10.
SAMPLING AND
LARGE SAMPLE TESTS
10.1
INTRODUCTION:
Before giving
the notion of sampling we will first define population. In a statistical
investigation the interest usually lies in the assessment of the general
magnitude and the study of variation with respect to one or more
characteristics relating to individuals belonging to a group. This group of
individuals under study is called population or universe, thus in statistics,
population is an aggregate of objects, animate or inanimate, under study. The
population may be finite or infinite.
It is obvious
that for any statistical investigation complete enumeration of the population
is rather impracticable. For example, if we want to have an idea of the average
per capita income of the people India, we will have to enumerate all the
earning individuals in the country, which is rather a very difficult task.
If the
population is infinite, complete enumeration is not possible. Also if the units
are destroyed in the course of inspection 100% inspection, through possible, is
not at all desirable. But even if the population is finite or the inspection is
not destructive, 100% inspection is not taken recourse to because of
multiplicity of causes, viz, administrative and financial implications, time
factor, etc., and we take the help of sampling.
A finite subset
of statistical individuals in a population is called a sample and the number of
individuals in a sample is called the sample size.
For the purpose
of determining population characteristics, instead of enumerating the entire
population, the individuals in the sample only are observed. Then the sample
characteristics are utilized to approximately determine or estimate the
population. For example, on examining the sample of a particular stuff we
arrive at a decision of purchasing or rejecting that stuff. The error involved
in such approximation is known as sampling error and is inherent and
unavoidable in any and every sampling scheme. But sampling results in
considerable gains, especially in time and cost not only in respect of making
observations of characteristics but also in the subsequent handling of the
data.
Sampling is
quite often used in our day to day practical life. For example, in a shop
we assess the quality of sugar, wheat or any other commodity by taking a
handful of it from the bag and then decide to purchase it or not. A housewife
normally tests the cooked products to find if they are properly cooked and
contain the proper quantity of salt.
10.2 TYPES OF SAMPLING:
Some of the
commonly known and frequently used types of sampling are:
(i)
Purposive
sampling
(ii)
Random
sampling
(iii)
Stratified
sampling
(iv)
Systematic
sampling
Below we will
precisely these terms, without entering into detailed discussion.
10.2.1
Purposive
sampling:
Purposive
sampling is one in which the sample units are selected with definite purpose in
view. For example, if we want to give the picture that the standard of living
has increased in the city of New Delhi, we may take individuals in the sample
from rich and posh localities like Defence Colony, South Extension, Golf Links,
Jor Bagh, Chanakyapuri, Greater Kailash etc. and ignore the localities where
low income group and the middle class families live. This sampling suffers from
the drawback of favouritism and nepotism and does not give a representative sample
of the population.
10.2.2
Random sampling:
In this case the
sample units are selected at random and the drawback of purposive sampling,
viz., favouritism or subjective element, is completely overcome. A random
sample is one in which each unit of population has an equal chance of being
included in it.
Suppose we take
a sample of size n from a finite population of size N. then there are NCn
possible samples. A sampling technique in which each of the NCn
samples has an equal chance of being selected is known as random sampling and
the sample obtained by this technique is termed as a random sample.
Proper care has
to be taken to ensure that the selected sample is random. Human bias, which
varies from individual to individual, is inherent in any sampling scheme administered
by human beings. Fairly good random samples can be obtained by the use of
Tippets random number tables or by throwing or a dice, draw of a lottery, etc.
The simplest
method, which is normally used, is the lottery system which is illustrated
below by means of an example.
Suppose we want
to select r candidates out of n. we assign the numbers one to n, one number
to each candidate and write these numbers on n slips which are made as
homogeneous as possible in shape, size, etc. these slips are then put in a bag
and thoroughly shuffled and then r slips are drawn one by one. The r
candidates corresponding to the numbers on the slips drawn will constitute the
random sample.
10.2.3
Simple sampling:
Simple sampling
is random sampling in which each unit of the population has an equal chance,
say p, of being included in the sample and that this probability is independent
of the previous drawings. Thus a simple sample of size n from a population may
be identified with a series of n independent trials with constant probability
p of success for each trial.
