, it is determined progressively on the basis of the information obtained during the survey. Mathematical decision rules are applied at different stages to decide whether additional observations are required or whether the survey should be terminated. This method is commonly used in statistical quality control under acceptance sampling plans.
Q. 10. What do you understand by the term 'parameter'?
Answer: A parameter is a statistical measure calculated from all the units of a population. It represents a characteristic of the entire population, such as the population mean or population standard deviation. A parameter is a fixed value for a given population and remains constant, although it is usually unknown and has to be estimated from sample data.
Q. 11. What is statistic?
Answer: A statistic is a statistical measure calculated from the observations of a sample. Examples include the sample mean and sample standard deviation. Unlike a parameter, a statistic varies from one sample to another because different samples may contain different observations. Statistics are used to estimate the corresponding population parameters.
Q. 12. What is Chi-square distribution?
Answer: The Chi-square (χ²) distribution is an important probability distribution obtained from the sum of the squares of mutually independent standard normal variables. If y₁, y₂, …, yᵥ are mutually independent standard normal variables, then the sum y₁² + y₂² + … + yᵥ² follows a Chi-square distribution with ν degrees of freedom. The Chi-square distribution is widely used in statistical inference, particularly in tests of goodness of fit, tests of independence, and estimation of population variance.
Q. 13. Distinguish between Sampling and Complete Enumeration.
Answer: Sampling and Complete Enumeration are two different methods of collecting data from a population.
Sampling refers to the process of selecting only a representative part of the population to obtain information about the whole population. Complete Enumeration, also known as Census, refers to collecting information from every member of the population.
Sampling requires less money, less time and fewer manpower compared to complete enumeration because only a part of the population is investigated. In contrast, complete enumeration is costly, time-consuming and labour-intensive as every unit of the population is covered.
Sampling is preferred when resources and time are limited. Complete enumeration is suitable only when adequate resources, time and manpower are available.
In sampling, better-trained investigators and closer supervision can often produce better-quality data. In complete enumeration, maintaining uniform quality throughout the survey is comparatively difficult because of its large scale.
Sampling provides an estimate of sampling errors, enabling the researcher to judge the accuracy of results. In complete enumeration, there is no way to estimate sampling errors because the entire population is surveyed.
Sampling is essential when the population is infinite, hypothetical or when testing destroys the unit under investigation. In such cases, complete enumeration is impossible.
Thus, sampling is generally more economical, quicker and practical, whereas complete enumeration provides complete information but requires much greater resources.
Q. 14. Distinguish between Sampling and Non-sampling Errors.
Answer: Errors in a survey are broadly classified into sampling errors and non-sampling errors.
Sampling errors arise because only a part of the population is selected for study. The sample characteristics may differ from the true characteristics of the population due to chance or improper sampling procedures. Non-sampling errors arise due to procedural mistakes during data collection, recording, observation or interviewing and may occur in both sample surveys and complete censuses.
Sampling errors occur because of sampling bias or random chance. Non-sampling errors arise from response errors, observational errors, non-response and interviewer's errors.
Sampling errors are absent in complete enumeration because every unit of the population is included. Non-sampling errors may occur even in complete enumeration due to mistakes made during data collection and processing.
Sampling errors can be reduced by adopting proper random sampling techniques and increasing the sample size. Non-sampling errors can be reduced through careful planning, proper questionnaire design, trained investigators, effective supervision and accurate recording of responses.
Examples of non-sampling errors include respondents giving incorrect information because of pride or personal interest, observers making estimation mistakes, respondents refusing to answer questions and interviewers allowing personal prejudice to influence data collection.
Therefore, sampling errors arise from the process of selecting a sample, whereas non-sampling errors arise from mistakes in collecting, recording and processing information.
Q. 15. Discuss the different types of sampling techniques.
Answer: Sampling techniques can broadly be classified into subjective sampling and objective sampling. Subjective sampling depends upon the personal judgment of the investigator, while objective sampling follows definite sampling rules. Objective sampling is further classified into probability sampling, non-probability sampling and mixed sampling. The important sampling techniques are discussed below.
(i) Deliberate or Judgement Sampling:
In this method, the researcher deliberately selects particular units of the population based on personal judgment. Convenience sampling, where units are selected because they are easily accessible, is also included under this category. It is a non-probability sampling technique.
(ii) Simple Random Sampling:
This is a probability sampling method in which every unit of the population has an equal chance of being selected. The sample may be selected through lottery or random number tables. It is one of the most widely used sampling methods.
(iii) Systematic Sampling:
In systematic sampling, after selecting the first unit randomly, every kth unit is selected from the population list until the required sample size is obtained. This method is simple and economical.
(iv) Stratified Sampling:
In this method, the population is divided into homogeneous groups called strata. Samples are then selected separately from each stratum, generally by simple random sampling. This technique is useful when the population is heterogeneous.
(v) Quota Sampling:
Quota sampling is a non-probability sampling method in which interviewers are assigned quotas from different strata. The actual selection is based on the interviewer's judgment rather than random selection.
(vi) Cluster or Area Sampling:
Here, the population is divided into clusters instead of individual units. Some clusters are selected randomly, and all units within the selected clusters are included in the sample. Area sampling is a special form of cluster sampling in which geographical areas are treated as clusters. It is particularly useful when a complete list of population units is unavailable.
(vii) Multi-stage Sampling:
Multi-stage sampling is an extension of cluster sampling. Selection is carried out in several stages, such as selecting states first, then districts, then towns and finally households. If random sampling is used at every stage, it becomes multi-stage random sampling.
(viii) Sequential Sampling:
In sequential sampling, the final sample size is not fixed in advance. The decision regarding the sample size is taken progressively as information is collected. This method is commonly used in statistical quality control and acceptance sampling.
Thus, different sampling techniques are chosen depending upon the nature of the population, objectives of the study, available resources and desired level of accuracy.
Q. 16. Narrate the different practical methods of selecting random sample.
Answer: Random samples can be selected by several practical methods. The unit describes three important methods.
(i) Lottery Method:
In the lottery method, each sampling unit is assigned an identification number or mark on a separate slip or ticket. All tickets are placed in a container, thoroughly mixed and shuffled. The required number of tickets is then drawn one by one at random. Since every ticket has an equal chance of being selected, this method provides a random sample.
(ii) Random Number Table Method:
Random number tables contain large sets of randomly arranged numbers that are independent and uncorrelated. These tables are prepared by experts such as Tippett, Fisher and Yates, and Kendall and Smith. A researcher may start from any row or column and select one-digit, two-digit, three-digit or four-digit numbers continuously according to the requirement until the desired sample size is obtained. This method is suitable for large populations and ensures randomness in selection.
(iii) Roulette Wheel Method:
The roulette wheel is a circular horizontal wheel with numbers written around its circumference. A fixed pointer is placed on the wheel, and the wheel is rotated. When it stops, the number indicated by the pointer is selected. By repeating the process, random numbers are generated and used to select the required sample.
These practical methods help ensure that every unit of the population has an equal chance of selection, thereby improving the reliability and objectivity of the sample survey.
Q. 17. Discuss the factors to be taken into consideration while determining the size of a sample.
Answer: Determining an appropriate sample size is one of the most important decisions in sampling. According to the study material, the following factors should be considered while deciding the size of a sample:
i. Nature of Population
The size of the sample depends on whether the population is homogeneous or heterogeneous. A homogeneous population generally requires a smaller sample, whereas a heterogeneous population requires a larger sample to represent all characteristics properly.
ii. Nature of Proposed Classes
If the study requires the formation of many classes, groups, or sub-groups, then the sample size should be sufficiently large so that each class contains an adequate number of observations.
iii. Nature of the Survey
The purpose of the survey also influences sample size. Technical or specialized surveys generally require a smaller sample, while general surveys usually require a larger sample to obtain representative results.
iv. Type of Sampling
The sampling method adopted affects the sample size. A properly selected small random sample is often more useful than a large sample selected through an inappropriate sampling technique.
v. Level of Accuracy and Confidence Level
If a researcher wants greater precision and a higher confidence level, a larger sample size is necessary. Higher confidence or lower level of significance demands a larger sample.
vi. Availability of Money
The amount of funds available for conducting the study influences sample size. Large samples require greater expenditure for data collection, processing and analysis.
vii. Other Factors
Several additional factors should also be considered, such as:
(a) Population size.
(b) Population variance.
(c) Nature of the sampling units.
(d) Size of the questionnaire.
(e) Training and efficiency of investigators.
(f) Time available for completing the study.
(g) Conditions under which the sampling is conducted.
Thus, the final sample size should be selected by considering statistical accuracy as well as practical limitations.
Q. 18. What are the features of sampling?
Answer: The important features of sampling are:
i. Sampling is the process of selecting only a part of the total population for investigation.
ii. The selected part of the population is called the sample, while the complete group is known as the population.
iii. The objective of sampling is to obtain reliable information about the entire population without studying every unit.
iv. Sampling saves both time and cost compared to complete enumeration or census.
v. A properly selected sample represents the characteristics of the whole population.
vi. Sampling is particularly useful when the population is very large or when complete enumeration is impractical.
vii. Sampling is based on two important principles:
(a) Validity, which ensures objective interpretation of results.
(b) Optimisation, which aims to achieve maximum efficiency for a given cost or minimum cost for a desired level of efficiency.
viii. Sampling forms the basis for statistical inference by enabling conclusions about the population from the sample observations.
Q. 19. State the features of quota sampling.
Answer: Quota sampling is an important non-probability sampling technique. Its main features are:
i. The population is divided into different strata or groups.
ii. Interviewers are assigned a fixed quota for each stratum.
iii. The interviewer selects respondents according to the assigned quota using personal judgment rather than random selection.
iv. Selection is based on convenience and judgment instead of probability.
v. It is less costly and quicker than stratified random sampling.
vi. The emphasis is on fulfilling the prescribed quota rather than ensuring randomness.
vii. Quota samples generally become judgment samples instead of random samples.
viii. Usually, the quota allotted to each stratum is proportional to the size of that stratum in the population.
