12. SAMPLING THEORY AND THEORY OF ESTIMATION · 3. Sampling & Sampling Theory Sampling : It is the...

J. K. SHAH CLASSES Sampling Theory and Theory of Estimation

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12. SAMPLING THEORY AND THEORY OF ESTIMATION

1. Population and Sample

Population or Universe. Population in statistics means the whole of the information

which comes under the purview of statistical investigation. It is the totality of all the

observations of a statistical experiment or enquiry.

A population may be finite or infinite according as the number of observations or items

in it are finite or infinite. The population of weights of students of class XII in a

government school is an example of a finite population. The population of pressure at

different points in the atmosphere is an example of an infinite population.

Sample. A part of the population selected for study is called a sample. In other words,

the selection of a group of individuals or items from a population in such a way that this

group represents the population, is called a sample.

2. Parameter and Statistic

There are various statistical measures in statistics such as mean, median, mode,

standard deviation, coefficient of variation etc. These statistical measures can be

computed both from population (or universe) data and sample data.

Parameter : Any statistical measure computed from population data is known as

parameter.

Statistics : Any statistical measure computed from sample data is known as statistic.

Thus a parameter is a statistical measure which relates to the population and is based

on population data, whereas a statistic is a statistical measure which relates to the

sample and is based on sample data. Thus a population mean, population median,

population variance, population coefficient of variation etc., are all parameters. Statistic

computed from a Sample such as sample mean, sample variance etc.

Notations

Statistical Measure Population Sample

Mean µµµµ x

Standard deviation σσσσ s

Proportion P p

Size N n


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3. Sampling & Sampling Theory

Sampling : It is the procedure or process of selecting a sample from a population. A

sampling can also be defined as the process of drawing a sample from a population.

Sampling Theory : The study of relationship existing between a population and the

samples drawn from population is called Sampling theory which is based on sampling.

SAMPLING WITH REPLACEMENT (SRSWR) : While selecting the units for a sample,

when a unit of sample selected is replaced before the next unit is selected then it is

called sampling with replacement.

In this case the total number of samples that can be drawn = (N)n For E.g.: Let

Population = {a, b, c}

N = 3, let n = 2

No. of samples = (N)n = (3)2 = 9

No. of samples = {(a, b) (a, c) (b, c) (b, a) (c, a) (c, b) (a, a) (b, b) (c, c)}

SAMPLING WITHOUT REPLACEMENT (SRSWOR) : While selecting the units for a

sample, when a unit of sample is selected but not replaced before the next unit is

selected then it is called Sampling Without Replacement.

In this case the total number of samples that can be drawn =

For E.g.: Let population = {a, b, c}

N = 3, let n = 2

No. of samples = NCn = 3C2 = 3C1 = 3

No. of samples = {(a, b), (a, c), (b, c)}

TYPES OF SAMPLING

A sample can be selected from a population in various ways. Different situations call for

different methods of sampling. There are three methods of Sampling:

1. Random Sampling or Probability Sampling Method

2. Non-Random Sampling or Non-Probability Sampling Method.

3. Mixed Sampling


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1 Random Sampling or Probability Sampling

Random Sampling: Random or Probability sampling is the scientific technique of drawing

samples from the population according to some laws of chance in which each unit in the

universe or population has some definite pre-assigned probability of being selected in the

sample. It is of two types.

(a) Simple Random Sampling (SRS):

It is the method of selection of a sample in such a way that each and every member

of population or universe has an equal chance or probability of being included in the

sample. Random sampling can be carried out in two ways.

1. Lottery Method: It is the simplest, most common and important method of

obtaining a random sample. Under this method, all the members of the

population or universe are serially numbered on small slips of a paper. They

are put in a drum and thoroughly mixed by vibrating the drum. After mixing, the

numbered slips are drawn out of the drum one by one according to the size of

the sample. The numbers of slips so drawn constitute a random sample.

2. Random Number Method: In this method, sampling is conducted on the basis

of random numbers which are available from the random number tables. The

various random number tables available are:

a. Trippet's Random Number Series;

b. Fisher's and Yates Random Number Series;

c. Kendall and Badington Random Number Series;

d. Rand Corporation Random Number Series;

One major disadvantage of random sampling is that all the members of the

population must be known and be serially numbered. It will entail a lot of difficulties

in case the population is of large size and will be impossible in case the population

is of infinite size.

(b) Restricted Random Sampling:

It is of three types

• Stratified Sampling

• Systematic Sampling(Mixed Sampling)

• Multi-stage Sampling


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Stratified Sampling: In stratified random sampling, the population is divided into

strata (groups) before the sample is drawn. Strata are so designed that they do not

overlap. An elementary unit from each stratum is drawn at random and the units so

drawn constitute a sample. Stratified sampling is suitable in those cases where the

population is hetrogeneous but there is homogeneity within each of the groups or

strata.

Advantages

(i) It is a representative sample of the hetrogeneous population.

(ii) It lessens the possibility of bias of one sidedness.

Disadvantages

(i) It may be difficult to divide population into hetrogeneous groups.

(ii) There may be over lapping of different strata of the population which will

provide an

unrepresentative Sample.

Systematic Sampling: In this method every elementary unit of the population is

arranged in order and the sample units are distributed at equal and regular intervals.

In other words, a sample of suitable size is obtained (from the orderly arranged

population) by taking every unit say tenth unit of the population. One of the first units

in this ordered arrangement is chosen at random and the sample is computed by

selecting every tenth unit (say) from the rest of the lot. If the first unit selected is 4,

then the other units constituting the sample will be 14, 24, 34, 44, and so on.