10.2.4
Stratified
sampling:
Here the entire
heterogeneous population is divided into a number of homogeneous groups,
usually termed as strata, which differ from one another but each of these
groups is homogenous within itself. Then units are sampled at random from each
of this stratum, the stratum in the population. The sample, which is the
aggregate of the sampled units of each of the stratum, is termed as stratified
sample and the technique of drawing this sample is known as stratified
sampling. Such a sample is by far the best and can safely be considered as
representative of the population from which it has been drawn.
10.3 PARAMETER AND STATISTIC:
In order to
avoid verbal confusion with the statistical constants of the population, viz.,
mean (μ), variance σ2, etc., which are usually referred to
as parameters, statistical measures computed from the sample observations
alone, eg., mean (x bar), variance s2, etc., have been termed by
Professor R.A Fisher as statistics.
In practice,
parameter values are not known and the estimates based on the sample values are
generally used. Thus statistic which may be regarded as an estimate of
parameter, obtained from the sample, is a function of the sample values only.
It may be pointed out that a statistic, as it is based on sample values and as
there are multiple choices of the samples that can be drawn from a population,
varies from sample to sample. The determination or the characterization of the
variation that may be attributed to chance or fluctuations or sampling is one
of the fundamental problems of the sampling theory.
10.3.1
Sampling
distribution of a statistic:
If we draw a
sample of size n from a given finite population of size N, then the total
number of possible samples is:
For each of
these k samples we can compute some statistic t = t(x1, x2,
xn), in particular the mean 
, the variance s2, etc., as given below:
|
Sample
number |
t |
|
s2 |
|
1 |
t1 |
|
s12 |
|
2 |
t2 |
|
s22 |
|
3 |
t3 |
|
S32 |
|
. |
. |
. |
. |
|
. |
. |
. |
. |
|
k |
tk |
|
sk2 |
The set of the
values of the statistic so obtained, one for each sample, constitutes what is
called the sampling distribution of the statistic, for example, the values t1,
t2,
tk and we can compute the various statistical
constants like mean, variance, skewness, kurtosis etc., for its distribution.
For example, the mean and variance of the sampling distribution of the
statistic t are given by:
10.4 TESTS OF SIGNIFICANCE:
A very important
aspect of the sampling theory is the study of the tests of significance, which
enable us to decide on the basis of the sample results, if
(i)
The
deviation between the observed sample statistic and the hypothetical parameter
value, or
(ii)
The
deviation between two independent sample statistics:
Is significant
or might be attributed to chance or the fluctuations of sampling is significant
or might be attributed to chance or the fluctuations of sampling.
Since, for large
n, almost all the distributions, eg., Binomial, Poisson, Negative binomial,
Hyper geometric, t, F, chi square can be approximated very closely by a normal
probability curve, we use the normal test of significance for large samples.
Some of the well known tests of significance for studying such differences for
small samples are t-test, F-test and Fishers z transformation.
10.5 NULL HYPOTHESIS:
The technique of
randomization used for the selection of sample units makes the test of
significance valid for us, for applying the test of significance we first set
up a hypothesis a definite statement about the population parameter. Such a
hypothesis, which is usually a hypothesis of no difference is called null
hypothesis and is usually denoted by H0. According to Prof.
R.A.Fisher, null hypothesis is the hypothesis which is tested for possible
rejection under the assumption that it is true.
For example, in
case of a single statistic, H0 will be that the sample statistic
does not differ significantly from the hypothetical parameter values and in the
case of two statistics, H0 will be that the sample statistics do not
differ significantly.
Having set up
the null hypothesis we compute the probability p that the deviation between the
observed sample statistic and the hypothetical parameter value might have
occurred due to fluctuations of sampling. If the deviation comes out to be
significant, null hypothesis is rejected at the particular level of significance
adopted and if the deviation is not significant, null hypothesis may be
retained at that level.
10.5.1
Alternative
Hypothesis:
Any hypothesis
which is complementary to the null hypothesis is called an alternative
hypothesis, usually denoted by H1. For example, if we want to test
the null hypothesis that the population has a specified mean μ0,
that is H0: μ = μ0, then the alternative
hypothesis could be
(i)
H1:
μ ≠ μ0
(ii)
H1:
μ > μ0
(iii)
H1:
μ < μ0
The alternative
hypothesis in (i) is known as a two tailed alternative and the alternative in
(ii) and (iii) are known as right tailed and left-tailed alternatives
respectively. The setting of alternative hypothesis is very important since it
enables us to decide whether we have to use as single-tailed or two tailed
test.