Q. 20. How do you prepare a sample applying stratified sampling?
Answer: In stratified sampling, the population is first divided into several non-overlapping groups known as strata. Each stratum consists of relatively homogeneous units, while there is considerable variation between different strata.
The procedure for preparing a stratified sample is as follows:
i. Identify the entire population to be studied.
ii. Divide the population into distinct and non-overlapping strata based on relevant characteristics such as age, income, occupation, gender or geographical region.
iii. Ensure that each stratum is internally homogeneous while different strata differ from one another.
iv. Decide the number of sample units to be selected from each stratum. Normally, the number selected is proportional to the size of each stratum.
v. Select the required sample units from every stratum by applying simple random sampling.
vi. Combine the selected units from all strata to obtain the final sample.
Stratified sampling is especially useful when the population is heterogeneous because it ensures that every important subgroup is adequately represented in the sample, thereby increasing the accuracy and reliability of the survey results.
Subject: Statistics for Business Decision (PGCO - VII)
Course: PGDP - NSOU M. Com.
Unit IV: Multiple Regression Analysis
Q. 1. What is partial correlation?
Answer: Partial correlation is a statistical measure that shows the degree of relationship between two variables after eliminating the effect of one or more other related variables. In a trivariate distribution, the partial correlation coefficient between x₁ and x₂, after removing the influence of x₃, is denoted by r₁₂.₃. It helps in measuring the true association between two variables by excluding the influence of the third variable. The value of a partial correlation coefficient always lies between –1 and +1.
Q. 2. What is multiple correlation?
Answer: Multiple correlation refers to the study of the relationship between one dependent variable and two or more independent variables considered jointly. It measures how strongly the dependent variable is related to the combined influence of the independent variables. In a trivariate distribution, the multiple correlation coefficient of x₁ on x₂ and x₃ is denoted by R₁.₂₃. The coefficient of multiple correlation ranges from 0 to +1, where higher values indicate a stronger joint relationship.
Q. 3. What does partial regression coefficient indicate?
Answer: A partial regression coefficient indicates the amount of change in the dependent variable due to a one-unit change in one independent variable while keeping the effect of the other independent variable constant. For example, b₁₂.₃ measures the change in x₁ for a one-unit change in x₂ after eliminating the influence of x₃ on x₁ and x₂. Thus, it represents the net effect of an independent variable on the dependent variable.
Q. 4. If r₁₃.₂ = –0.83, then what does it reflect?
Answer: The value r₁₃.₂ = –0.83 reflects a high degree of negative relationship between x₁ and x₃ after eliminating the influence of x₂ on both variables. Since the coefficient is negative and close to –1, it indicates a strong inverse association between x₁ and x₃ after controlling for x₂.
Q. 5. If r₂₃.₁ = 0.02, then what does it indicate?
Answer: The value r₂₃.₁ = 0.02 indicates a very low or negligible positive relationship between x₂ and x₃ after eliminating the influence of x₁. Since the coefficient is very close to zero, it suggests that there is almost no linear association between x₂ and x₃ after controlling for x₁.
Q. 6. If R₁.₂₃ = 0.57, then what does it measure?
Answer: The value R₁.₂₃ = 0.57 measures the degree of relationship between x₁ and the joint influence of x₂ and x₃. A coefficient of 0.57 indicates a moderate positive relationship between the dependent variable x₁ and the combined effect of the independent variables x₂ and x₃.
Q. 7. If R²₂.₁₃ = 0.81, then what does it imply?
Answer: The value R²₂.₁₃ = 0.81 implies that 81% of the variation in x₂ is explained jointly by x₁ and x₃ through the multiple regression model. The remaining 19% of the variation is due to other unexplained factors or random error. This indicates that x₁ and x₃ together have a very strong explanatory power for x₂.
Q. 8. If b₂₃.₁ = –2.81, then what does it indicate?
Answer: The value b₂₃.₁ = –2.81 indicates that, after eliminating the influence of x₁, a one-unit increase in x₃ causes x₂ to decrease by 2.81 units on average. The negative sign shows an inverse relationship between x₂ and x₃ after controlling for x₁, while the magnitude of 2.81 represents the rate of change in the dependent variable.
Q. 9. Distinguish between partial correlation and multiple correlation.
Answer: Partial correlation and multiple correlation are two different statistical techniques used when three or more variables are involved. According to the study material, the differences are as follows:
i. Partial correlation measures the relationship between two variables after eliminating the influence of one or more other related variables, whereas multiple correlation measures the relationship between one dependent variable and the joint effect of two or more independent variables.
ii. In partial correlation, the effect of other variables is controlled or removed before measuring the association. In multiple correlation, the combined influence of all independent variables is considered simultaneously.
iii. Partial correlation studies the net relationship between two variables only. Multiple correlation studies the overall relationship between one variable and a group of variables.
iv. Partial correlation coefficient is represented by symbols such as r₁₂.₃, r₁₃.₂ and r₂₃.₁. Multiple correlation coefficient is represented by R₁.₂₃, R₂.₁₃ and R₃.₁₂.
v. The value of the partial correlation coefficient lies between –1 and +1, whereas the value of the multiple correlation coefficient lies between 0 and +1.
vi. Partial correlation is mainly used to measure the independent association between two variables after removing the influence of another variable. Multiple correlation is used to measure how strongly a dependent variable is related to the combined influence of several independent variables.
Q. 10. Distinguish between partial correlation coefficient and partial regression coefficient.
Answer: The differences between partial correlation coefficient and partial regression coefficient are as follows:
i. A partial correlation coefficient measures the degree and direction of relationship between two variables after eliminating the influence of other variables. A partial regression coefficient measures the change in the dependent variable due to a one-unit change in an independent variable while keeping the other independent variables constant.
ii. Partial correlation coefficient measures only the strength of association. Partial regression coefficient measures the rate of change or regression effect.
iii. Partial correlation coefficient is denoted by symbols such as r₁₂.₃, r₁₃.₂ and r₂₃.₁. Partial regression coefficient is denoted by symbols such as b₁₂.₃, b₁₃.₂, b₂₁.₃, etc.
iv. The value of a partial correlation coefficient always lies between –1 and +1. A partial regression coefficient has no fixed limits and may take any positive or negative value.
v. Partial correlation coefficient is a dimensionless measure and is not affected by the units of measurement. Partial regression coefficient depends on the units in which the variables are measured.
vi. Partial correlation coefficient is used to determine the closeness of relationship between variables. Partial regression coefficient is used in constructing multiple regression equations and estimating the value of the dependent variable.
Q. 11. Write a short note on partial correlation coefficient.
Answer: The partial correlation coefficient measures the degree of relationship between two variables after eliminating the influence of one or more other related variables. It is useful when three or more variables are involved and the researcher wants to identify the actual relationship between any two variables by controlling the effects of the remaining variables.
In a trivariate distribution consisting of variables x₁, x₂ and x₃, the partial correlation coefficient between x₁ and x₂ after eliminating the influence of x₃ is denoted by r₁₂.₃. Similarly, r₁₃.₂ measures the relationship between x₁ and x₃ after removing the effect of x₂, and r₂₃.₁ measures the relationship between x₂ and x₃ after removing the effect of x₁.
The coefficient is calculated using the simple correlation coefficients among the variables. It provides the net relationship between two variables after excluding the effect of the third variable.
The value of the partial correlation coefficient always lies between –1 and +1. A positive value indicates a positive relationship, a negative value indicates a negative relationship, and a value close to zero indicates a weak relationship after controlling the influence of the other variable. It is widely used in statistical analysis to understand the independent relationship between variables.
Q. 12. Write a short note on multiple correlation coefficient.
Answer: The multiple correlation coefficient measures the degree of relationship between one dependent variable and the joint effect of two or more independent variables. It is used when several independent variables simultaneously influence a dependent variable.
In a trivariate distribution, the multiple correlation coefficient of x₁ on x₂ and x₃ is denoted by R₁.₂₃. Likewise, R₂.₁₃ measures the relationship between x₂ and the joint influence of x₁ and x₃, while R₃.₁₂ measures the relationship between x₃ and the combined influence of x₁ and x₂.
Multiple correlation analysis helps determine how strongly a dependent variable is associated with a group of explanatory variables taken together. It is an important tool in multiple regression analysis for evaluating the overall effectiveness of the independent variables in explaining the dependent variable.
The value of the multiple correlation coefficient always lies between 0 and +1. A value close to 1 indicates a very strong relationship, while a value close to 0 indicates a weak relationship between the dependent variable and the combined effect of the independent variables.
Q. 13. Discuss the major problems associated with the multiple regression analysis. Mention the ways by which these problems can be reduced / removed.
Answer: According to the study material, multiple regression analysis is associated with several important problems. The major problems and their remedies are discussed below.
i. Standard Error:
While estimating regression coefficients, the standard deviation of the estimates is known as the standard error. A large standard error reduces the precision of the estimated coefficients.
The problem can be reduced by increasing the sample size, which improves the accuracy and reliability of the estimated regression coefficients.
ii. Multicollinearity:
Multicollinearity arises when there is a high degree of correlation among the independent variables included in the regression model. As a result, the individual effect of each independent variable cannot be estimated accurately.
This problem can be reduced by removing the highly correlated independent variables from the regression equation. It can also be minimized by increasing the sample size, which reduces the extent of the relationship among highly correlated explanatory variables.
iii. Autocorrelation:
Autocorrelation or serial correlation occurs when the value of a variable in the current period is related to its values in previous periods. In such cases, the least squares estimates become inefficient.
The problem can be reduced by taking the first difference of the time series data when first-order autocorrelation exists. Another method is to use the Generalized Least Squares (GLS) estimation technique.
iv. Heteroscedasticity:
Heteroscedasticity occurs when the assumption of constant variance of the random error term is violated. It generally arises when the regression equation is not properly specified and the estimated coefficients vary widely.