Advantages: It is most suitable where the population units are serially numbered or

serially arranged.

Disadvantages: It may not provide a desirable result due to large variation in the

items selected.

Multi-stage Sampling: In this sampling method, sample of elementary units is

selected in stages. Firstly a sample of cluster is selected and from among them a

sample of elementary units is selected. It is suitable in those cases where

population size is very big and it contains a large number of units.

2 Non-Random Sampling or Non-Probability Sampling Method

A sample of elementary units that is being selected on the basis of personal judgment is

called a non-probability sampling. It is of five types.


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• Purposive Sampling;

• Cluster Sampling;

• Quota Sampling;

• Convenience Sampling;

• Sequential Sampling.

Purposive Sampling: Purposive sampling is the method of sampling by which a sample

is drawn from a population based entirely on the personal judgement of the investigator. It

is also known as Judgement Sampling or Deliberate Sampling. A randomness finds no

place in it and so the sample drawn under this method cannot be subjected to

mathematical concepts used in computing sampling error.

Cluster Sampling: Cluster Sampling involves arranging elementary items in a population

into hetrogeneous subgroups that are representative of the overall population. One such

group constitutes a sample for study.

Quota Sampling: In quota sampling method, quotas are fixed according to the basic

parameters of the population determined earlier and each field investigator is assigned

with quotas of number of elementary units to be interviewed.

Convenience Sampling: In convenience sampling, a sample is obtained by selecting

convenient population elements from the population.

Sequential Sampling: In sequential sampling a number of sample lots are drawn one

after another from the population depending on the results of the earlier samples draw

from the same population. Sequential sampling is very useful in Statistical Quality

Control. If the first sample is acceptable, then no further sample is drawn. On the other

hand if the initial lot is completely unacceptable, it is rejected straightway. But if the initial

lot is of doubtful and marginal character falling in the area lying between the acceptance

and rejection limits, a second sample is drawn and if need be a third sample of bigger

size may be drawn in order to arrive at a decision on the final acceptance or rejection of

the lot. Such sampling can be based on any of the random or non-random method of

selection.

3. Mixed Sampling : It is partly probabilistic and partly Non- probabilistic in

nature.

Systematic sampling comes under the category of Mixed Sampling

Advantages of Random (OR Probability) Sampling


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1. Random sampling is objective and unbiased. As a 'result, it is defensible before the

superiors or even before the court of law. 8

2. The size of sample depends on demonstrable statistical method and therefore, it

has a justification for the expenditure involved.

3. Statistical measures, i.e. parameters based on the population can be estimated and

evaluated by sample statistic in terms of certain degree of precision required.

4. It provides a more accurate method of drawing conclusions about the characteristics

of the population as parameters.

5. It is used to draw the statistical inferences.

6. The samples may be combined and evaluated, even though accomplished by

different individuals.

7. The results obtained can be assessed in terms of probability, and the sample is

accepted or rejected on a consideration of the extent to which it can be considered

representative.

4. Sampling Distribution of a Statistic

From a population of size N, number of samples of size n can be drawn. These samples

will give different values of a statistic. E.g. if different samples of size n are drawn from a

population, different values of sample mean are obtained. The various values of a

statistic thus obtained, can be arranged in the form of a frequency distribution known as

Sampling Distribution. Thus we can have sampling distribution of sample mean x ,

sampling distribution of sample proportion p etc.

5. Errors in Sampling

Any statistical measure say, mean of the sample, may not be equal to the

corresponding statistical measure (mean) of the population from which the sample has

been drawn. Thus there can be discrepancies in the statistical measure of population,

i.e., parameter and the statistical measures of sample drawn from the same population

i.e., statistic. These discrepancies are known as Errors in Sampling.

6. Standard Error of a Statistic : Standard error is used to measure the variability of the

values of a statistic computed from the samples of the same size drawn from the

population, whereas standard deviation is used to measure the variability of the

observations of the population itself.


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The standard deviation of the sample statistics is called standard error of that statistic.

E.g. if different samples of the same size n are drawn from a population, we get different

values of sample mean x . The S.D. of x . is called standard error of x . . It is obvious

that the standard error of x . will depend upon the size of the sample and the variability

of the population.

i) Standard error of sample mean SE ( x ) = s

orn n

σ

σ=Population S.D

and s=Sample S.D

ii) Standard error of proportion SE (p) = ( ) 1 P P

n

− or

p(1 )

n

p−

Where P=Population proportion

P=Sample proportion

If i) Population size is Finite and the Sampling Fraction n

N≥.05

And ii) Samples are drawn Without Replacement(SRSWOR)

Then , each of the above formula for Standard Error will be multiplied by the factor

1

N n

N

−

−( Finite Population correction or Finite Population Multiplier)FPC

• Formula for standard Error when i) n<30( small sample)

ii) Population S.D σ is unknown

in such a case SE( x )=1

s

n −

The following table will provide us a better understanding of the situations while

calculating SE ( x )

Sample Size Parameter Formula

Large (n ≥ 30) SD is known x

SEn

σ=

Large (n ≥ 30) SD is unknown x

sSE

n=


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Small (n < 30) SD is known x

SEn

σ=

Small (n < 30) SD is unknown x

sSE

n-1=

Rule of multiplying FPC will remain unaltered in a cases

7. Estimation of Population Parameters : The basic purpose of this chapter is to know

about the population parameters with the help of Sample Statistic. If the population is

completely unknown, and we find the population parameters using the knowledge of

sample values only then it is called Estimation of Population Parameters.