10.6 ERRORS IN SAMPLING:
The main objective in sampling theory is to
draw valid inferences about the population parameters on the basis of the
sample results. In practice we decide to accept or reject the lot after
examining a sample from it. As such we are liable to commit the following two
types of errors:
Type I Error:
Reject H0 when it is true.
Type II Error:
Accept H0 when it is wrong, that is accept H0 when H1
is true.
If we write.
P { Reject H0
when it is true} = P{ Reject H0| H0} =α
and P { Accept H0
when it is wrong} = P{ accept H0| H1} =β
then α and
β are called the sizes of type I error and type II error, respectively.
In practice,
type I error amounts to rejecting a lot when it is good and type II error may
be regarded as accepting the lot when it is bad.
Thus P { Reject
a lot when it is good} = α
and P { Accept a
lot when it is bad } = β
where α and
β are referred to as Producers risk and consumers risk respectively.
10.7 CIRITICAL REGION AND LEVEL OF SIGNIFICANCE:
A region in the
sample space S which amounts to rejection of H0 is termed as
critical region or region of rejection. If ω is the critical region and if
t = t(x1, x2,
xn) is the value of the
statistic based on a random sample of size n, then
P( t Є
ω| H0 ) = α,
P(t Є 
| H1 ) = β
Where , the complementary set of ω, is called the
acceptance region.
We have ω
=S and ω
=φ
The probability
α that a random value of the statistic t belongs to the critical region
is known as the level of significance. In other words level of significance is
the size of the type I error. The levels of significance usually employed in
testing of hypothesis are 5 % and 1%. The level of significance is always fixed
in advance before collecting the sample information.
10.7.1
One tailed and
two tailed tests:
In any test, the
critical region is represented by a portion of the area under the probability
curve of the sampling distribution of the test statistic.
A test of any
statistical hypothesis where the alternative hypothesis is one tailed is called
a one tailed test. For example, a test for testing the mean of a population
H0:
μ = μ0
Against the
alternative hypothesis:
H1:
μ > μ0 (right tailed) or H1: μ <
μ0 (left tailed)
Is a single
tailed test. In the right test (H1: μ > μ0),
the critical region lies entirely in the right tail of the sampling
distribution or , while for the left tail test (H1: μ
< μ0), the critical region is entirely in the left tail or
the distribution.
A test of
statistical hypothesis where the alternative hypothesis is two tailed such as:
H0:
μ = μ0, against the alternative hypothesis H1:
μ ≠ μ0 is known as two tailed test and in such a
case the critical region is given by the
portion of the area lying in both the tails of the probability curve of
the test statistic.
In a particular
problem, whether one tailed or two tailed test is to be applied depends
entirely on the nature of the alternative hypothesis. If the alternative
hypothesis is two tailed we apply two tailed test and if alternative hypothesis
is one tailed, we apply one tailed test.
10.7.2
Critical values
or significant values:
The value of
test statistic which separates the critical region and the acceptance region is
called the critical value or significant value. It depends upon:
(i)
The
level of significance used, and
(ii)
The
alternative hypothesis, whether it is two tailed or single tailed.
As has been
pointed out earlier, for large samples, the standardized variable corresponding
to the statistic t viz. :
------------------------(a)
Asymptotically
as n ΰ
∞. The value of Z given by (a) under the null hypothesis is known as test
statistic. The critical value of the test statistics at level of significance
α for a two- tailed test is given by Zα where Zα
is determined by the equation
P(|Z|> Zα)
= α ------------------(1)
That is Zα
is the value so that the total area of the critical region on both tails
is α. Since normal probability curve is a symmetrical curve, from (1), we
get
P(Z > Zα)
+ P(Z < -Zα) = α
θ P (Z > Zα)
+ P(Z >Zα) = α
θ 2P (Z > Zα)
= α
θ P( Z > Zα)
= α/2
That is the area
of each tail is α/2. Thus Zα is the value such that area
to the right of Zα is α/2 and to the left of - Zα
is α/2.
In case of
single tail alternative, the critical value Zα is determined so
that total area to the right of it is α and for left tailed test the total
area to the left of - Zα is α.