The problem can be reduced by applying suitable transformations of the variables, such as logarithmic transformation, square root transformation, cube root transformation and negative reciprocal transformation.
v. Non-normality of Residuals:
This problem arises when the residuals do not follow a normal distribution or when the assumption of constant variance is violated.
The problem can be removed by applying the Generalized Linear Model. It can also be reduced by transforming the dependent variable so that the violation of the normality assumption of residuals is minimized.
Thus, the major problems of multiple regression analysis are standard error, multicollinearity, autocorrelation, heteroscedasticity and non-normality of residuals. Appropriate statistical techniques and suitable transformations help reduce or eliminate these problems and improve the reliability of the regression model.
Subject: Statistics for Business Decision (PGCO - VII)
Course: PGDP - NSOU M. Com.
Unit V: Theory of Attributes
Q. 1. What is independence of attributes?
Answer: Independence of attributes refers to a situation where the presence or absence of one attribute does not influence the presence or absence of another attribute. In other words, the occurrence of one attribute is completely unrelated to the occurrence of the other. For example, if criminality and the ability to play cricket are independent attributes, then the proportion of cricketers among criminals should be the same as the proportion of cricketers among non-criminals. Thus, neither attribute has any effect on the other.
Q. 2. What is association of attributes?
Answer: Association of attributes exists when the presence or absence of one attribute depends upon the presence or absence of another attribute. In such cases, the two qualitative characteristics are related to each other instead of occurring independently. The main objective of studying the association of attributes is to determine whether two qualitative characteristics are related, the nature of their relationship, and the degree of their association. This forms an important part of the statistical analysis of qualitative data.
Q. 3. What is positive association of attributes?
Answer: Two attributes are said to have positive association when the presence of one attribute is associated with the presence of the other, and the absence of one attribute is associated with the absence of the other. This means that individuals possessing one attribute are more likely to possess the other attribute as well. Likewise, those lacking one attribute are also likely to lack the other.
Examples include:
(a) Illiteracy and criminality.
(b) Educational attainment and employment.
Thus, in positive association, both attributes tend to occur together.
Q. 4. What is negative association of attributes?
Answer: Two attributes are said to have negative association when the presence of one attribute is associated with the absence of the other attribute. This means that the occurrence of one attribute reduces the likelihood of the occurrence of the other.
Examples include:
(a) Vaccination and attack of smallpox.
(b) Literacy and poverty.
Hence, negatively associated attributes tend to occur in opposite directions, where the presence of one is linked with the absence of the other.
Q. 5. What does the term 'complete association' imply?
Answer: Complete association is a perfect form of positive association between two attributes. Two attributes A and B are said to be in complete association if A cannot occur without B, although B may occur without A, or vice versa.
In other words, complete association exists if:
All A's are B's, that is,
(AB) = (A)
or
All B's are A's, that is,
(AB) = (B)
Thus, whenever one attribute is present, the other attribute is also necessarily present, indicating a perfect positive relationship between them.
Q. 6. What do you mean by the term 'complete dissociation'?
Answer: Complete dissociation is the perfect form of negative association between two attributes. Two attributes A and B are said to be in complete dissociation when they never occur together.
Complete dissociation exists if either:
No A's are B's, that is,
(AB) = 0
or
No α's are β's, that is,
(αβ) = 0
Therefore, complete dissociation indicates that the simultaneous occurrence of the two attributes is impossible.
Q. 7. What is Yule's coefficient of association?
Answer: Yule's coefficient of association is one of the most popular methods of measuring the relationship between two qualitative attributes. It measures both the nature and the degree of association between two attributes. It is denoted by the symbol Q.
The formula is:
Q = [(AB)(αβ) – (Aβ)(αB)] / [(AB)(αβ) + (Aβ)(αB)]
The value of Q lies between +1 and –1.
Interpretation:
(i) If Q = +1, the attributes are completely associated (perfect positive association).
(ii) If Q = –1, the attributes are completely dissociated (perfect negative association).
(iii) If Q = 0, the attributes are completely independent.
A positive value of Q indicates positive association.
A negative value of Q indicates negative association.
Thus, Yule's coefficient provides both the direction and strength of association between two qualitative attributes.
Q. 8. What is Yule's coefficient of colligation?
Answer: Yule's coefficient of colligation is another measure used to determine the degree of association between two qualitative attributes. It is represented by the symbol Y and is closely related to Yule's coefficient of association.
The coefficient of colligation is calculated as:
Y = [√{(AB)(αβ)} − √{(Aβ)(αB)}] / [√{(AB)(αβ)} + √{(Aβ)(αB)}]
The value of Y also lies between +1 and –1.
Interpretation:
(a) A positive value indicates positive association.
(b) A negative value indicates negative association.
(c) A value of zero indicates independence of attributes.
(d) A value of +1 indicates complete positive association.
(e) A value of –1 indicates complete negative association.
Yule's coefficient of colligation provides an alternative measure of association and has a direct relationship with Yule's coefficient of association, making it useful in the analysis of qualitative data.
Q. 9. Explain the following terms: i) Order of a class, ii) Positive class frequencies, iii) Negative class frequencies, iv) Relationship between class frequencies of various orders, v) Consistency of given data.
Answer:
i) Order of a Class:
The order of a class refers to the number of attributes represented in that class. It depends upon the number of attributes taken together in the classification of the universe. The universe without any specified attribute is called the class of zero order and is represented by N.
A class containing one attribute is called a class of the first order. Examples are A, B, α and β.
A class containing two attributes is called a class of the second order. Examples are AB, Aβ, αB and αβ.
Similarly, a class containing three attributes is called a class of the third order, and in general, a class represented by n attributes is known as a class of the nth order.
ii) Positive Class Frequencies:
A positive class is a class that contains one or more positive attributes. The frequencies corresponding to such classes are known as positive class frequencies. Examples include:
(A) – Frequency of objects possessing attribute A.
(AB) – Frequency of objects possessing both attributes A and B.
(ABC) – Frequency of objects possessing attributes A, B and C simultaneously.
Thus, positive class frequencies represent the number of observations having the specified positive attributes.
iii) Negative Class Frequencies:
A negative class is one that contains one or more negative attributes, representing the absence of the corresponding positive attributes. The frequencies of such classes are known as negative class frequencies. Examples include:
(α) – Frequency of absence of attribute A.
(αβ) – Frequency of absence of both A and B.
(αβγ) – Frequency of absence of attributes A, B and C.
These frequencies measure the number of observations that do not possess the specified attributes.
iv) Relationship between Class Frequencies of Various Orders:
Class frequencies of different orders are interrelated. A class frequency of a lower order can always be expressed as the sum of the corresponding higher-order class frequencies.
For one attribute:
N = (A) + (α)
When a second attribute B is introduced:
(A) = (AB) + (Aβ)
(α) = (αB) + (αβ)
(B) = (AB) + (αB)
(β) = (Aβ) + (αβ)
Also,
N = (A) + (α)
= (B) + (β)
= (AB) + (Aβ) + (αB) + (αβ)
These relationships help determine missing class frequencies whenever some frequencies are known.
v) Consistency of Given Data:
Consistency of data refers to the property that none of the class frequencies should be negative.
If every class frequency is zero or positive, the data are said to be consistent.
If even one class frequency becomes negative, the data are inconsistent because the frequency of a class can never be negative.
Thus, consistency is tested by calculating all missing class frequencies and verifying that each of them is non-negative.
Q. 10. State, with reasons, whether in each of the following cases the data are consistent or not.
Answer:
Case I: There is only one attribute A which is being studied in respect of a universe. The class frequency (A) has been shown less than zero.
The data are inconsistent.
Reason:
For one attribute,
N = (A) + (α)
Both (A) and (α) represent frequencies, and frequencies can never be negative.
Since (A) is less than zero, the given data violate the basic principle of class frequencies.
Therefore, the data are inconsistent.
Case II: There is only one attribute B which is being studied in respect of a universe. The class frequency (B) has been shown greater than N.
The data are inconsistent.
Reason:
For one attribute,
N = (B) + (β)
Since (β) cannot be negative,
(B) cannot exceed the total number of observations N.
If (B) is greater than N, then (β) would become negative, which is impossible.
Hence, the given data are inconsistent.
Case III: There are two attributes A and B which are being studied together in respect of a universe. The class frequency (AB) has been shown less than zero.
The data are inconsistent.
Reason:
(AB) represents the number of objects possessing both attributes A and B simultaneously.
A class frequency can never be negative.
Therefore, if (AB) is less than zero, it violates the condition of consistency.
Hence, the given data are inconsistent.
Q. 11. What is meant by association of two attributes?
Answer: Association of two attributes refers to the relationship between two qualitative characteristics in such a way that the presence or absence of one attribute depends upon the presence or absence of the other attribute. In other words, two attributes are said to be associated when they are not independent of each other. For example, literacy and employment, education and income, or vaccination and immunity may exhibit association. The purpose of studying association is to determine whether the occurrence of one attribute has any influence on the occurrence of another attribute and to measure the degree of such relationship. If the occurrence of one attribute has no effect on the occurrence of the other, the attributes are said to be independent and there is no association between them.
Q. 12. Distinguish between association and correlation as the terms are used in Statistics.
Answer: Association and correlation are both used to study relationships between characteristics, but they differ in several important respects.
Association is concerned with qualitative characteristics or attributes such as literacy, employment, intelligence, criminality, or vaccination, which cannot be measured numerically. Correlation, on the other hand, deals with quantitative variables such as income, height, weight, marks, or age, which can be measured in numerical terms.
Association studies whether two attributes occur together or separately and determines whether they are positively associated, negatively associated, or independent. Correlation measures both the direction and the degree of relationship between two measurable variables.
The analysis of association is based on class frequencies, contingency tables, and methods such as comparison of observed and expected frequencies, proportion method, and Yule's coefficient of association. Correlation is generally measured by statistical coefficients such as Karl Pearson's coefficient of correlation or Spearman's rank correlation coefficient.
