Sampling and
Sampling Distributions
A statistical population is the aggregate of all the units
pertaining to a study.
i.e. it is the set of all elements about which we wish to
make inferences.
A sample is a subset of a population.
The process of drawing a sample from a large
population is called sampling.
STATISTIC: Characteristic or measure obtained from a
sample.
PARAMETER: Characteristic or measure obtained
from a population.
A sampling distribution is the probability distribution,
under repeated sampling of the population, of a given
statistic.
 Consider a very large population.
 Assume we repeatedly take samples
of a given size from the population
and calculate the sample mean for
each sample.
 Different samples will lead to
different sample means.
 The distribution of these means is
the “sampling distribution of the
sample mean”.
When all of the possible sample means are
computed, then the following properties are true:
The mean of the sample means will be the mean
of the population (μ).
The variance of the sample means will be the
variance of the population divided by the sample
size (σ2/n).
The standard deviation of the distribution of a sample
statistic is known as the standard error of the statistic.
The nature of the sampling distribution depends on
the distribution of the population and/or the
statistic being considered and the sample size
used.
A population comprises of four numbers:
3, 5, 7 and 9
(a) List all possible samples of size 2 that
can be drawn from the population without
replacement.
(b) Show that the mean of the sampling
distribution of sample means is equal to the
population mean.
(c) Calculate the standard deviation of the
sampling distribution of sample means and
hence, show that it is less than the
population standard deviation.
Testing of Hypothesis
Hypothesis is an assumption about a population
A few examples are as follows:
1. Mean purchases made by females (μ1) is more than
or equal to the mean purchases made by males (μ2)
in a textile stores (μ1 > μ2).
2. Mean age of female shoppers (μ1) is less than or
equal to that of male shoppers (μ2) in a book
exhibition (μ1 < μ2).
3. Mean monthly income of buyers (μ) in a shop is
more than or equal to Rs 10000\- (μ > 10000).
4. The mean stay-over time of customers (μ) in a shop
is at most 45 minutes (μ < 45).
Definitions
Parameter: It is a function of population values.
Statistic: It is a function of sample values.
Null Hypothesis: It is an assumption about the
population parameter which the statement of no
change. It is denoted by H0.
Alternate Hypothesis: It is the statement of
assumption which can be considered to be the
alternative to the null hypothesis is called the
alternative hypothesis. It is denoted by H1.
As long as there is no apparent contradiction to
the null hypothesis, we retain this belief. But,
when we find observations contradicting it, there
is a reason to suspect the validity of this null
hypothesis and the problem of testing the null
hypothesis arises.
When we proceed to test H0, we must be aware
of the assumption that is expected to be valid if
null hypothesis turns out to be valid if null
hypothesis turns out to be invalid. This
assumption is known as alternative hypothesis.
H0: The mean I.Q. of all persons in a city is 105
 H1: The mean I.Q. of all persons in the city is 100
(if it is known that the mean I.Q. is 105 or 100 and
nothing else)
OR
 H1: The mean I.Q. of all the persons in the city is less
than 105
(if it is known that the mean I.Q. is not more than 105)
OR
 H1: The mean I.Q. of all the persons in the city is more
than 105
(if it is known that the mean I.Q. is not less than 105)
OR
 H1: The mean I.Q. of all the persons is not equal to 105
(if any information is absent)
The first thing to do when given a claim is to
write the claim mathematically (if possible), and
decide whether the given claim is the null or
alternative hypothesis.
If the given claim contains equality, or a
statement of no change from the given or
accepted condition, then it is the null hypothesis,
otherwise, if it represents change, it is the
alternative hypothesis.
Example
"He's dead,” said Dr. X to Captain K.
Mr. S, as the science officer, is put in charge of
statistically determining the correctness of Xs'
statement and deciding the fate of the crew member
(to vaporize or try to revive)
His first step is to arrive at the hypothesis to be
tested.
Does the statement represent a change in previous
condition?
Yes, there is change, thus it is the alternative
hypothesis, H1
No, there is no change, therefore is the null
hypothesis, H0
The correct answer is that there is change.
Dead represents a change from the accepted
state of alive.
The null hypothesis always represents no
change.
Therefore, the hypotheses are:
 H0: Patient is alive.
 H1: Patient is not alive (dead).
PROCEDURE IN HYPOTHESIS TESTING
1.Formulate the Hypothesis: Set up a null hypothesis based
on the belief and an appropriate alternate hypothesis.
2. Set up a Suitable Significance Level: The confidence with
which a null hypothesis is rejected or accepted depends upon
the significance level used for the purpose.
