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Sampling Theory The procedure for drawing a random sample a distribution is that numbers 1, 2, are a

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Title: Sampling Theory The procedure for drawing a random sample a distribution is that numbers 1, 2, are a


1
Sampling TheoryThe procedure for drawing a
random sample a distribution is that numbers 1,
2, are assigned to the elements of the
distribution and tables of random numbers are
then used to decide which elements are included
in the sample. If the same element can not be
selected more than once, we say that the sample
is drawn without replacement otherwise, the
sample is said to be drawn with replacement.The
usual convention in sampling is that lower case
letters are used to designate the sample
characteristics, with capital letters being used
for the parent population. Thus if the sample
size is n, its elements are designated, x1, x2,
, xn, its mean is x and its modified variance is
s2 å (xi - x )2 / (n - 1).The corresponding
parent population characteristics are N (or
infinity), X and S2.Suppose that we repeatedly
draw random samples of size n (with replacement)
from a distribution with mean m and variance s2.
Let x1, x2, be the collection of sample
averages and let xi xi - m (i 1, 2,
) s / Ö n The collection x1, x2, is
called the sampling distribution of
means.Central Limit Theorem. In the limit, as
n tends to infinity, the sampling distribution of
means has a standard normal distribution.
2
Attribute and Proportionate SamplingIf the
sample elements are a measurement of some
characteristic, we are said to have attribute
sampling. On the other hand if all the sample
elements are 1 or 0 (success/failure,agree/
no-not-agree), we have proportionate sampling.
For proportionate sampling, the sample average x
and the sample proportion p are synonymous, just
as are the mean m and proportion P for the parent
population. From our results on the binomial
distribution, the sample variance is p (1 - p)
and the variance of the parent distribution is P
(1 - P).We can generalise the concept of the
sampling distribution of means to get the
sampling distribution of any statistic. We say
that a sample characteristic is an unbiased
estimator of the parent population
characteristic, is the mean of the corresponding
sampling distribution is equal to the parent
characteristic.Lemma. The sample average
(proportion ) is an unbiased estimator of the
parent average
(proportion) E x m E p P.The
quantity Ö ( N - n) / ( N - 1) is called the
finite population correction (fpc). If the parent
population is infinite or we have sampling with
replacement the fpc 1.Lemma. E s S fpc.
3
Confidence IntervalsFrom the statistical tables
for a standard normal distribution, we note
that Area Under From To Density
Function 0.90 -1.64 1.64 0.95 -1.96 1.96 0.99
-2.58 2.58 From the central limit theorem,
if x and s2 are the mean and variance of a random
sample of size n (with n greater than 25) drawn
from a large parent population, then we can make
the following statement about the unknown parent
mean m Prob -1.64 x - m 1.64)
0.90 s / Ö ni.e. Prob x - 1.64 s
/ Ö n m x 1.64 s / Ö n 0.90The
range x 1.64 s / Ö n is called a 90
confidence interval for the population mean m.
Example Attribute SamplingA random sample of
size 25 has x 15 and s 2. Then a 95
confidence interval for m is 15 1.96 (2 / 5)
(i.e.) 14.22 to 15.78Example Proportionate
SamplingA random sample of size n 1000 has p
0.40 Þ 1.96 Ö p (1 - p) / (n) 0.03.A 95
confidence interval for P is 0.40 0.03
(i.e.) 0.37 to 0.43.
n (0,1)
0.95
0
-1.96
1.96
4
Small Sampling TheoryFor reference purposes, it
is useful to regard the expression x 1.96 s
/ Ö nas the default formula for a confidence
interval and to modify it to suit particular
circumstances. O If we are dealing with
proportionate sampling, the sample proportion is
the sample mean and the standard error
(s.e.) term s / Ö n simplifies as follows
x -gt p and s / Ö n -gt Ö p(1 - p) / (n). O
A 90 confidence interval will bring about the
swap 1.96 -gt 1.64. O If the
sample size n is less than 25, the normal
distribution must be replaced by
Students t n - 1 distribution.
O For sampling without replacement from a
finite population, a fpc term must be
used.The width of the confidence interval
band increases with the confidence
level.Example. A random sample of size n 10,
drawn from a large parent population, has a mean
x 12 and a standard deviation s 2. Then a 99
confidence interval for the parent mean is x
3.25 s / Ö n (i.e.) 12 3.25 (2)/3.2 (i.e.)
9.83 to 14.17and a 95 confidence interval
for the parent mean is x 2.262 s / Ö n
(i.e.) 12 2.262 (2)/3.2 (i.e.) 10.492 to
13.508.Note that for n 1000, 1.96 Ö p (1 - p)
/ n 0.03 for values of p between 0.3 and 0.7.
This gives rise to the statement that public
opinion polls have an inherent error of 3.
This simplifies calculations in the case of
public opinion polls for large political
parties.
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