Thus the
significant or critical value of Z for a single tailed test (left or right) at
level of significance α is same as the critical value of Z for a two
tailed test at level of significance 2α.
10.7.3
Procedure for
testing of hypothesis:
We now summarise
below the various steps in testing of a statistical hypothesis in a systematic
manner.
1.
Null
hypothesis: set up the null hypothesis H0.
2.
Alternative
hypothesis: set up the alternative hypothesis H1. This will enable
us to decide whether we have to use a single tailed test or two tailed test.
3.
Level
of significance: choose the appropriate level of significance (α)
depending on the reliability of the estimates and permissible risk. This is to
be decided before sample is drawn, that is α is fixed in advance.
4.
Test
statistic: compute the test statistic
Under the null
hypothesis.
5.
Conclusion:
we compare z the computed value of z in step 4 with the significant value zα,
at the given level of significance, α.
If |Z| < zα,
that is if the calculated value of Z is less than zα we say it
is not significant. By this we mean that the difference t E(t) is just due
to fluctuations of sampling and the
sample data do not provide us sufficient evidence against the null hypothesis
which may therefore, be accepted.
If |Z| > zα,
that is if the computed value of test statistic is greater than the critical or
significant value, then we say that it is significant and the null hypothesis
is rejected at level of significance α that is with confidence coefficient
(1- α).
10.8 TEST OF SIGNIFICANCE FOR LARGE SAMPLES:
In this section
we will discuss the test of significance when samples are large. We have seen
that for large values of n, the number of trials, almost all the distributions,
eg., binomial, Poisson, Negative binomial, etc., are very closely approximated
by normal distribution. Thus in this case we apply the normal test, which is
based upon the following fundamental property of the normal probability curve.
If X ≈ n(
μ, σ2), then z =
Thus from the
normal probability tables, we have
P( -3 ≤ Z
≤ 3) = 0.9973 that is P(|Z|≤ 3 ) = 0.9973
θ P(|Z|> 3 ) =
1- P( |Z| ≤ 3) = 0.0027
That is in all
probability we should expect a standard normal variate to lie between ± 3.
Also from the
normal probability tables, we get
P ( -1.96
≤ Z ≤ 1.96 ) = 0.95 that is P(|Z| ≤ 1.96) = 0.95
θ P(|Z| > 1.96
) = 1 0.95 = 0.05
θ P(|Z| ≤
2.58 ) = 0.99
θ P(|Z| > 2.58
) = 0.01
Thus the
significant values of Z at 5% and 1% level of significance for a two tailed test are 1.96 and 2.58
respectively.
Thus the steps
to be used in the normal test are as follows:
(i)
Compute
the test statistic Z under H0.
(ii)
If
|Z| >3, H0 is always rejected.
(iii)
If
|Z| ≤3, we test its significance at certain level of significance,
usually at 5% and sometimes at 1% level of significance. Thus, for a two tailed
test if |Z| >1.96, H0 is rejected at 5% level of significance.
Similarly if |Z|
> 2.58, H0 is contradicted at 1% level of significance and if |Z|
≤ 2.58, H0 may be accepted at 1% level of significance.
From the normal
probability tables, we have:
P(Z >1.645) =
0.5 P (0 ≤Z ≤1.645)
= 0.5 0.45
= 0.05
P(Z> 2.33) =
0.5 - P(0 ≤Z ≤2.33)
= 0.5 0.49
= 0.01
Hence for a
single tail test we compare the computed value of |Z| with 1.645 and 2.33 and
accept or reject H0 accordingly.
10.9 SAMPLING OF ATTRIBUTES:
Here we shall
consider sampling from a population which is divided into two mutually
exclusive and collectively exhaustive classes one class possessing a particular
attribute, say A, and the other class not possessing that attribute, and then
note down the number of persons in the sample of sizen, possessing that
attribute. The presence of an attribute in sampled unit may be termed as
success and its absence as failure. In this case a sample of n observations is
identified with that of a series of n independent Bernoulli trials with
constant probability P of success for each trial. Then the probability of x
successes in n trials, as given by the binomial probability distribution is
10.9.1
Test
for single proportion:
If X is the
number of successes in n independent trials with constant probability P of
success for each trial
E(X) = nP and
V(X) = n PQ
Where Q = 1- P,
is the probability of failure.