Association is used in the analysis of attributes, whereas correlation is used in the analysis of variables.
Thus, association applies to qualitative data, while correlation applies to quantitative data.
Q. 13. When are two attributes said to be:
a) Positively associated
Answer:
Two attributes are said to be positively associated when the presence of one attribute is associated with the presence of the other attribute and the absence of one attribute is associated with the absence of the other. In other words, both attributes tend to occur together. Examples include the association between educational attainment and employment or between illiteracy and criminality.
b) Negatively associated
Answer:
Two attributes are said to be negatively associated when the presence of one attribute is associated with the absence of the other attribute. This means that as one attribute is present, the other tends to be absent. Examples include vaccination and attack of smallpox, or literacy and poverty.
c) Independent
Answer:
Two attributes are said to be independent when the presence or absence of one attribute is not influenced by the presence or absence of the other attribute. In such a case, the occurrence of one attribute has no effect on the occurrence of the other. For example, if criminality and the ability to play cricket are independent, the proportion of cricketers among criminals and non-criminals will be the same. Thus, there is no association between the two attributes.
Q. 14. What are the various methods of finding whether two attributes are associated, dissociated or independent?
Answer: According to the Theory of Attributes, the following methods are used to determine whether two attributes are positively associated, negatively associated (dissociated) or independent. The first two methods indicate only the nature of association, while the last two also measure its degree.
i. Comparison of Observed and Expected Frequencies Method:
In this method, the observed frequency of the joint occurrence of two attributes is compared with the expected frequency under the assumption of independence.
The expected frequency of (AB) is:
Expected (AB) = (A)(B)/N
Decision rule:
If (AB) > (A)(B)/N, the attributes are positively associated.
If (AB) < (A)(B)/N, the attributes are negatively associated or dissociated.
If (AB) = (A)(B)/N, the attributes are independent.
This method identifies only the nature of association and not its magnitude.
ii. Proportion Method:
Under this method, the proportions of the concerned attributes are compared.
Decision rule:
If (AB)/(B) > (Aβ)/(β), the attributes are positively associated.
If (AB)/(B) < (Aβ)/(β), the attributes are negatively associated.
If (AB)/(B) = (Aβ)/(β), the attributes are independent.
Like the previous method, this method only determines the nature of association.
iii. Yule's Coefficient of Association (Q):
This is the most widely used method for studying association between two attributes. It indicates both the direction and the degree of association.
Formula:
Q = [(AB)(αβ) – (Aβ)(αB)] / [(AB)(αβ) + (Aβ)(αB)]
Interpretation:
Q = +1 indicates complete positive association.
Q = –1 indicates complete negative association or complete dissociation.
Q = 0 indicates independence.
Values between –1 and +1 indicate varying degrees of association.
iv. Yule's Coefficient of Colligation (Y):
This method also measures both the nature and degree of association.
It is based on the square roots of the cross products of the four cell frequencies.
The coefficient varies between –1 and +1.
Positive values indicate positive association.
Negative values indicate negative association.
Zero indicates independence.
Thus, the various methods used to determine whether two attributes are associated, dissociated or independent are the Comparison of Observed and Expected Frequencies Method, Proportion Method, Yule's Coefficient of Association and Yule's Coefficient of Colligation.
Q. 15. Distinguish between coefficient of correlation and coefficient of association.
Answer: The coefficient of correlation measures the relationship between two quantitative variables, whereas the coefficient of association measures the relationship between two qualitative attributes.
The coefficient of correlation is applicable when numerical data such as income, price, height or weight are available. The coefficient of association is used when the data relate to qualitative characteristics such as literacy, employment, sickness or marriage.
The coefficient of correlation is calculated from the actual numerical values of observations. In contrast, the coefficient of association is calculated from the frequencies of a two-by-two contingency table based on the presence and absence of attributes.
The coefficient of correlation measures the strength and direction of a linear relationship between variables. The coefficient of association measures the degree and direction of association between qualitative attributes.
The coefficient of correlation has several measures such as Karl Pearson's coefficient and Spearman's rank correlation. The coefficient of association is generally measured by Yule's coefficient of association (Q) and Yule's coefficient of colligation (Y).
Although both coefficients range from –1 to +1, their fields of application are different. Correlation is used for measurable variables, whereas association is used for attributes that cannot be measured quantitatively.
Q. 16. Define Yule's coefficient of association (Q) and coefficient of colligation (Y).
Answer:
Yule's Coefficient of Association (Q):
Yule's coefficient of association is the most popular method of studying association between two qualitative attributes. It measures both the nature and the degree of association between the attributes. It is denoted by the Roman capital letter Q.
Its formula is:
Q = [(AB)(αβ) – (Aβ)(αB)] / [(AB)(αβ) + (Aβ)(αB)]
The value of Q lies between –1 and +1.
Q = +1 indicates complete positive association.
Q = –1 indicates complete negative association or complete dissociation.
Q = 0 indicates complete independence.
Values between –1 and +1 indicate different degrees of positive or negative association.
Yule's Coefficient of Colligation (Y):
Yule's coefficient of colligation is another measure of association between two qualitative attributes. It also measures both the direction and the degree of association. It is represented by the letter Y.
Its formula is:
Y = [√(AB × αβ) – √(Aβ × αB)] / [√(AB × αβ) + √(Aβ × αB)]
The value of Y also ranges between –1 and +1.
Y = +1 indicates complete positive association.
Y = –1 indicates complete negative association.
Y = 0 indicates independence.
Thus, both Q and Y are important measures for studying the direction as well as the degree of association between two qualitative attributes.
Subject: Statistics for Business Decision (PGCO - VII)
Course: PGDP - NSOU M. Com.
Unit VI: Theory of Hypothesis
Q. 1. What is test of significance?
Answer: A test of significance, also known as a test of hypothesis, is a statistical procedure used to determine whether a hypothesis regarding a population parameter is acceptable in the light of sample data. In many practical situations, complete information about the population is unavailable. Therefore, decisions are made on the basis of information obtained from a sample.
The procedure begins with a statistical hypothesis concerning a population parameter. The validity of this hypothesis is then examined by analysing the sample observations. Depending on the result of the statistical test, the hypothesis is either accepted or rejected.
The test of significance helps researchers make objective decisions by comparing the observed sample results with the expected results under the assumption that the null hypothesis is true. It provides a scientific basis for deciding whether the observed differences are due to chance or indicate a significant effect.
Thus, a test of significance is a statistical method that enables us to decide whether a particular hypothesis about a population is true or false based on sample evidence.
Q. 2. Explain the procedure generally followed in testing of hypothesis.
Answer: The general procedure for testing a hypothesis consists of the following steps:
Step 1: Formulation of Hypotheses
The first step is to formulate the null hypothesis (H₀) and the alternative hypothesis (H₁). The null hypothesis generally states that there is no difference or no effect, whereas the alternative hypothesis contradicts the null hypothesis.
Step 2: Selection of the Test Statistic
An appropriate test statistic is selected depending on the nature of the sample, sample size, and whether the population standard deviation is known or unknown. The test statistic should have a known sampling distribution under the null hypothesis.
Step 3: Fixing the Level of Significance
A suitable level of significance (α), generally 5% or 1%, is chosen. This represents the maximum probability of committing a Type I error, that is, rejecting a true null hypothesis.
Step 4: Determination of the Critical Region
Based on the level of significance and the form of the alternative hypothesis, the critical region (rejection region) and the acceptance region are determined.
Step 5: Calculation of the Test Statistic
The value of the selected test statistic is calculated using the sample data.
Step 6: Decision Rule
The calculated value of the test statistic is compared with the critical value.
If the calculated value falls within the critical region, the null hypothesis is rejected.
If the calculated value falls within the acceptance region, the null hypothesis is accepted.
Step 7: Drawing the Conclusion
Finally, the conclusion is stated in the context of the problem, indicating whether the sample provides sufficient evidence to reject or accept the null hypothesis.
Q. 3. Distinguish between: critical region and acceptance region
Answer: The critical region is the set of values of the test statistic that leads to the rejection of the null hypothesis. If the calculated value of the test statistic falls within this region, it indicates that the observed sample result is sufficiently different from what is expected under the null hypothesis, and therefore the null hypothesis is rejected.
The acceptance region is the set of values of the test statistic that leads to the acceptance of the null hypothesis. When the calculated test statistic lies within this region, the observed sample result is considered consistent with the null hypothesis, and hence the null hypothesis is accepted.
The critical region is also known as the rejection region, whereas the acceptance region consists of all values outside the critical region. The critical region is determined by the chosen level of significance, while the acceptance region represents the remaining probability under the sampling distribution.
Q. 4. Distinguish between: null hypothesis and alternative hypothesis
Answer: The null hypothesis is the hypothesis that is tested for possible rejection. It is denoted by H₀. It generally states that there is no difference, no association, or no significant effect between the population parameter and the specified value. During hypothesis testing, it is assumed to be true unless sufficient evidence is obtained against it.
The alternative hypothesis is the hypothesis that contradicts the null hypothesis and is denoted by H₁. It states that there is a difference, an association, or a significant effect. The alternative hypothesis is accepted only when the null hypothesis is rejected.
The null hypothesis forms the basis of statistical testing, whereas the alternative hypothesis represents the research claim that the investigator wishes to establish. The null hypothesis is accepted when the sample evidence is insufficient to reject it, while the alternative hypothesis is accepted only when the sample evidence strongly supports it.
Q. 5. Distinguish between: one-tailed and two-tailed test
Answer: A one-tailed test is used when the alternative hypothesis specifies the direction of the difference. It tests whether the population parameter is either greater than or less than a specified value. Consequently, the critical region is located entirely in one tail of the probability distribution, either the left tail or the right tail.