A level of significance say 5% means the risk of making a
wrong decision is only in 5 out of 100 cases. Level of
significance widely used is 5% or 1%. Thus, a 1% level of
significance provides greater confidence to the decision than a
5% significance level as the risk of making wrong decision is
only in 1 out of 100 cases. It is denoted by a Greek alphabet
alpha (α). Where (1 – α) is the CONFIDENCE LEVEL.
3. Select Test Criterion: The test criterion is selected
on the basis of sample size. If the sample is large (n 
30), the z-test implying normal distribution is used;
whereas if the sample size is small (n < 30), the t-test
is more suitable. The most commonly used tests are z,
t, F and χ2.
A corresponding TEST STATISTIC is calculated.
4. Decision Criterion: The Test Statistic calculated in
the previous step is now classified to fall within the
acceptance region or the rejection region at the given
level of significance. Accordingly the null hypothesis
is accepted or rejected.
5. Conclusion: On the basis of the decision the
conclusion is stated.
ERRORS IN DECISION MAKING
The problem of testing of a hypothesis is
actually a problem of deciding whether to
accept or to reject the null hypothesis H0, in
favor of alternate hypothesis H1.
The decision of rejecting or accepting of the
null hypothesis is taken on the basis of
observations made only on a sample of units
selected from the population. This decision
cannot be always correct. When this decision
is not correct, an error is said to occur.
States of nature are something that you, as a
decision maker has no control over.
Either it is, or it isn't. This represents the true
nature of things.
Possible states of nature (Based on H0)
 Crew member is alive (H0 true /H1 false )
 Crew member is dead (H0 false / H1 true)
Decisions are something that you have control
over.
You may make a correct decision or an incorrect
decision.
It depends on the state of nature as to whether
your decision is correct or incorrect.
Possible decisions (Based on H0) / conclusions
(Based on claim)
 Reject H0 if sufficient evidence to say patient
is dead, is available
 Fail to Reject H0 if sufficient evidence to
say patient is dead, is not available
Statistically speaking
State of Nature
Decision
Reject H0
Fail to
reject H0
H0 True
Patient is alive,
Sufficient evidence
of death
Patient is alive,
Insufficient evidence
of death
H0 False
Patient is dead,
Sufficient evidence
of death
Patient is dead,
Insufficient evidence
of death
State of Nature
Decision
Crew member alive Crew member dead
Vaporize the Error
Right decision
crew member
Try to revive Right decision
crew member
Error
Following table gives the
possibilities that exist in reality.
Null Hypothesis H0 is
True
Decision
Not True
Reject H0
Type I Error
No Error
Do not reject H0
No Error
Type II Error
Type I Error
Reject H0, when H0 is True
Type II Error
Do Not Reject H0, when H0 is Not True
Which of the two errors is more serious?
Type I or Type II?
Level of significance
To design a good test we would like to arrive at a
decision criterion in such a way that none of the two
errors, (Type I Error and Type II Error) occur.
But when P(Type I Error) → 0, P(Type II Error) → 1
& when P(Type II Error) → 0, P(Type I Error) → 1
Hence, no test can be perfect. We therefore design a
test such that one of the two probabilities is restricted
to a small value α (0 < α < 1 and α is closer to 0) and
then minimize probability of the other error.
The error in rejecting H0, when it is true (Type I
Error) is more serious error than (Type II Error),
therefore an upper limit is put on P(Type I Error)
and P(Type II Error) is simultaneously
minimized. This upper limit is known as level of
significance.
Thus, a test is so designed that
P(Type I Error) < α
then α is called level of significance
Hence, α = Max. P(Type I Error).
DECISION CRITERION
In p-value of the test
statistic is less than the
level of significance α,
reject H0.
Distributions used in
testing of hypothesis
In order to test different parameters, for
different sample sizes and comparisons of
such parameters for multiple populations,
different statistical distributions are used.
Testing of Hypotheses
Testing mean
Testing variance
Testing mean
Two
samples
Single
sample
Sample
size ≥30
Z test
Sample
size <30
t test
Sample
size ≥30
Z test
More than
two samples
Sample
size <30
t test
ANOVA
One way
Assuming
unequal
variances
Two way
without
replication
Assuming
equal
variances
Two way with
replication
Paired t test
Testing
variance
Single sample
Two samples
Chi Square test
F test
For testing association between two variables
Chi-Square test for Independence of
Attributes is used.
Expected frequencies are calculated using the
following formula:
RT  CT
E=
N
O= Observed frequencies
For fitting a distribution to a given data
Chi-Square test for Goodness of Fit is used
Expected
frequencies
are
depending upon the distribution.
calculated
Thank You