Concept of Sampling Distribution

7/27/2019 Concept of Sampling Distribution

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Concept of Sampling Distribution


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BASIC DEFINITIONS

1. Universe or Population: An aggregate of

items about which we obtain information.

It can be finite e.g. number of students in a

college etc.

It can be infinite e.g. number of hair on

the head.

2. Sample: A part of population is called as a

sample.


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There are two methods to collect

statistical data:

1. CENSUS METHOD: Data is collected from

each and every unit of the population under

investigation i.e. Complete Enumeration is

done.

2. SAMPLING METHOD: Data is collected from

the sample of items selected from

population.


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IMPORTANCE OF SAMPLING METHOD

1. Saving of time.

2. Saving of money.

3. Intensive study.4. Organizational Convenience.

5. More reliable results.


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DIFFERENCE BETWEEN CENSUS AND

SAMPLE METHOD

S No. CENSUS METHOD SAMPLING METHOD

1 SCOPE All items relating to

universe are

investigated.

Only few items are

inquired.

2 COST Expensive Economical

3 FIELD OF INVESTIGATON Suitable for limited field. Suitable for large field.

4 HOMOGENEITY Useful where units of

population are

heterogeneous

Useful where units of

population are

homogeneous.

5 TYPE OF UNIVERSE Each and every unit of

universe is necessary,

census method is more

appropriate.

When population is

infinite or vast this

method is more

appropriate.


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SAMPLING

METHODSPROBABILITY

SAMPLINGMETHODS

SimpleRandomsampling

Stratifiedrandom

sampling

Systematicrandom

sampling

Multistagerandom

sampling

Cluster sampling

NON PROBABILITYSAMPLINGMETHODS

Judgme

nt

Samplin

g

Quota

Sampling

Convenienc

esamplin

g

Extensiv

e

samplin

g


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SAMPLING ERRORS

Faulty selection of the sampling method.

Substituting one sample for the sample due

to the difficulties in collecting the sample.

Faulty demarcation of sampling units.


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NON SAMPLING ERRORS


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IMPORTANT TERMS

PARAMETERS STATISTICS1. DEFINITION: Any statistical measures

computed from the population data is

known as parameter.

1. DEFINITION: Any statistical measure

computed from sample data is known as

statistic.

2. Parameters are denoted by Greekletters

2. Statistics are denoted by Roman letters

Population mean Sample mean X

Population

standard deviation2 Sample standard

deviations

Population Variance Sample Variance s2

Population

proportionP Sample

proportionp


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IMPORTANT TERMS

SAMPLING WITH

REPLACEMENT

SAMPLING WITHOUT

REPLACEMENT

Sampling where each unit

of population may be

chosen more than once iscalled sampling with

replacement.

And if each unit can not

chosen more than once is

called sampling withoutreplacement.

In this case, total number of

possible samples each of

size n is drawn from a

population of size N is Nn.

In this case, total number of

possible samples will be NCn

= m (say)


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An important property of sampling

distribution

Random samples of large size (n > 30) are

taken from a population which may or may

not be normally distributed or not, then the

sampling distribution of the statistic will

approach a normal distribution.


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Standard error of a statistic

The standard deviation of the sampling distribution ofa statistic is known as the standard error of thestatistic.

In sampling distribution instead of standard deviation formeasuring variation, we use the term standard error ofmean.

The standard error of mean measure the extent to whichthe sample mean differ from the population mean.

Like the standard error of mean, we could have standarderror of median, standard deviation, proportion,variance etc.


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UTILITY OF STANDARD ERROR

1. Reliability of a Sample: Standard error isinversely proportional to reliability of a sample.

2. Tests of significance: In large sample ( n > 30), ifthe difference between observed and expected

value is greater than 1.96 Standard error., thenwe reject the hypothesis at 5% and concludethat sample differs widely from the population.But if the difference between the observed andthe expected value is greater than 2.58 S.E, thenwe reject the null hypothesis at 1% andconclude that the sample differs widely from thepopulation.


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UTILITY OF STANDARD ERROR

3. To determine the confidence limits of the unknown

population mean: The standard error enables us in

determining the confidence limits within which a

population parameter is expected to lie with a

certain degree of confidence. The confidence limitsof population mean are given by:

LARGE SAMPLE SMALL SAMPLE

95% confidence limits for 95% confidence limits for

x 1.96 S.E and x + 1.96 S.E X t.05 S.E and x + t.05 S.E

99% confidence limits for 99% confidence limits for

X 2.58 S.E and x + 2.58 S.E X t.01 S.E and x + t.01 S.E


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Sampling distribution of means

PROPERTIES:

1. The mean of the sampling distribution of means is equal to thepopulation mean ().

i.e. x =

2. The standard error of the sampling distribution of means is:

S.E x = /n (sampling is with replacement)

S.E = /n *(N-n)/(N-1)]

3. To find the probability of the sampling distribution of means:Z = X/ S.E x


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Q1. Consider a population consisting of three values: 2, 5 and 8.

Draw all possible sample of size 2 with replacement from the

population. Construct sampling distribution of means. Also find

the mean and standard error of the distribution.

Sample No Sample Values Sample mean

Solution:


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Mean = fx/ f

Sample

means (x)

f fx d = x - d2 fd2

Variance = fd2/ f

S.E =


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Q2. Suppose a population consist of values 1,2,3,4, and 5. Take

all possible sample of size 2 (without replacement) and construct

a sampling distribution of mean. Show that mean of sampling

distribution of mean is equal to the population mean.


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ESTIMATES


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POINT ESTIMATES


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INTERVAL ESTIMATES

Concept of Sampling Distribution

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