A two-tailed test is used when the alternative hypothesis simply states that the population parameter is different from the specified value without indicating the direction of the difference. In this case, the critical region is divided equally between the two tails of the probability distribution.
In a one-tailed test, the entire level of significance is placed in one tail of the distribution, whereas in a two-tailed test the level of significance is equally divided between both tails. A one-tailed test is appropriate for directional hypotheses, while a two-tailed test is appropriate for non-directional hypotheses.
Q. 6. Distinguish between Type I Error and Type II Error
Answer: Type I Error is committed when the null hypothesis (H₀) is rejected even though it is actually true. It is also known as the error of rejecting a true hypothesis. The probability of committing a Type I Error is denoted by α, which is called the level of significance. While designing a statistical test, the critical region is selected in such a way that the probability of Type I Error does not exceed the chosen level of significance. Mathematically,
Probability of Type I Error = P(x ∈ Critical Region | H₀ is true)
Type II Error is committed when the null hypothesis is accepted even though it is actually false. It is the error of failing to reject a false null hypothesis. The probability of committing a Type II Error is denoted by β. Mathematically,
Probability of Type II Error = P(x ∈ Acceptance Region | H₁ is true)
The main differences are as follows. Type I Error occurs when a true null hypothesis is rejected, whereas Type II Error occurs when a false null hypothesis is accepted. The probability of Type I Error is represented by α, while the probability of Type II Error is represented by β. The level of significance controls the probability of Type I Error, whereas the probability of Type II Error depends upon the sampling distribution and the power of the test. The power of a statistical test is equal to 1 – β, which represents the probability of correctly rejecting a false null hypothesis.
Q. 7. Explain clearly the procedure of testing hypothesis.
Answer: The procedure of hypothesis testing consists of the following steps:
Step 1. Formulation of Hypotheses
The first step is to formulate the null hypothesis (H₀) and the alternative hypothesis (H₁) according to the problem under consideration. The null hypothesis generally states that there is no difference or no effect, whereas the alternative hypothesis contradicts the null hypothesis.
Step 2. Selection of the Appropriate Test Statistic
An appropriate test statistic is selected depending on the nature of the data, sample size, and knowledge of the population parameters. For large samples with known population standard deviation, the Z-statistic is generally used. For small samples with unknown population standard deviation, the t-statistic is used.
Step 3. Selection of the Level of Significance
Choose the level of significance (α), usually 5% or 1%. This determines the probability of committing a Type I Error and helps in defining the critical region.
Step 4. Determine the Critical Region
Using the chosen level of significance and the appropriate probability distribution, determine the critical value or critical region. The critical region is the set of values of the test statistic that leads to rejection of the null hypothesis.
Step 5. Compute the Test Statistic
Calculate the value of the test statistic from the sample observations using the appropriate statistical formula.
Step 6. Decision Making
Compare the calculated value of the test statistic with the critical value.
If the calculated value falls in the critical region, reject the null hypothesis.
If the calculated value falls in the acceptance region, accept (or fail to reject) the null hypothesis.
Step 7. Draw the Conclusion
Finally, interpret the statistical decision in terms of the original problem and state the conclusion clearly.
Q. 8. Point out the assumptions in hypothesis testing in large samples.
Answer: The assumptions of hypothesis testing in large samples are:
i. The sample size should be sufficiently large, generally greater than 30.
ii. The sample should be selected randomly from the population.
iii. The observations included in the sample should be independent of one another.
iv. The sampling distribution of the sample mean is assumed to be approximately normal due to the Central Limit Theorem.
v. If the population standard deviation is known, the standard normal (Z) test is used directly.
vi. If the population standard deviation is unknown, its unbiased estimate obtained from the sample may be used because, for large samples, the difference between n and (n – 1) is negligible.
vii. In large sample testing, the parent population need not necessarily be normally distributed, since the sampling distribution of the sample mean approaches normality for sufficiently large samples.
Q. 9. How does small sampling theory differ from large sampling theory?
Answer: Small sampling theory is applicable when the sample size is 30 or less, whereas large sampling theory is applicable when the sample size exceeds 30.
In small sampling theory, the parent population is assumed to follow a normal distribution. In large sampling theory, the parent population need not necessarily be normal because the Central Limit Theorem ensures that the sampling distribution is approximately normal.
Small sampling theory generally uses the Student's t-distribution for hypothesis testing when the population standard deviation is unknown. Large sampling theory uses the standard normal (Z) distribution.
In small samples, the sample standard deviation is calculated using (n – 1) degrees of freedom, and this correction is important. In large samples, the difference between n and (n – 1) is negligible, so the sample standard deviation provides a satisfactory estimate of the population standard deviation.
Critical values in small sampling theory are obtained from the t-table and depend on degrees of freedom. In large sampling theory, critical values are obtained from the standard normal distribution and are independent of degrees of freedom.
Small sampling theory is more restrictive because it requires normality of the parent population, whereas large sampling theory is less restrictive due to the large sample approximation.
Q. 10. Explain the following terms:
(i) Test Statistic
A test statistic is a statistic calculated from the sample observations. It is used to decide whether the null hypothesis (H₀) should be accepted or rejected. The choice of the test statistic depends on the nature of the data, sample size and assumptions regarding the population. The sampling distribution of the chosen statistic under the null hypothesis is known in advance. If the calculated value falls in the critical region, the null hypothesis is rejected; otherwise it is accepted.
Examples of test statistics are:
(a) Z-statistic for large samples,
(b) t-statistic for small samples,
(c) F-statistic for comparison of variances, and
(d) Chi-square (χ²) statistic for categorical data and goodness-of-fit tests.
(ii) Level of Significance
The level of significance is the probability of rejecting the null hypothesis when it is actually true. It is denoted by α (alpha). It determines the size of the critical region and represents the maximum probability of committing a Type I error. Commonly used significance levels are 5% (α = 0.05) and 1% (α = 0.01). A smaller value of α provides stronger evidence before rejecting the null hypothesis.
(iii) Confidence Level
The confidence level is the probability that the confidence interval constructed from a sample contains the true population parameter. It is equal to (1 − α).
For example:
(a) 95% confidence level corresponds to α = 0.05.
(b) 99% confidence level corresponds to α = 0.01.
A higher confidence level gives greater assurance that the interval contains the true parameter but generally results in a wider confidence interval.
(iv) Power of a Test
The power of a test is the probability of rejecting the null hypothesis when it is actually false. It measures the effectiveness of a statistical test in detecting a false null hypothesis.
Power of a Test = 1 – Probability of Type II Error (β)
A powerful test has a greater chance of identifying a real difference or effect. The power increases with larger sample size, higher significance level and greater difference between the hypothesized and actual parameter values.
Q. 11. Give some important applications of t-test and explain how it helps in business decision making.
Answer: The t-test is mainly used when the sample size is small and the population standard deviation is unknown. It helps determine whether observed differences are statistically significant.
Important applications of the t-test are:
i. Testing a single population mean.
It is used to determine whether the sample mean differs significantly from a specified population mean.
ii. Comparing the means of two independent samples.
It helps determine whether two populations have significantly different average values.
iii. Comparing paired observations.
It is used for before-and-after studies, such as employee performance before and after training.
iv. Quality control.
Manufacturers use the t-test to determine whether production processes meet quality standards.
v. Medical and social science research.
Researchers compare treatment effects and experimental results using the t-test.
How it helps in business decision making:
The t-test supports business decisions in several ways:
• It helps managers compare the effectiveness of two marketing strategies.
• It evaluates whether employee training improves productivity.
• It compares customer satisfaction before and after introducing new services.
• It helps determine whether a new production process improves product quality.
• It assists financial analysts in comparing investment returns.
• It reduces the risk of making decisions based on random sample fluctuations.
Thus, the t-test provides scientific evidence for making reliable business decisions rather than depending on intuition alone.
Q. 12. Discuss the F-test for testing the equality of two variances.
Answer: The F-test is used to determine whether the variances of two independent normal populations are equal.
Assumptions
(a) The two samples are randomly selected.
(b) The populations are normally distributed.
(c) The samples are independent.
Hypotheses
Null Hypothesis:
H₀ : σ₁² = σ₂²
Alternative Hypothesis:
H₁ : σ₁² ≠ σ₂²
Test Statistic
The F-statistic is
F = Larger Sample Variance / Smaller Sample Variance
or
F = s₁² / s₂²
The calculated F-value follows the F-distribution with degrees of freedom:
(n₁ − 1) and (n₂ − 1)
Decision Rule
Calculate the sample variances.
Compute the F-statistic.
Compare the calculated value with the tabulated critical F-value.
If the calculated value exceeds the critical value, reject the null hypothesis.
Otherwise, accept the null hypothesis and conclude that the variances are equal.
Applications
The F-test is widely used in analysis of variance (ANOVA), quality control, industrial experiments, agricultural research and business studies where comparison of variability is important.
Q. 13. What is chi-square test?
Answer: The Chi-square (χ²) test is a non-parametric statistical test used to compare observed frequencies with expected frequencies under a specified hypothesis. It determines whether the difference between observed and expected frequencies is due to chance or is statistically significant.
The test statistic is
χ² = Σ (Observed Frequency − Expected Frequency)² / Expected Frequency
The calculated χ² value is compared with the critical value from the Chi-square distribution table using appropriate degrees of freedom. If the calculated value exceeds the critical value, the null hypothesis is rejected.
The unit mainly discusses two forms of the Chi-square test:
Goodness of Fit Test.
Test of Independence of Attributes.
Q. 14. Explain important uses of chi-square test with the help of examples.
Answer: The Chi-square test has several important practical applications.
i. Test of Goodness of Fit:
It examines whether the observed data fit a theoretical probability distribution.
Example:
A dice is thrown several times to determine whether it is fair.
ii. Test of Independence of Attributes:
It determines whether two categorical variables are associated or independent.
Example:
Testing whether customer satisfaction is independent of gender.
iii. Testing Homogeneity:
It checks whether different populations have the same distribution.