Types of Estimation : Estimation is divided into two groups :

(i) Point Estimation (ii) Interval Estimation

Point Estimation : Point estimation a single statistic is used to provide an estimate of

the population parameter. In other words, the estimate of a population parameter given

by a single number is called the point estimation of the parameter. In point estimation,

we find a statistic which may be used for to replace an unknown parameter of the

population for all practical purposes.

Interval Estimation : There are situations where the point estimation is not desirable

and we are interested in finding such limits within which with a known probability or to a

known degree of reliability, the value of the population parameter is expected to lie.

Such a process of estimation is called the interval estimation.

Confidence Level Confidence Coefficient + (Z)

90% 1.64

95% 1.96

98% 2.33

99% 2.58

Almost Certain(99.73%) 3

Most commonly used confidence level is 95% when no reference given use Z = 3


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Interval estimate of population mean

x - z SE ( x ) , x + z SE ( x )

Interval estimate of population proportion

p - Z SE (p) , p + Z SE (p)

8. Determination of Sample Size

The need for determination of the proper size of the sample is very great for practical

use in business where either the standard error is known on the basis of past

experience or where a given absolute level of accuracy is desired. If the sample size is

too large, more money and time have to be spent but the result obtained from the large

sample may not be more accurate than that from a smaller sample. On the other hand,

a valid conclusion may not be reached if the sample size too small. The method of

determining a proper size is given for the following two cases ;

(a) Sample size for Estimating a Population Mean :

∴∴∴∴ 2 2

2

Zn

E

σ=

Here E = | µµµµ - x . |, i.e. the difference between sample mean x . and the

Population mean µµµµ.

E is called Permissible Sampling Error

(b) Sample size for Estimating a Population Proportion :

∴∴∴∴ n = ( )2

2

P 1 PZ

E

−

Here E = |P - p| i.e. the difference between the sample proportion p and the

population proportion P.

E is called Permissible Sampling Error


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NUMERICAL PROBLEMS ON SAMPLING

Simple random sampling with/ without replacement and estimation of sample proportion:

1. A population comprises 3 members 1, 5, 3. Draw all possible samples of size two (i) with

replacement (ii) without replacement

Find the sampling distribution of sample mean in both cases.

2. In simple random sampling with replacement, the total number of possible sample with

distinct permutation of member is:

(N = Size of Population, n = Sample size)

a) N x n

b) Nn

c) N

d) n

3. In simple random sampling without replacement, the total number of possible sample with

distinct permutation of member is:

(N = Size of Population, n = Sample size)

a) Nn

b) P(N, n)

c) C(N,n)

d) None of the above

4. If from a population with 20 members, a random sample without replacement of 2

members is taken, the number of all such samples is :

a) 400

b) 190

c) 210

d) 200

5. If from a population with 25 members, a random sample with replacement of 2 members

is taken, the number of all such samples is:

a) 50

b) 300


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c) 625

d) 125

6. If from a population with 25 members, a random sample without replacement of 3

members is taken, the number of all such sample is:

a) 3400

b) 1250

c) 3400

d) 2300

7. A random sample of 200 articles taken from a large batch of articles contains 25 defective

articles. What is the estimate of the proportion of defective articles in the entire batch?

a) 0.125

b) 0.075

c) 0.250

d) 0.025

8. A random sample of 200 articles taken from a large batch contains 15 defective articles.

What is the estimate of the sample proportions of defective articles?

a) 0.075

b) 0.02

c) 0.03

d) 0.06

Calculation of Standard Error of sample Mean( x ) and sample proportion (p)

9. If a random sample of size 5 is drawn from a finite population of 41 units without replacement,

then find the standard error of sample mean if the population SD is 10 .

a) 1.00

b) 1.341

c) 1.258


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d) 1.589

10. A simple random sample of size 64 is drawn from a finite population consisting of 122

units. If the population standard deviation is 16.8. Find the standard error of sample mean

when the sample is drawn with replacement.

a) 2.1

b) 2.9

c) 4.2


11. A simple random sample of size 64 drawn from a finite population consisting of 122 units.

If the population standard deviation is 16.8, find the standard error of sample mean when

the sample is drawn without replacement.

a) 2.1

b) 2.9

c) 2.07

d) 1.45

12. A random sample of 400 oranges was taken from a large consignment and 52 were

found to be defective. The standard error of the population of defective ones in a sample

of this size is nearly .

a) 0.17

b) 0.0017

c) 0.017

d) 1.700

13. A simple random sample of size 9 is drawn without replacement from a finite population

consisting of 25 units. If the number of defective units in the sample be 5, then the

standard error of the sample proportion of defectives is:

a) 0.67

b) 0.53

c) 0.11

d) 0.135


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Calculation of SE( x ) when n<30 and population of SD σ is unknown

14. A random sample of size 5 is taken from a population containing 100 units. If the sample

observations are 10, 13, 18, 7, 12, find an estimate of the standard error of sample mean,

if sampling is done with replacement.

a) 1.00

b) 1.82

c) 1.28

d) 1.78

15. A random sample of size 5 is taken from a population containing 100 units. If the sample

observations are 18, 7, 13, 10, 12 find an estimate of the standard error of sample mean,

if sampling is done without replacement.

a) 1.85

b) 1.00

c) 1.28

d) 1.78

Interval Estimation- Determination of confidence limit of population mean µµµµ.

16. A random sample of size 100 has mean 15, the population variance being 25. Find the

interval estimate of the population mean with a confidence level of (i) 99% and (ii) 95%.

