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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
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