Example:
Comparing brand preferences among customers from different cities.
iv. Quality Control:
Manufacturers use the Chi-square test to examine whether the proportion of defective products differs from expected standards.
Example:
Checking whether the number of defective items produced each day matches the expected defect rate.
v. Market Research:
Marketing firms use the test to analyse consumer preferences and purchasing behaviour.
Example:
Determining whether product preference depends on age group.
vi. Medical and Biological Research:
Researchers use the Chi-square test to study the relationship between diseases and treatment methods.
Example:
Testing whether recovery depends on the type of medicine administered.
vii. Educational Research:
It helps determine whether examination performance is associated with teaching methods.
Example:
Testing whether students' success depends on the method of classroom instruction.
Thus, the Chi-square test is an important statistical tool for analysing categorical data and making decisions in business, economics, social sciences and scientific research.
Subject: Statistics for Business Decision (PGCO - VII)
Course: PGDP - NSOU M. Com.
Unit VII: Analysis of Variance
Q. 1. What do you mean by ANOVA?
Answer: ANOVA (Analysis of Variance) is a statistical technique used to test whether the means of three or more populations are equal. When only two populations are compared, the t-test or z-test is generally used. However, when the number of populations exceeds two, ANOVA based on the F-distribution is applied.
The basic principle of ANOVA is to compare the variation between the sample means with the variation within the samples. If the variation between the sample means is significantly larger than the variation within the samples, it indicates that at least one population mean differs from the others.
ANOVA tests the following hypotheses:
(a) Null Hypothesis (H₀): All population means are equal.
(b) Alternative Hypothesis (H₁): At least two population means are not equal.
The technique was introduced by R. A. Fisher and is widely used in scientific research, business, agriculture, economics, and social sciences for comparing multiple groups simultaneously.
Q. 2. What are the assumptions in ANOVA?
Answer: The validity of ANOVA depends on the following assumptions:
i. Independent Samples:
The samples should be selected randomly from the populations, and they must be independent of one another. The observations in one sample should not influence those in another sample.
ii. Normal Populations:
Each sample should be drawn from a population that follows a normal distribution. This assumption ensures the reliability of the F-test used in ANOVA.
iii. Equal Variances:
The populations under consideration should have equal variances. This assumption is also known as the homogeneity of variances and ensures that the comparison among sample means is meaningful.
These assumptions must be satisfied before applying the ANOVA technique for hypothesis testing.
Q. 3. What do you mean by One-Way ANOVA?
Answer: One-Way ANOVA is a type of Analysis of Variance in which the data are classified and analyzed based on only one factor or one variable. For example, in an agricultural experiment, crop yield may be classified according to different varieties of seeds or according to different fertilizers, but not both simultaneously. Since only one factor is considered for classification, the technique is called One-Way ANOVA. The objective of One-Way ANOVA is to determine whether there is any significant difference among the means of three or more populations based on a single classification factor. One-Way ANOVA always uses a right-tailed F-test. The rejection region lies in the right tail of the F-distribution. If the calculated F-value exceeds the critical F-value, the null hypothesis of equal means is rejected.
Q. 4. Discuss about the Methodology for One-Way ANOVA.
Answer: Suppose there are k samples with sample sizes n₁, n₂, ..., nₖ.
The following notations are used:
nᵢ = Size of the ith sample
xᵢⱼ = jth observation of the ith sample
Tᵢ = Total of the observations of the ith sample
T = Grand total of all observations
n = Total number of observations
x̄ᵢ = Mean of the ith sample
The methodology of One-Way ANOVA consists of the following steps:
Step 1:
Calculate the mean of each sample.
x̄₁, x̄₂, ..., x̄ₖ
Step 2:
Calculate the mean of the sample means.
X̄ = (x̄₁ + x̄₂ + ... + x̄ₖ) / k
Step 3:
Calculate the Sum of Squares Between Samples (SSB).
SSB measures the variation due to differences among the sample means.
Then calculate the Mean Square Between Samples (MSB).
MSB = SSB / (k − 1)
The degrees of freedom between samples are (k − 1).
Step 4:
Calculate the Sum of Squares Within Samples (SSW).
SSW measures the variation within the individual samples.
Then calculate the Mean Square Within Samples (MSW).
MSW = SSW / (n − k)
The degrees of freedom within samples are (n − k).
Step 5:
Calculate the Total Sum of Squares (SST).
SST = SSB + SSW
The total degrees of freedom are (n − 1), which is equal to the sum of the degrees of freedom between and within samples.
Step 6:
Compute the F-statistic.
F = MSB / MSW
The F-statistic follows an F-distribution with degrees of freedom (k − 1, n − k).
Step 7:
Compare the calculated F-value with the tabulated F-value at the chosen level of significance.
If Fcal ≥ Ftab, reject the null hypothesis and conclude that the population means are not equal.
If Fcal < Ftab, accept the null hypothesis and conclude that there is no significant difference among the population means.
Q. 5. What do you mean by One-Way ANOVA Table?
Answer: A One-Way ANOVA Table is a systematic presentation of all the calculations required in One-Way Analysis of Variance. It summarizes the total variation in the data by dividing it into two components: variation between the samples and variation within the samples. It also records the corresponding sums of squares, degrees of freedom, mean squares, and the calculated F-statistic. The table consists of three sources of variation:
Between Samples:
This represents the variation due to differences among the sample means. The sum of squares between samples (SSB) is calculated, and its degrees of freedom are (k − 1). The mean square between samples (MSB) is obtained by dividing SSB by (k − 1).
Within Samples:
This represents the variation within each sample due to random or chance factors. The sum of squares within samples (SSW) is calculated, and its degrees of freedom are (n − k). The mean square within samples (MSW) is obtained by dividing SSW by (n − k).
Total:
This represents the overall variation in all the observations combined. The total sum of squares (SST) is equal to the sum of SSB and SSW, and its degrees of freedom are (n − 1).
The calculated F-statistic is obtained by dividing MSB by MSW. This value is then compared with the critical F-value at the chosen level of significance. If the calculated F-value is greater than or equal to the tabulated F-value, the null hypothesis is rejected, indicating that there is a significant difference among the population means. Otherwise, the null hypothesis is accepted, indicating that there is no significant difference among the population means.
Q. 6. Discuss about the Hypotheses for One-Way ANOVA and Conclusion.
Answer: In One-Way ANOVA, only one factor or classification criterion is considered for comparing the means of different populations. The purpose is to determine whether the differences among the sample means are statistically significant.
Null Hypothesis (H₀)
The null hypothesis states that the means of all the populations are equal.
H₀ : μ₁ = μ₂ = μ₃ = … = μₖ
This means that there is no significant difference among the population means and any observed difference is due to random sampling fluctuations.
Alternative Hypothesis (H₁)
The alternative hypothesis states that at least two population means are not equal.
H₁ : At least two means are not equal.
This implies that a significant difference exists among the population means.
Conclusion
The test statistic in One-Way ANOVA is
F = MSB / MSW
where
MSB = Mean Square Between Samples
MSW = Mean Square Within Samples
Decision Rule
If the calculated F-value is greater than or equal to the tabulated (critical) F-value at the chosen level of significance with degrees of freedom (k−1, n−k), reject the null hypothesis. This indicates that the population means are not equal.
If the calculated F-value is less than the tabulated F-value, accept (fail to reject) the null hypothesis. This indicates that there is no significant difference among the population means and they may be considered equal.
Q. 7. What do you mean by Two-Way ANOVA?
Answer: Two-Way Analysis of Variance (Two-Way ANOVA) is an extension of One-Way ANOVA in which the observations are classified according to two different factors simultaneously. It is used to examine the effect of two independent factors on a dependent variable.
Unlike One-Way ANOVA, where only one source of variation is considered, Two-Way ANOVA separates the total variation into variation due to the first factor, variation due to the second factor, and random error.
The main objectives of Two-Way ANOVA are:
(i) To study the effect of the first factor.
(ii) To study the effect of the second factor.
(iii) To determine whether the differences among the group means are statistically significant.
Two-Way ANOVA is especially useful when data are arranged in rows and columns representing two different classifications.
Q. 8. Discuss about the Methodology for Two-Way ANOVA.
Answer: The methodology of Two-Way ANOVA consists of the following steps:
Step 1
Arrange the observations according to two factors in a row-column format.
Step 2
Calculate the total of each row and each column.
Step 3
Find the grand total of all observations.
Step 4
Calculate the Correction Factor (CF):
CF = T² / N
where
T = Grand total of all observations
N = Total number of observations
Step 5
Calculate the Total Sum of Squares (SST).
SST = Σx² − CF
Step 6
Calculate the Sum of Squares due to Rows (SSR).
SSR = Σ(Rᵢ² / c) − CF
where
Rᵢ = Total of the ith row
c = Number of columns
Step 7
Calculate the Sum of Squares due to Columns (SSC).
SSC = Σ(Cⱼ² / r) − CF
where
Cⱼ = Total of the jth column
r = Number of rows
Step 8
Calculate the Error Sum of Squares (SSE).
SSE = SST − SSR − SSC
Step 9
Determine the degrees of freedom.
Rows = r − 1
Columns = c − 1
Error = (r − 1)(c − 1)
Total = rc − 1
Step 10
Calculate the Mean Squares.
MSR = SSR / (r − 1)
MSC = SSC / (c − 1)
MSE = SSE / [(r − 1)(c − 1)]
Step 11
Compute the F-ratios.
For rows:
F = MSR / MSE
For columns:
F = MSC / MSE
Step 12
Compare the calculated F-values with the corresponding tabulated F-values. Accept or reject the null hypotheses accordingly.
Q. 9. What do you mean by Two-Way ANOVA Table?
Answer: The Two-Way ANOVA Table is a systematic tabular presentation used to summarize the analysis of variance when two factors are involved. It shows the different sources of variation along with their sum of squares, degrees of freedom, mean squares and calculated F-values. A standard Two-Way ANOVA table contains the following components:
Source of Variation
Rows
Columns
Error
Total
Sum of Squares (SS)
SSR
SSC
SSE
SST
Degrees of Freedom (d.f.)