17. A random sample of 50 items drawn by a particular population has a mean 30 with a S.D.

2.8, construct a 98% confidence interval estimate of the population mean.

18. The quality control manager of a tyre company has sample of 100 tyres and has found

the mean life time to be 30,214 km. The population S.D. is 860. Construct a 95%

confidence interval for the mean life time for this particular brand of tyres.

19. A pharmaceutical company wants to estimate the mean life of a particular drug under

typical weather conditions. Following results were obtained from a sample random

sample of 100 bottles of the drug.

Sample mean = 18 months


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Population s.d. = 5 months

Sample size = 100

Find interval estimate with a confidence level of (a) 90% (b) 95% (c) 99%

Determination of Confidence Limit For Population Proportion(P):

20. A factory produces 60000 pairs of shoes on a daily basis. From a sample of 600 pairs, 3

per cent were found to be of inferior quality. Estimate the number of pairs that can be

reasonably expected to be spoiled in the daily production process at 95% level of

confidence.

21. A random sample of 800 units from a large consignment showed that 200 were damaged.

Find 95% confidence limits for the population proportion of damaged units in the

consignment.

22. Out of 300 households in a town 123 have T.V. sets. Find 95% confidence limits to the

true value of the proportion of the households with T.V. sets in the whole town.

23. A factory is producing 50,000 pairs of shoes daily. From a sample of 500 pairs 2% were

found to be of substandard quality. Estimate the number of pairs that can be reasonably

expected to be spoiled in the daily production and assign limits at 95% level of

confidence.

Determination of Sample size from Confidence limit of µµµµ and P

24. It is known that the population standard deviation in waiting time for L.P.G. gas cylinder in

Delhi is 15 days. How large a sample should be chosen to be 95% confident, the waiting

time is within 7 days of true average.

25. A manufacturing concern wants to estimate the average amount of purchase of its

product in a month by the customers whose standard deviation is Rs.10. Find the sample

size if the maximum error is not to exceed Rs.3 with a probability of 0.99.

26. Mr. X wants to determine on the basis of sample study, the mean time required to

complete a certain job so that he may be 95% confident that the mean may remain within

+ 2 days of the true mean. As per the available records the population variance is 64

days. How large should be sample be for his study?


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27. In measuring reaction time, a psychologist estimated that the standard deviation is 1.08

seconds. What should be the size of the sample in order to be 99% confident that the

error of her estimates of mean would not exceed 0.18 seconds?

28. The incidence of a particular disease in an area in such that 20 per cent people of that

area suffers from it. What size of sample should be taken so as to ensure that the error of

estimation of the proportion should not be more than 5 per cent with 95 per cent

confidence?

� Important theoretical aspect in Sampling:

Introduction:

1 Sampling is a process whereby we judge the characteristics or draw inference about the

totality or Universe (known as population) on the basis of judging the characteristics of a

selected portion taken from that totality (known as sample).

2. Population is the aggregate or totality of the data forming a subject of investigation

Types of Population:

a) Finite Population: When the items in the population are fixed and limited.

Example : No. of students in the class

b) Infinite Population: If a population consist of infinite no. of items its an

infinite population. If a sample is known to have been drawn from a continuous

probability distribution, then the population is infinite. Example : Population of all real

numbers lying between 5 and 20.

c) Real Population: A Population consisting of the items which are all present

physically is termed as real population.

d) Hypothetical Population: The Population consists of the results of the

repeated trails is named as hypothetical population The tossing of a coin

repeatedly results into a hypothetical population of heads and tails.

3. Sample: Sample is the part of population selected on some basis it is a finite

subset of the population.

4. Sample Units : Units forming the samples are called Sample Units.

5. Sample Frame : A complete list of sampling units is called Sample Frame


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6. Sample Fraction : n

Nis called Sampling Fraction where n = Sample Size and N =

Population Size.

7. Complete enumeration or census : In case of enumeration, information is collected for

each and every unit the aggregate of all the units under consideration is called the

‘population’ or the ‘universe’. The results are more accurate and reliable but it involves lot

of time, money and man power.

Basic principle of Sample Survey:

a) Law of Statistical Regularity : It states that a reasonably larger number of items

selected at random from a large group of items, will on the average, represent the

characteristics of the group.

b) Law of Inertia of Large Numbers : This law states that other things same, as the

sample size increases, the results tend to be more reliable and accurate.

c) Principle of Optimization : The principle of optimization ensures that an optimum level

of efficiency at a minimum cost or the maximum efficiency at the given level of cost can

be achieved with the selection of an appropriate sampling design.

d) 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.

• Sampling and Non sampling Errors

i) Sampling Errors: Sampling Errors have their origin in sampling and arise due to the

fact that only a part of the population (i.e. sample) has been used to estimate

population parameters and draw inference about them. As such the sampling errors

are totally absent in a census enumeration.

Sampling errors can never be completely eliminated but can be minimize by choosing a

proper sample of adequate size.

ii) Non Sampling Errors or Bias: As distinct from sampling errors, the non-sampling

errors primarily arise at the stages of observation, approximation and processing of

the data and are thus present in both the complete enumeration and the sample

survey. These error usually arise due to faulty planning, defective schedule of

questionnaire from non-response from the respondents.


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iii) Sampling error is totally absent in “Complete Enumeration” or “Census”

But, Non-Sampling errors are present in both “Complete Enumeration” and “Sample

survey”

• Parameter is a statistical measure on population.

Statistic is a statistical measure on sample.

