Sampling Theory

CPT Section D Quantitative Aptitude Chapter 15

Prof. Bharat Koshti

Learning Objectives :

The different procedures of sampling which will be best representative of the population.

The Concept of Sampling Distribution

The techniques of construction of Class Interval & its interpretation.

How to determine the sample size with pre defined degree of precision?.

Why Sampling?

We come across situations whose where we would like to know about vast population.

But the factors like time, money, cost and large size of the population make it almost impossible to go for the complete enumeration of all the population.

Sampling Theory

Instead we can select a representative or part of the population (known as sample) & infer about the entire population on the basis of our knowledge about the sample.

This is the basis of Sampling Theory.

What is sampling ?

It is a process of learning about a population on the basis of samples drawn from it.

Some important terms in sampling theory:

• (A) Population • (B) Sample • (C) Parameter • (D) Statistic

A)Population

Population: It is a set of all units under consideration. e.g.

(a) Population of Income-tax Payers in India

(b) Population of all people living in Kolkata

(c) Population of employees working in a certain factory

(d) Population of cars produced by a certain company

(e) Population of flowers in a certain town.

Population Size:

The no. of members in the population is called as the size of the population. It is denoted by N.

Different types of Population:

• Population consists of finite no. of elements is called as finite population. e.g. In the previous example all (a) to (d) are the examples of finite population.

Finite Population:

• Population consists of infinite or uncountable number of units is known as infinite population. e.g. (e)

Infinite Population:

Different types of Population:

• Population consists of real objects is called as existent population. e.g. In the previous example all (a) to (e) are the examples of existent population.

Existent Population :

• A population exists just hypothetically is known as hypothetical population. e.g. Population of heads when a coin is tossed infinitely many times.

Hypothetical(Imaginary) Population:

(B) Sample

(B) Sample : It is a part of the population which is selected at random from the population. The number of members/units in the sample is known as the sample size & is denoted by ‘n’.

-These units are known as “Sampling Units”.

-A detailed & complete list of sampling units is known as “Sampling frame”. It must be updated.

Principles of Sample Survey

Sample Survey is the study of unknown population on the basis of a proper representative sample drawn from it.

This study is based on the following principles:

• 1.Law of statistical regularity • 2. Principle of Inertia • 3. Principle of Optimization • 4. Principle of Validity

1.Law of Statistical Regularity:

According to the law if a sample of fairly large size is drawn from the population at random, then on an average the sample would posses the characteristics of that population.

The reliability of a statistic in estimating a population characteristics varies as the square root of the sample size.

2. Principle of Inertia :

According to this law as the results derived from the sample are more likely to be reliable, accurate and precise as the sample size increases, provided other factors are kept constant.

3.Principle of Optimization:

The principle of optimization ensures that an optimum level of efficiency at a minimum cost or the maximum efficiency at a given level of cost can be achieved if the selection of an appropriate sampling design is done.

4. Principle of Validity :

The principle of validity states that a sampling design is valid only if it is possible to obtain valid estimates and valid tests about population parameters. Only a probability sampling ensures this validity.

Sample Survey And Complete Enumeration

When complete information is collected for all the units belonging to a population, it is defined as complete enumeration or census. We prefer sample survey to complete enumeration due to the following factors: • Speed • Cost • Reliability • Accuracy

Errors In Sample Survey

Every sampling is subjected to what is known as sampling fluctuation which is called as sampling error. However, errors due to recording observations, biases on the part of the enumerators, wrong and faulty interpretation of data etc. are present in both sampling and census and this type of error is termed as non-sampling errors.

(C) Parameter:

A parameter may be defined as a characteristic of a population based on all the units of the population. It is unknown and is estimated from the sample observations taken from the population. e.g. • Population Mean(µ) • Population Proportion(P) • Population Variance(σ2)

(D) Statistic

A statistic may be defined as a function of sample observations. If the sample observations are denoted by (x1x2 x3……….. xn) then a statistic T may be expressed as T = f(x1x2 x3……….. xn)

A statistic is used to estimate a particular population parameter. The estimates of population mean, variance and population proportion are given by Sample Mean(x̅),Sample Variance(s) and Sample Proportion(p).