Rows = r − 1
Columns = c − 1
Error = (r − 1)(c − 1)
Total = rc − 1
Mean Squares (MS)
MSR = SSR / (r − 1)
MSC = SSC / (c − 1)
MSE = SSE / [(r − 1)(c − 1)]
F-Statistics
For rows:
F = MSR / MSE
For columns:
F = MSC / MSE
The ANOVA table helps compare the calculated F-values with the critical F-values and provides the basis for statistical decision-making regarding the significance of the two factors.
Q. 10. Discuss about the Hypotheses for Two-Way ANOVA and Conclusion.
Answer: In Two-Way ANOVA, separate hypotheses are tested for each of the two factors.
Hypotheses for Rows
Null Hypothesis (H₀)
There is no significant difference among the row means.
Alternative Hypothesis (H₁)
At least one row mean differs significantly.
Hypotheses for Columns
Null Hypothesis (H₀)
There is no significant difference among the column means.
Alternative Hypothesis (H₁)
At least one column mean differs significantly.
Conclusion
After calculating the F-values for rows and columns separately:
If the calculated F-value is greater than or equal to the corresponding critical F-value, reject the null hypothesis. This indicates that the respective factor has a significant effect on the observations.
If the calculated F-value is less than the critical F-value, accept (fail to reject) the null hypothesis. This indicates that the corresponding factor has no significant effect, and the observed differences are not statistically significant.
Thus, Two-Way ANOVA enables the researcher to examine the influence of two different factors simultaneously and determine whether each factor contributes significantly to the variation in the data.
Subject: Statistics for Business Decision (PGCO - VII)
Course: PGDP - NSOU M. Com.
Unit VIII: Statistical Quality Control
Q. 1. What is Statistical Quality Control (SQC)?
Answer: Statistical Quality Control (SQC) refers to the application of statistical techniques for controlling and maintaining the quality of products during the manufacturing process. It helps determine whether the desired quality standards are being achieved without inspecting every item produced. Instead, it relies on statistical sampling and analysis to evaluate product quality.
Statistical Quality Control is defined as the statistical techniques employed for determining the extent to which quality goals are being met, for indicating whether the variations occurring in the manufacturing process exceed normal expectations, and for taking decisions regarding the acceptance or rejection of products.
Alford and Beatly define SQC as an industrial management technique by means of which products of uniform acceptable quality are manufactured. The emphasis of SQC is on producing quality products rather than merely identifying and rejecting defective ones after production.
Thus, SQC serves as an effective quality management tool that helps manufacturers maintain consistency, reduce defects, improve efficiency, and ensure customer satisfaction.
Q. 2. Discuss the need for Statistical Quality Control.
Answer: Statistical Quality Control has become an essential part of modern manufacturing because it helps organizations maintain quality standards and remain competitive. The study material highlights the following needs for SQC:
i. Assurance of Quality to Customers:
SQC assures customers that the products they purchase conform to established quality standards. This increases customer confidence and satisfaction.
ii. Development of Quality Consciousness:
The implementation of SQC makes employees more quality conscious. Workers pay greater attention to maintaining the required quality standards during production.
iii. Protection Against Heavy Losses:
SQC protects both manufacturers and customers from heavy losses resulting from the rejection of large quantities of defective products. Early detection of quality problems minimizes wastage.
iv. Guidance for Future Quality Standards:
SQC acts as a guide for establishing future quality standards whenever existing control limits need revision or improvement. It provides valuable statistical information for quality planning.
v. Creation of Goodwill:
Consistent production of quality products enhances the goodwill and reputation of the manufacturer. Customers develop greater trust in the company's products, leading to increased market acceptance.
Therefore, Statistical Quality Control is essential for maintaining product quality, reducing production costs, improving operational efficiency, and strengthening the firm's competitive position.
Q. 3. "Variations in the quality of products in a manufacturing process are attributed to two distinct types of causes". Explain.
Answer: In practice, it is almost impossible for every product produced in a manufacturing process to be exactly identical. Although products are manufactured according to prescribed specifications, slight differences in quality inevitably occur. Some products may fall slightly below the standard, while others may exceed it.
According to the study material, these variations arise due to two distinct types of causes:
(i) Chance Causes:
Chance causes are natural and unavoidable variations that occur because of numerous small and independent factors present in the production process. These variations are inherent in manufacturing and cannot be completely eliminated. When only chance causes are present, the production process is considered to be under statistical control.
(ii) Assignable Causes:
Assignable causes are specific and identifiable reasons for variation arising from defects or faults in the production design or manufacturing process. These variations are not inherent and can be prevented or corrected. Statistical Quality Control aims to identify such causes quickly so that corrective measures can be taken before a large number of defective products are produced.
Thus, quality variation in manufacturing is classified into chance causes and assignable causes. While chance causes are unavoidable and acceptable within control limits, assignable causes indicate abnormal conditions requiring immediate corrective action.
Q. 4. What do you mean by chance causes in the context of SQC?
Answer: Chance causes refer to the small and unavoidable variations that naturally occur during the manufacturing process. These variations arise from numerous minor and independent factors that are inherent in production and cannot be completely eliminated.
If the variations in product quality are caused solely by chance causes, the manufacturing process is regarded as being under control or in a state of statistical control.
Characteristics of chance causes include:
(i) They are inherent in every manufacturing process.
(ii) They are produced by several independent and minor factors.
(iii) They cannot be prevented altogether.
(iv) They result in only small variations in product quality.
(v) Their presence indicates that the production process is operating normally within statistical limits.
Therefore, chance causes represent normal process variation and do not require corrective action as long as the process remains within established control limits.
Q. 5. Define assignable causes in the context of SQC.
Answer: Assignable causes are the specific and identifiable factors responsible for major variations in the quality of products. These variations generally arise due to defects or faults in production design, machinery, materials, workers, or manufacturing methods.
Unlike chance causes, assignable causes are not inherent in the production process. They can be detected, investigated, and eliminated through appropriate corrective measures.
According to the study material, the importance of Statistical Quality Control lies in its ability to detect assignable causes quickly so that corrective action can be taken before products become defective or before large-scale quality problems develop.
Characteristics of assignable causes include:
(i) They produce major variations in product quality.
(ii) They arise due to identifiable defects or faults.
(iii) They are preventable and controllable.
(iv) They indicate that the production process is not under statistical control.
(v) They require immediate investigation and corrective action.
Thus, assignable causes represent abnormal variations in manufacturing and must be identified and removed to restore the production process to a state of statistical control.
Q. 6. What is Control Chart?
Answer: A control chart is a graphical display used to monitor the quality of a production process over time. It helps quality control engineers identify whether the manufacturing process is operating under statistical control or whether any unusual variation has occurred. The concept of the control chart was developed by Dr. Walter A. Shewhart of Bell Laboratories, USA.
A control chart consists of three horizontal lines:
• Central Line (CL): Represents the desired or average quality level of the process.
• Upper Control Limit (UCL): Represents the upper permissible limit of variation.
• Lower Control Limit (LCL): Represents the lower permissible limit of variation.
Generally, the UCL and LCL are placed at three standard deviations (±3σ) from the central line and are therefore known as three-sigma control limits. If all sample points lie within these limits, the process is considered to be under statistical control. If any point falls outside the control limits, it indicates the presence of assignable causes of variation, requiring immediate investigation and corrective action. Control charts are based on the normal distribution and are widely used in statistical quality control for maintaining product quality.
Q. 7. What is Mean Chart?
Answer: A Mean Chart, also known as the X̄-chart, is a control chart used to monitor the average value of a quality characteristic in different samples drawn from a production process. It is prepared using rational sub-groups, generally consisting of 4 or 5 units, with at least 25 such sub-groups.
The preparation of a mean chart involves the following steps:
i. Calculate the mean of each sample.
ii. Compute the average of all sample means, which becomes the central line.
iii. Determine the upper and lower control limits using the standard error of the sample means or conversion factors.
iv. Plot the sample means on graph paper against the control limits.
The control limits are:
UCL = X̄ + 3σx
LCL = X̄ − 3σx
When conversion factors are used:
UCL = X̄ + A₂R̄
LCL = X̄ − A₂R̄
The mean chart is mainly used to detect changes in the process average and helps determine whether the production process is maintaining the desired quality level.
Q. 8. What is Range Chart?
Answer: A Range Chart, also called the R-chart, is a control chart used to study the variability or dispersion in a production process, especially when the sample size is small. Instead of monitoring averages, it examines the spread of observations within each sample.
The preparation of an R-chart involves:
i. Calculating the range of each sample by subtracting the smallest observation from the largest observation.
ii. Finding the average of all sample ranges (R̄), which serves as the central line.
iii. Computing the upper and lower control limits.
The control limits are:
UCLR = D₄R̄
LCLR = D₃R̄
where D₃ and D₄ are standard control chart constants depending on the sample size.
If the calculated lower control limit is negative, it is taken as zero because the range cannot be negative. The R-chart is useful in determining whether the process variability remains stable and whether the process is under statistical control.
Q. 9. What is Control Chart for Standard Deviation?
Answer: The Standard Deviation Control Chart, also called the σ-chart, is a control chart used to monitor the variability of a production process using sample standard deviations. Although the R-chart is easier to use, the σ-chart is considered a more accurate and theoretically superior measure of process dispersion.
The σ-chart is similar to the R-chart in construction. The central line represents the average of the sample standard deviations (σ̄), and the control limits are determined using standard chart constants.
The control limits are:
UCLσ = B₄σ̄
LCLσ = B₃σ̄
where:
• σ̄ is the average of the sample standard deviations.
• B₃ and B₄ are control chart constants based on sample size.