Sampling introduction: 1. The aggregate or totality of statistical data forming a subject of investigation is known as :

a) Sample b) Population c) Both a) and b) above d) None of the above

2. Population is also known as: a) Universe b) Range c) Area d) Region

3. A portion of the population which is examined with a view to estimating the characteristics

of the population are known as:

a) Sample

b) Universe

c) Population

d) Statistic

4. If a sample is known to have been drawn from a continuous probability distribution then

the population is .

a) Large

b) Finite

c) Infinite

d) Nothing can be said about the population

5. The population of tea drinkers in Kolkata City is an example of:

a) A hypothetical population

b) An infinite population

c) A finite population

d) Nothing can be said about the population


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6. The possibility of reaching valid conclusions concerning a population by means of a

properly chosen sample is based on which of the following laws?

a) Law of Inertia

b) Law of Large Number

c) Law of Statistical Regularity

d) All of the above

7. A sample is a study of a __________ of the population.

a) parameters

b) statistics

c) part

d) none of the above

8. A population is the __________ of units under study.

a) totality

b) part

c) subset


9. “A sample is less expensive than a cenus’’

a) The statement is incorrect.

b) The statement is correct.

c) The given statement is based on nature of sample.

d) None of the above.

10. When the population is infinite we should use the:

a) Sample Method

b) Census Method

c) Either Sample or Census Method



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11. A complete list of all the units in a finite population, properly numbered for identification, is

called a:

a) Universe

b) Sampling Data

c) Sampling Units

d) Sampling Frame

12. Statistical data may be collected by complete enumeration called

a) Sample Enquiry

b) Census Enquiry

c) Both a) and b) above

d) Neither a) nor b) above

13. A border patrol checkpoint which stops every passenger van is utilizing :

a) simple random sampling.

b) systematic sampling

c) systematic sampling.

d) complete enumeration

14. A population consisting of all the items which are physically present is called :

a) hypothetical

b) normal population

c) existent population


15. A population consisting of all real numbers is an example of :

a) an infinite population

b) a finite population

c) an imaginary



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16. The population of roses in Salt Lake City is an example of

a) a hypothetical population

b) an infinite population

c) a finite population

d) an imaginary population

17. Value of ________ is different for different sample

a) Statistic

b) Population



18. A statistic is a __________variable.

a) Compound

b) Simple

c) Random

d) Both a) and c) above

Laws of sample survey :

19. Law of Statistical Regularity states that:

a) A sample of reasonably small size when selected at random, is almost not sure to represent the characteristics of the population

b) A sample of reasonably large size when selected, is almost not sure to represent the characteristics of the population.

c) A sample of reasonably large size when selected at random, is almost sure to represent the characteristics of the population, on an average


20. Law of Inertia is also known as:

a) Law of Statistics.

b) Law of Large Number

c) Law of Balance



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21. Law of Inertia states that:

a) Sample of high size show a high degree of stability.

b) Sample of low size shows a high degree of stability.

c) Results obtained from sample of high size are expected to be very far.


22. Increase in reliability and accuracy of results from a sampling study with the increase in sample size is known as the principle of:

a) statistical regularity

b) optimization.

c) law of increasing returns.

d) inertia of large numbers.

23. Sampling error increases with an increase in the size of the sample.

a) The above statement is true.

b) The above statement is not true.

c) Sampling error do not depends upon the sample size


24. Two basic Statistical laws concerning a population are

a) the law of statistical irregularity and the law of inertia of large numbers.

b) the law of statistical regularity and the law of inertia of large numbers.

c) the law of statistical regularity and the law of inertia of small numbers.

d) the law of statistical irregularity and the law of inertia of small numbers.

Errors in Sample survey:

25. How many different kind of errors can one find in sampling process?

a) One

b) Two

c) Three

d) Many

26. A sample survey is prone to:

a) Non-sampling errors

b) Sampling errors

c) Either a) or b)

d) Both a) and b)


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27. Bias is also known as:

a) Sampling Error

b) Non-Sampling Error

c) Error


28. Sampling error are:

a) Particularly detectfull

b) Can be corrected

c) Arise because the information collected relates only to a part of the population.

d) All of the above.

29. Sample Error is completely absent in:

a) Complete Enumeration

b) Census



30. __________Can occur in census.

a) Standard Error

b) Sampling Error

c) Bias


31. Non- Sampling Errors include :

a) bias

b) mistakes

c) both bias and mistake

d) none of these

32. ________Errors are likely to be more in case of complete enumeration:

a) Sampling errors

b) Probability errors

c) Non sampling errors



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33. “Non - sampling errors are present both in census as well as a sample survey.’’ - State whether the given statement is correct or not.

a) Correct

b) Incorrect

c) Nothing cannot be said


34. There are more chances of non-sampling errors than sampling errors in case of :

a) studies of large samples.

b) complete enumeration.

c) inefficient investigators.

d) all of the above

Concept of Sampling Distribution of Statistic and Standard Error:

� Samples can be drawn with or without replacement � Probability distribution of a statistic is called sampling of statistic. Example:

sampling distribution of ( x )., sampling distribution of (p) � Standard deviation of the sampling distribution of the sampling is called Standard

Error of statistic � As sample size increases standard error decreases proportionately. � Precision of the sample is reciprocal to standard Errors.. � Standard Error measures sampling fluctuations. i.e fluctuations in the value of

statistics due to sampling

Related MCQ’s

35. Concept of Sampling Distribution is offen talked about in context of.

a) Statistical

b) Quantitative Analysis

c) Sampling Analysis

d) None of these

36. Values of a particular statistic with their relative frequencies will constitute

a) Probability Distribution

b) Sampling Distribution

c) Theoretical Distribution

d) None of these


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37. We can have sampling distribution of:

a) mean only.