It has been
proved that for large n, the binomial distribution tends to normal
distribution. Hence for large n, X≈ N(nP, nPQ) that is
And we apply the
normal test.
10.9.2
Test of
significance for difference of proportions:
Suppose we want
to compare two distinct populations with respect to the prevalence of a certain
attribute, say A, among their members. Let X1, X2 be the
number of persons possessing the given attributes respectively. Then sample
proportions are given by
and 
If P1
and P2 are the population proportions, then
E(p1)
= P1, E(p2) = P2
and 
Since for large
sample, p1 and p2 are asymptotically normally
distributed (p1- p2) is also normally distributed. Then
the standard variable corresponding to the difference (p1- p2)
is given by
Thus the test
statistic in this case is
10.10
SAMPLING OF VARIABLES:
In the case of
sampling variables each member of the population provides the value of the
variable and the aggregate of these values forms the frequency distribution of
the population. From the population, a random sample of size n can be drawn by
any of the sampling methods discussed before which is same as choosing n values
of the given variables from the distribution.
10.11
UNBIASED
ESTIMATE FOR POPULATION MEAN (μ) AND VARIANCE (σ2):
Let x1,
x2,
xn be a random sample of size n from a large
population X1, X2,
. XN with mean μ
and variance σ2. Then the sample mean (
) and variance (s2) are given by
and
Now E () = 
Since xi
is a sample observation from the population Xi, (I = 1, 2,
N) it
can take any one of the values X1, X2,
. XN
each with equal probability 1/N.
Thus the sample
mean is an unbiased estimate of the population mean μ.
Now 
----------------------------(1)
We have V(xi)
= E[xi-E(xi)]2 = E(xi-μ]2
Also we know
that
In particular,
-----------------------(2)
,
where 
is the population variance.
-----------------------------(3)
Substituting
from (2) and (3) in (1) we get
-------------(4)
Since 
, sample variance is not an unbiased estimate of
population variance.
From (4), we get
Where 
Therefore S2
is an unbiased estimate of the population variance σ2.
10.12STANDARD ERROR OF SAMPLE MEAN:
The variance of
the sample mean is σ2/n, where σ is the population
standard deviation and n is the size of the random sample.
The standard
error of mean of a random sample of size n from a population with variance
σ2 is 
Proof:
Let x1,
x2,
xn be a random sample of size n from a population
with variance σ2, then the sample mean is given by
The covariance
terms vanish since the sample observations are independent
But 
Standard error 
10.12
TEST OF SIGNIFICANCE FOR SINGLE MEAN:
we have proved that if xi, (I = 1, 2,
, n ) is a random sample
of size n from a normal population with mean μ and variance σ2,
then the sample mean is distributed normally with mean σ2/n
that is 
. However, this result holds, that is , even in random sampling from non-normal population
provided the sample size n is large by central limit theorem.
Thus for large
samples, the standard normal variate corresponding to is:
Under the null
hypothesis, H0that the sample has been drawn from a population with
mean μ and variance σ2 that is there is no significant
difference between the sample mean 
and population mean (μ), the test statistic is:
10.14 TEST OF SIGNIFICANE FOR DIFFERENCE OF MEANS:
Let 
be the mean of a random sample of size n1
from a population with mean μ1 and variance σ12
and let 
be the mean of an independent random sample of size n2
from another population with mean μ2 and variance σ22.
Then, since sample sizes are large,
and 
Also 
, being the difference of two independent normal
variates is also a normal variate. The Z standard normal variable corresponding
to is given by
Under the null
hypothesis H0: μ1 = μ2, that is
there is no significant difference between the sample means, we get
The covariance term
vanishes, since the sample mean 
and are independent.
Thus under H0:
μ1 = μ2, the test statistic becomes
10.15 TEST OF SIGNIFICANCE FOR THE DIFFERENCE OF
STANDARD DEVIATIONS:
If s1 and
s2 are the standard deviations of two independent samples, then
under null hypothesis, H0: σ1 = σ2
that is the sample standard deviations dont differ significantly, the
statistic
for
large samples.
But in case of
large samples, the standard error(S.E.) of the difference of the sample
standard deviations is given by
σ12
and σ22 are usually unknown and for large samples,
we use their estimates given by the corresponding sample variances. Hence the
test statistic reduces to