The σ-chart is interpreted in the same way as the R-chart. If all plotted points lie within the control limits, the process variability is considered to be under statistical control. If any point falls outside the limits, it indicates the existence of assignable causes that require investigation.
Q. 10. What is p-Chart?
Answer: The p-chart, or Fraction Defective Chart, is a control chart used for quality characteristics measured by attributes. It is suitable when each product is classified into only two categories: defective and non-defective.
The p-chart is based on the proportion of defective items in each sample. Since the number of defectives follows the binomial distribution, the sample proportion approaches the normal distribution when the sample size is large.
The central line is the average fraction defective:
p̄ = Total number of defectives in all samples ÷ Total number of items inspected in all samples
The control limits are:
UCL = p̄ + 3√[p̄(1−p̄)/n]
LCL = p̄ − 3√[p̄(1−p̄)/n]
where n is the sample size.
The p-chart is used to determine whether the proportion of defective items remains consistent from sample to sample. If many sample points fall outside the control limits, the production process is considered to be out of statistical control.
Q. 11. What is c-Chart?
Answer: The c-chart, or Control Chart for Number of Defects, is used to monitor the number of defects or imperfections occurring in a single unit or item. It is appropriate when the quality characteristic is the count of defects rather than whether an item is defective or not.
The c-chart assumes that the number of defects follows the Poisson distribution, where the mean and variance are equal. The central line is the average number of defects per unit, denoted by c̄.
The control limits are:
UCLc = c̄ + 3√c̄
LCLc = c̄ − 3√c̄
where c̄ is the average number of defects per item.
If the calculated lower control limit is negative, it is taken as zero because the number of defects cannot be negative. The c-chart helps monitor process quality by identifying unusual increases or decreases in the number of defects and enables corrective action whenever the process goes out of control.
Q. 12. What do you mean by acceptance sampling?
Answer: Acceptance sampling is a statistical quality control technique used to decide whether a lot or batch of products should be accepted or rejected on the basis of the inspection of only a sample selected from the lot instead of examining every item. It is mainly used after the production process is completed and before the products are released for sale. This technique helps determine whether the quality of the entire lot is satisfactory.
Acceptance sampling is a method of product control. It saves time, labour and inspection cost because only a representative sample is inspected. Based on the number of defective items found in the sample, the entire lot is either accepted or rejected. The unit also discusses different acceptance sampling plans such as single sampling, double sampling and multiple sampling inspection plans.
Q. 13. What do you mean by the term 'rational sub-group'?
Answer: A rational sub-group is a carefully formed sample or group of items selected in such a way that the variation in quality among the items within the same group is as small as possible, while the variation between different groups is as large as possible.
The concept of rational sub-grouping is essential for the effective use of control charts. The purpose is to ensure that the differences observed within a group are mainly due to chance causes, while differences between groups can reveal assignable causes if they exist.
Proper rational sub-grouping enables quality control engineers to detect abnormal variations quickly and accurately, making control charts an effective tool for monitoring production quality.
Q. 14. What are control charts?
Answer: A control chart is a graphical display of measurements of an industrial process over time. It is used to monitor whether a production process is operating under statistical control. The control chart was developed by Dr. Walter A. Shewhart and is widely used in statistical quality control.
A control chart consists of three horizontal lines:
Central Line (CL): It represents the desired or average quality level of the process.
Upper Control Limit (UCL): It represents the upper limit of acceptable variation.
Lower Control Limit (LCL): It represents the lower limit of acceptable variation.
Normally, the UCL and LCL are placed three standard deviations above and below the central line respectively. These are known as three-sigma control limits.
When all sample points lie within the control limits, the production process is considered to be under statistical control. If any point falls outside these limits, it indicates the presence of assignable causes of variation. The concerned authority should investigate the causes and take corrective action to restore the process to a state of control.
Q. 15. State the different types of control charts for variables and attributes.
Answer: Control charts are classified into two broad categories.
(i) Control Charts for Variables:
These charts are used when the quality characteristic can be measured quantitatively in specific units. The three commonly used control charts are:
Mean Chart (X̄ Chart): It is used to control the average quality level of the process.
Range Chart (R Chart): It is used to monitor the variability or dispersion in the production process.
Standard Deviation Chart (σ Chart): It is used to control the process variability by measuring the standard deviation of the samples.
(ii) Control Charts for Attributes:
These charts are used when quality characteristics cannot be measured numerically but are judged by the presence or absence of defects.
Fraction Defective Chart (p Chart): It measures the proportion or fraction of defective items in a sample.
Number of Defects Chart (c Chart): It records and controls the number of defects or imperfections per unit.
Q. 16. What is acceptance sampling? Why is it used? State the conditions for its use.
Answer: Acceptance sampling is a statistical method of quality control in which only a sample selected from a production lot is inspected to decide whether the entire lot should be accepted or rejected. It is a product control technique applied after the manufacturing process is completed.
Acceptance sampling is used because:
(i) It reduces the cost of inspection.
(ii) It saves time and labour by avoiding complete inspection.
(iii) It is suitable when testing every item is expensive or destructive.
(iv) It provides a scientific basis for accepting or rejecting production lots.
(v) It ensures that products released for sale satisfy the required quality standards while maintaining inspection efficiency.
The conditions for the use of acceptance sampling are:
(i) The products should be produced in clearly identifiable lots or batches.
(ii) The sample should be selected randomly so that every item has an equal chance of being included.
(iii) The sample should be representative of the entire lot.
(iv) Predetermined acceptance and rejection criteria should be fixed before inspection.
(v) The quality characteristics and inspection procedures should be clearly defined.
The lot should be accepted or rejected solely on the basis of the inspection results obtained from the selected sample according to the prescribed sampling plan. Acceptance sampling is widely adopted in industrial quality control because it provides an economical and reliable method for quality inspection without examining every item in the lot.
Q. 17. Narrate the different types of sampling inspection plan popularly used in practice.
Answer: Sampling inspection is a technique in which only a sample of items is inspected instead of examining every item in a lot. On the basis of the inspection results, the entire lot is accepted or rejected. The unit explains three popular attribute sampling inspection plans.
Single Sampling Inspection Plan
In a single sampling inspection plan, only one sample of predetermined size is selected from the lot. The number of defective items in the sample is counted. If the number of defectives is less than or equal to the acceptance number, the entire lot is accepted. If the number of defectives exceeds the acceptance number, the lot is rejected. This method is simple, easy to operate and widely used when inspection costs are low.
Double Sampling Inspection Plan
Under the double sampling inspection plan, a first sample is inspected. If the results clearly indicate acceptance, the lot is accepted immediately. If the results clearly indicate rejection, the lot is rejected. However, if the results are inconclusive, a second sample is drawn and inspected. The decision to accept or reject the lot is then taken on the basis of the combined results of both samples. This plan reduces the average amount of inspection compared with the single sampling plan.
Multiple Sampling Inspection Plan
In the multiple sampling inspection plan, inspection is carried out through a series of small samples. After examining each sample, a decision is taken either to accept the lot, reject the lot or continue inspection by drawing another sample. This process continues until a final decision is reached. This method generally requires the least average sample size but is comparatively more complicated to administer and therefore requires skilled supervision.
Thus, the three popular sampling inspection plans are single sampling, double sampling and multiple sampling inspection plans. The choice among them depends on inspection cost, desired accuracy and the nature of production.
Q. 18. Write short notes on: Fraction defective chart.
Answer: The Fraction Defective Chart, also known as the p-chart, is a control chart used for attributes when each manufactured item is classified as either defective or non-defective. It is applied to determine whether the proportion of defective items in different samples remains under statistical control.
The number of defective items in a sample follows a binomial distribution. For sufficiently large sample sizes, the sample proportion approaches a normal distribution according to the Central Limit Theorem, making the p-chart suitable for practical applications.
The central line of the p-chart is the average fraction defective, denoted by p̄, which is calculated as:
p̄ = Total number of defectives in all samples ÷ Total number of items in all samples
The control limits are:
UCL = p̄ + 3√[p̄(1 − p̄)/n]
LCL = p̄ − 3√[p̄(1 − p̄)/n]
If many sample points fall outside these control limits, the production process is considered to be out of statistical control and corrective action is required. The p-chart is widely used in quality control for monitoring the proportion of defective products in manufacturing industries.
Q. 19. Write short notes on: Control chart for number of defects.
Answer: The Control Chart for Number of Defects, also known as the c-chart, is used to control and monitor the number of defects or imperfections occurring in a product or unit. For example, in the manufacture of glass bottles, the number of air bubbles present in each bottle may be monitored through a c-chart.
The chart assumes that the number of defects follows a Poisson distribution, in which the mean and variance are equal. For large sample sizes, the Poisson distribution can be approximated by the normal distribution.
The central line of the chart is the average number of defects per unit, represented by c̄.
The control limits are:
UCL = c̄ + 3√c̄
LCL = c̄ − 3√c̄
Since the average number of defects cannot be negative, if the calculated lower control limit is negative, it is taken as zero.
The c-chart helps management identify unusual variations in the number of defects. If observations remain within the control limits, the production process is considered to be under statistical control. If observations fall outside the limits, assignable causes should be investigated and corrective measures taken immediately.
Q. 20. Distinguish between product control and process control.
Answer: Product control refers to the quality control activity carried out after the production process has been completed. The products are inspected when they are ready for sale or delivery to customers. The primary objective of product control is to determine whether the finished products meet the prescribed quality standards and whether the lot should be accepted or rejected. Acceptance sampling plans are generally used for product control.
Process control refers to the quality control activity performed during the manufacturing process itself. In this method, quality is monitored while the products are still in the semi-finished stage. The objective is to detect variations in the production process at an early stage so that corrective measures can be taken before defective products are produced. Control charts such as mean charts, range charts, p-charts and c-charts are commonly used for process control.
Thus, product control emphasizes inspection of finished products and acceptance or rejection of lots, whereas process control emphasizes continuous monitoring of the manufacturing process to prevent defects and maintain consistent quality standards.
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