b) standard deviation

c) both mean and standard deviation

d) any statistical measure

38. In general, mean of the sampling distribution is _______ as the mean of the population.

a) more than

b) less than

c) same


39. The standard deviation of sampling distribution is commonly known as:

a) probability error

b) human bias

c) simple error

d) standard error

40. The population standard deviation describes the variation among elements of the

universe, whereas, the standard error measures the:

a) variability in a statistic due to universe

b) variabillity in a statistic due to sampling

c) variablity in a parameter due to universe

d) variablity in a statistic due to parameter

41. As the units selected in two or more samples drawn from a population are not the same,

the value of a_______ varies from sample to sample, but the _________always remains

constant.

a) mean, standard deviation

b) statistic, standard deviation

c) statistic, parameter

d) parameter, statistic


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42. The greater the value of standard error implies:

a) More the departure of actual frequencies from the expected ones.

b) More unreliability of the sample.



43. Standard error can be described as:

a) The error committed in sample survey

b) The error committed in estimating a parameter

c) Standard deviation of a statistic

d) The error committed in sampling.

44. The reciprocal of the standard error is:

a) Precision of the sample

b) Error of the sample

c) Error of the Universe


45. Precision of random sample:

a) increases directly with increase in sample size

b) increases with the increase in sample size

c) increases proportionately with sample size

d) none of these.

46. The standard error of the_______ is the standard deviation of sample means.

a) Population

b) Sample

c) Mean

d) Median

47. Sampling Fluctuations may be described as :

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.


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Theory of estimation, confidence limits and determination of sample size.

Theory of Estimation

1. It is a process whereby, we estimate the Parameter from the given value of a Statistic.

For Example, we estimate Population Mean µµµµ from Sample Mean x , Population S.D σσσσ

from sample S.D s, Population proportion P from Sample proportion p

.2 For the purpose of Estimation, parameter denoted by ‘��’ and statistic by ‘t’

For example: x=t and µµµµ==== θθθθ

PROPERTIES OF A GOODS ESTIMATOR

1. UNBIASEDNESS

A statistic ‘t’ is said to be an unbiased estimator of parameter θθθθ if E(t) = θθθθ

i.e. Mean of Statistic = Parameter.

a) If E(t) -θ = 0 or If E(t)= θθθθ

Then, ‘t’ is an unbiased estimator of θθθθ

b) The magnitude of biasness is: { E(t) - θθθθ}

c) E( x )= µ µ µ µ ((((Hence ( x ) is an Unbiased Estimator of µµµµ))))

d) E(p)=P (Hence, p is an unbiased estimator of the Parameter P)

e) But, E(s2)≠ �� (i.e mean of sample variance is not population variance)

(hence s2 is a biased estimator of the population variance 2σ .)

2. Consistency :

Accuracy should increase with increase in sample size. That is, as n will increase, it

should be closer to the parameter. Symbolically as n increases,

i) E(statistic) ͢͢ ͢͢→Parameter and

ii) V(statistic) →0

3. Efficiency and Minimum Variance:

If there are two consistent estimators, among them we should choose the estimator with minimum variance, that is, the statistic with smaller sample variance.

If t and t’ are two consistent estimators and var.(t) < var. (t’) we shall choose t as the estimator as it is more efficient in choosing θθθθ.


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4. Sufficiency : It is said to be a sufficient estimator of a parameter θθθθ, if it contains all information about

the parameter θθθθ. NOTE 1:

1. Minimum Variance Unbiased Estimator (MVUE)

(Minimum variance along with 0 bias)

A statistic, which is unbiased and has also minimum variance, that is most efficient, is

said to be MVUE.

NOTE 2:

Best Asymptotically Normal Estimator or Best Linear Unbiased Estimator (BLUE)

If the estimator is unbiased along with minimum variance and at the same time it is a linear

function of θθθθ, that is,

t = f(θθθθ), then it is called Best Linear unbiased estimator (BLUE)

• In interval estimation the confidence limits are the upper and lower limits of the

interval estimate.

Concept of level of significance ∝∝∝∝ : Level of significance is defined as the

significance of error in evaluating the confidence limits. Thus, Level of

significance, the Compliment of Level of Confidence.

Example 1: 5% level of significance =95% level of confidence

Example 2: 1% level of significance =99% level of confidence

Symbolically we expressed the fact that a parameter � lies between two values C1

and C2 is 1- ∝∝∝∝

1- That is , P(C1 < θ < C2 )=1-∝∝∝∝

where “∝∝∝∝” is called the “ level of significance” and 1- ∝∝∝∝ is called the “level of

confidence”.

• Sample size is determined from C.L. (Confidence Limits) of mean and proportions

Related MCQ’s:

48. The limits within which the parameter values are expected to lie can be determined by

using which of the following concepts:

a) Sampling Distribution

b) Probability Distribution

c) Standard Deviation

d) Standard Error.