Sampling Fluctuation

The variation in the values of a statistic is termed as “Sampling Fluctuation”.

If we compute the value of a statistic, say mean, it is quite natural that the value of the sample mean may vary from sample to sample as the sampling units of one sample may be different from that of another sample.

e.g. If we take a random sample of 5 observations i.e. {20,27,38,30,35} from the population of 100 units. Then its sample mean (x̅) is 30.

Now if we take another sample of 5 observations from the same population as {25,36,33,37,44} then its sample mean (x̅) is 35. The difference between these two sample means i.e. 35-30 = 5 is called as sampling fluctuation.

Sampling Distribution

We can obtain values of statistic from all possible samples of given size with corresponding probabilities. This is called as sampling distribution of a statistic.

• The mean of the statistic, obtained from its sampling distribution, is known as “Expectation” and the standard deviation of the statistic T is known as the “Standard Error (SE)“ of T.

• SE can be regarded as a measure of precision achieved by sampling. • SE is inversely proportional to the square root of sample size. Thus

S.E.(T) α 1/√n̅

Standard Error for Mean

( )X

N nSE (for SRS WOR)N 1n

σ −=

−

Standard Error(SE) of Mean

For sampling with replacement (SRSWR) SE(x̅) = σ / √n̅

For sampling without replacement(SRSWOR)

SE for Sample Proportion (P)

• SE for sample proportion (p)

Parameter

µ - Population Mean s - Sample Standard Deviation

x̅ - Sample Mean

Statistic

σ – Population Standard deviation

P - Population Proportion p - Sample Proportion

Notations in Parameter V/S Statistic

Number of Samples can be Drawn

In case of sampling with replacement the total no. of samples that can be drawn is Nn

In case of sampling without replacement the total no. of samples that can be drawn is NCn

Question Time MCQ's

1. Statistical data may be collected by complete enumeration is called

(a) Census inquiry

(b) Sample inquiry

(c) both

(d) none

Answer: A

2. Sampling can be described as a statistical procedure (a) To infer about the unknown universe from a knowledge of any

sample.

(b) To infer about the known universe from a knowledge of a sample drawn from it.

(c) To infer about the unknown universe from a knowledge of a random sample drawn from it.

(d) Both (a) and (b).

Answer: C

3. Statistical decision about an unknown universe is taken on the basis of

(a) Sample observations

(b) A sampling frame

(c) Sample survey

(d) Complete enumeration

Answer: A

4. A Statistic is

(a) a function of sample observations.

(b) a function of a population units.

(c) a characteristic of a population.

(d) a part of a population.

Answer: A

5.The law of Statistical Regularity says that

(a) Sample drawn from the population under discussion possesses the characteristics of the population.

(b) A large sample drawn at random from the population would possesses the characteristics of the population.

(c) A large sample drawn at random from the population would posses the characteristics of the population on an average.

(d) An optimum level of efficiency can be attained at a minimum cost.

Answer: C

(a) The distribution of sample observations.

(b) The distribution of random sample.

(c) The distribution of a parameter.

(d) The probability distribution of a statistic.

Answer: d

MCQ.6:The sampling distribution is

(a) The variation in the values of a statistic.

(b) The variation in the values of a sample.

(c) The differences in the values of a parameter.

(d) The variation in the values of observations.

Answer: A

MCQ.7:Sampling fluctuations may be described as

(a) Error in statistic

(b) Absolute error

(c) Percentage error

(d) Relative error.

Answer: A

MCQ. 8:The difference of the actual value and the expected value using a model is

(a) square root of(N-1)/(N-n)

(b) square root of(N-n)/(N-1)

(c) square of(N-1)/(N-n)

(d) square of(N-n)/(N-1)

Answer: B

MCQ.9:Finite population multiplier is

(a) n/N

(b) N/n

(c) (n+1)/N

(d) (N+1)/n

Answer: A

MCQ.10:Sampling fraction is