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49. Within ± 3SE, we have % of the area under normal curve.

a) 99.73

b) 99

c) 98

d) 100

50. Give the formula for computing the standard error of mean in case of population is finite. (With the symbols having the usual meaning)

a) n

σ

b) 1N

N nn

σ −

−

c) 1

N n

Nn

σ −

−


51. A single number that is used to estimate an unknown population parameter is known as:

a) Interval Estimate

b) Point Estimate

c) Estimate

d) Statistics

52. ________is a range of values used in making estimation of a population parameter.

a) Interval Estimate

b) Parameter

c) Statistics

d) Point Estimate

53. The 95% confidence interval for the sample mean is given by: (symbols having the usual meaning)

a) 1.96n

σµ ±

b) 1.675n

σµ ±

c) 0.96xn

σ±

d) 1.96xn

σ±


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54. Which of the following(s) are the criterion for an ideal estimator:

I. Unbiased ness and Minimum Variance

II. Sufficiency

III. Consistency

IV. Efficiency

a) I, II and III above

b) I, II, and IV above

c) All of the above

d) I and II only.

55. A statistic T is known to be minimum variance unbiased estimator (MVUE) of θ if:

a) T is unbiased for θ

b) T has the minimum variance among all the unbiased estimators of θ



56. Why unbiased ness is considered along with minimum variance?

a) Because for a parameter θ, there exists a good number of unbiased statistics.

b) Because for a parameter θ, there doesn’t exists a goods number of unbiased statistics.

c) Because for a parameter θ, there are less number of unbiased statistics.


57. A statistic T is known to be consistent estimator of the parameter θ if the difference between T and θ can be made smaller and smaller by taking ______the sample size n.

a) smaller and smaller

b) larger and larger

c) nothing can be said

d) it doesn’t depends on the sample size

58. A statistic T is known to be an efficient estimator of θ if T has the________ standard error among all the estimators of θ when the sample size is kept fixed.

a) maximum

b) worst

c) minimum



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59. If statistics T contains all the information about θ, then T is known to be a estimator of θ.

a) Efficient

b) Sufficient

c) Consistent

d) Systematic

60. For an unknown parameter, how many interval estimates exist?

a) Only one

b) Many

c) Two

d) Three

61. The most commonly used confidence interval is:

a) 90%

b) 93%

c) 95%

d) 99%

62. The sample standard deviation is:

a) An unbiased estimator

b) A biased estimator

c) A biased estimator for population variance

d) A biased estimator for population SD

63. If x1, x2 ...., xn is a simple random sample of size n from a finite population of N units with

mean xand variance σ2, then which of the following is true:

a) E ( x ) = µ2

b) Var ( x ) = σ

c) E ( x ) = µ

d) All of the above are incorrect


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64. The estimate which is used in making estimation of a population parameter is :

a) point estimation

b) interval estimation

c) both a) and b) above

d) none of these

65. The permissible sampling error required to determine sample size for

a) estimating a mean

b) estimating a proportion



66. The criteria for an ideal estimation are :

a) unbiasedness, consistency, efficiency and sufficiency

b) unbiasedness, expectation, sampling and estimation.

c) estimation consistency, sufficiency and efficiency

d) estimation, expectation, unbiasedness and sufficiency.

67. An unbiased estimator

a) has the smallest variance among all estimators.

b) is always the best estimator

c) has an expected value equal to the true parameter value

d) always generates the value of the parameter.

68. An efficient estimator

a) has a small variance.

b) gets closer to the true parameter value as the sample size increases


d) always generates the true value of the parameter.

69. When choosing an estimator of a population parameter, one should consider :

a) sufficiency

b) efficiency




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70. A consistent estimator

a) has the smallest variance among all estimators :

b) gets closer to the true parameter value as the sample size increases.


d) is always sufficient.

71. The standard deviation is required to determine sample size for :

a) estimating a mean

b) estimating a proportion

c) both of a) and b) above

d) none of these

72. The difference between sample S.D. and the estimate of population S.D. is negligible if the sample size is :

a) small

b) moderate

c) sufficiently large

d) none of these

73. The difference between the estimate from the sample and the parameter to be estimated is:

a) sampling error

b) permissible sampling error.

c) confidence level

d) none of these.

74. The interval bounded by upper and lower limits is known as :

a) estimate interval

b) confidence interval

c) point interval

d) none of these.

75. A sufficient statistic

a) is consistent

b) is unbiased

c) uses all information a sample contains about the parameter to be estimated

d) is always efficient.


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76. An interval estimate is :

a) a range of values used to estimate the population parameter.

b) a single value that is used estimate the population parameter.

c) always unbiased

d) always a sufficient statistic

77. Another name of best asymptotically normal estimator is :

a) minimum variance unbiased estimator

b) best linear unbiased estimator

c) consistent asymptotically normal efficient estimator.

d) none of these.

78. The values of a characteristics x of a population containing six units are given by 2, 6, 5, 1, 7, 3. Take all possible samples of size two and find the mean of the sample means.

a) 1

b) 4

c) 2

d) 3

79. Let the five numbers 3, 4, 5, 6, 7 constitute a universe. Select all samples of size three and compute the mean of the sample means. The value thus obtained is:

a) 3

b) 4

c) 5

d) 6

80. Which of the following(s) are the important sampling distribution often used in statistical analysis?

a) Chi-square distribution

b) F-distribution

c) t-distribution

d) All of the above

Determination of confidence Limits for small samples (n<30 and population S.D σ is unknown)

• In such a case the distribution follows T-distribution with (n-1) degrees of freedom an accordingly

i) SE( x )=1

s

n −and

ii) Instead of using “Z” values we shall be using t values from the table for (n-1) d.f. and at the desired level of confidence or significance whre n = sample size

• C.L()=1

sx

n±

−.�� ,� ��


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81. Which of the following distribution is used to test the significance if size of the sample is less than 30 and variance of the population is unknown?

a) Standard Normal Distribution

b) Normal Distribution

c) “t’’ distribution

d) Chi-Square distribution

THEORETICAL QUESTIONS ON TYPES OF SAMPLING

82. Which Sampling provides separate estimates for populations means for different

purposes and also an over all estimate? a) Multistage sampling b) Simple random Sampling c) Systematic sampling d) Stratified sampling

83. When we have an idea that the error might be involved, we use:

a) Point Estimate b) Interval Estimate c) Both (a)and (b) d) None of these

84. Which sampling adds flexibility to the sampling process?

a) Systematic sampling b) Multistage sampling c) Stratified sampling d) Simple random sampling

85. When every member in population has an equal chance of being selection ,then that

sampling is called_______________ a) Restrictive b) Purposive c) Subjective d) Non- restrictive

86. If every 9th unit is selected from universal set then this type of sampling is known as:

a) Quota sampling b) Systematic sampling c) Stratified sampling d) None of these

87. The sampling is said to be large sampling if the size of the sample is:

a) Greater than or equal to 30 b) Less than 30 c) Less than or equal to 35 d) Less than 25

88. The method of sampling in which each unit of the population has an equal chance of being selected in the sample is a) Random Sampling b) Stratified sampling c) Systematic sampling d) None of the above

89. A selection Procedure of a sample, having no involvement of probability is known as:

a) Purposive sampling b) Judgement sampling c) Subjective Sampling d) All of the above


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90. In order to test the quality of chalks, the best suitable method wil be__________ a) Complete enumeration b) Simple random sampling c) Systematic sampling d) Stratified sampling

91. In factory there are 48 employees with employee code from 1 to 48 the employer

desires to take the sample of every sixth employee under the systematic sampling technique the sample size will be: a) 6 b) 8 c) 10 d) 7

92. Which of the following is non-probability sampling method?

a) Systematic Sampling b) Quota Sampling c) Random Sampling d) Stratified Sampling

93. Method used to rest the human blood is called in Statistical terminology________

a) Census Investigation b) Blood Investigation c) Sample Investigation d) None of these

94. Area sampling is similar to___________

a) Quota Sampling b) Cluster Sampling c) Judgement Sampling d) None of these.

95. 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 96. Simple random sampling is very effective if (a) The population is not very large. (b) The population is not much heterogenous (c) The population is partitioned into several sections. (d) Both (a) and (b) 97. Simple random sampling is (a) A probability sampling (b) A non-probabilistic sampling (c) A mixed sampling (d) Both (a) and (b) 98. Which sampling is subjected to the Discretion of the sampler?

(a) Systematic sampling (b) Simple random sampling (c) Purposive sampling (d) Quota sampling

99. By using sampling methods we have

(a) the error estimation & less quality data. (b) less quality data & lower costs. (c) the error estimation & higher quality data. (d) higher quality data & higher costs.

100. Which would you prefer for ___________”The universe is large”

(a) Full enumeration (b) sampling (c) both (d) none.


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101. Which would you prefer for ___________”Where testing destroys the quality of the product”

(a) Full enumeration (b) sampling (c) both (d) none. 102. In cost accounting operation Statistical Sampling methods are used.

(a) true (b) false (c) both (d) none 103. In control of book keeping and clerical errors Statistical sampling methods are used. (a) true (b) false (c) both (d) none 104. Deliberate sampling is free from bias.

(a) True (b) false (c) both (d) none.

105. Which would you prefer___________when a higher degree of confidence is desired (a) Larger sample (b) Small sample (c) both (d) none

106. Which would you prefer ____________when previous experience reveals a low rate of

error. (a) Larger sample (b) Small sample (c) both (d) none 107. Cluster sampling is ideal in case the data are widely scattered. (a) True (b) false (c) both (d) none. 108. Stratified random sampling is appropriate when the universe isnot homogenous.


109. In Stratified sampling, the sampling is subdivided into several parts, called (a) Strata (b) Strati (c) Start (d) none 110. The ways of selecting a sample are .

(a) Random sampling (b) multi –stage sampling (c) both (d) none

111. _______________sampling is the most appropriate in cases when the population is

more or less homogenous with respect to the characteristics under study (a) Multi –stage (b) Stratified (c) Random (d) none

112. Random sampling is called lottery sampling



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Theory Answers

1 B 2 A 3 A 4 C 5 C 6 D 7 A 8 A

9 B 10 A 11 A 12 B 13 D 14 C 15 A 16 B

17 A 18 C 19 C 20 B 21 A 22 D 23 B 24 B

25 B 26 D 27 C 28 D 29 C 30 C 31 C 32 C

33 A 34 D 35 C 36 B 37 D 38 C 39 D 40 B

41 A 42 B 43 C 44 A 45 C 46 C 47 A 48 D

49 A 50 C 51 B 52 A 53 D 54 C 55 C 56 A

57 B 58 C 59 B 60 B 61 C 62 D 63 C 64 C

65 C 66 A 67 C 68 A 69 C 70 B 71 A 72 C

73 B 74 B 75 C 76 A 77 B 78 B 79 C 80 D

81 C 82 D 83 B 84 D 85 B 86 B 87 A 88 A

89 D 90 B 91 B 92 B 93 B 94 C 95 B 96 B

97 A 98 C 99 C 100 B 101 B 102 A 103 A 104 B

105 A 106 B 107 A 108 A 109 A 110 C 111 C 112 A

12. SAMPLING THEORY AND THEORY OF ESTIMATION · 3. Sampling & Sampling Theory Sampling : It is the...

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Transcript of 12. SAMPLING THEORY AND THEORY OF ESTIMATION · 3. Sampling & Sampling Theory Sampling : It is the...