Download PPT - Ahmadu Bello University

BY
Mrs. J. O Odengle
Aliyu Garba
[email protected]
Institute of Computing & ICT
Ahmadu Bello University, Zaria
October, 2014
Important statistical terms
Population:
a set which includes all
measurements of interest
to the researcher
(The collection of all
responses, measurements,
counts that are of interest)
or
Sample:
A subset of the population
Hypothesis:
A hypothesis is a kind of truth claim about some
aspect of the world: for instance
Important statistical
terms(Cont…)
Hypothesis:
A hypothesis is a kind of truth claim about some
aspect of the world: for instance, the attitudes of
patients or the prevalence of a disease in a
population. Research sets out to try to prove this
truth claim (or, more properly, to reject the null
hypothesis - a truth claim phrased as a negative).
For example, let us think about the following
hypothesis:
Levels of ill-health are affected by deprivation.
and the related null hypothesis:
Levels of ill-health are not affected by
deprivation.
Why sampling?
Get information about large populations

Less costs

Less field time

More accuracy i.e. Can Do A Better Job of
Data Collection

When it’s impossible to study the whole
population
Target Population:
The population to be studied/ to which the
investigator wants to generalize his results
Sampling Unit:
smallest unit from which sample can be selected
Sampling frame
List of all the sampling units from which sample is
drawn
Sampling scheme
Method of selecting sampling units from sampling
frame
Types of sampling

Non-probability samples

Probability samples
Non probability samples


As they are not truly representative, nonprobability samples are less desirable than
probability samples. However, a researcher
may not be able to obtain a random or
stratified sample, or it may be too expensive.
A researcher may not care about generalizing
to a larger population. The validity of nonprobability samples can be increased by trying
to approximate random selection, and by
eliminating as many sources of bias as
possible.
Non probability samples
Quota sample
The defining characteristic of a quota
sample is that the researcher deliberately
sets the proportions of levels or strata within
the sample. This is generally done to insure
the inclusion of a particular segment of the
population. The proportions may or may not
differ dramatically from the actual proportion
in the population. The researcher sets
a quota, independent of population
characteristics

Non probability samples

Quota sample
Example: A researcher is interested in the
attitudes of members of different religions
towards the death penalty in Anambra. In Lower
a random sample might miss Muslims (because
there are not many in that state). To be sure of
their inclusion, a researcher could set a quota of
3% Muslim for the sample. However, the sample
will no longer be representative of the actual
proportions in the population. This may limit
generalizing to the state population. But the
quota will guarantee that the views of Muslims
are represented in the survey
Non probability samples
Purposive sample
A purposive sample is a non-representative
subset of some larger population, and is
constructed to serve a very specific need or
purpose. A researcher may have a specific
group in mind, such as high level business
executives. It may not be possible to specify the
population -- they would not all be known, and
access will be difficult. The researcher will
attempt to zero in on the target group,
interviewing whomever is available

Non probability samples
Purposive sample
A subset of a purposive sample is
a snowball sample -- so named because
one picks up the sample along the way,
analogous to a snowball accumulating
snow. A snowball sample is achieved by
asking a participant to suggest someone
else who might be willing or appropriate
for the study. Snowball samples are
particularly useful in hard-to-track
populations, such as truants, drug users,
etc
Non probability samples
Purposive sample
Convenience sample
A convenience sample is a matter of taking
what you can get. It is an accidental sample.
Although selection may be unguided, it
probably is not random, using the correct
definition of everyone in the population
having an equal chance of being selected.
Volunteers would constitute a convenience
sample.

Non probability samples
Non-probability samples are limited with
regard to generalization. Because they
do not truly represent a population, we
cannot make valid inferences about the
larger group from which they are drawn.
Validity
can
be
increased
by
approximating random selection as much
as possible, and making every attempt to
avoid introducing bias into sample
selection
Probability samples

Random sampling


Each subject has a known probability of
being selected
Allows application of statistical sampling
theory to results to:
Generalise
 Test hypotheses

Conclusions

Probability samples are the best

Ensure
Representativeness
 Precision

Methods used in probability
samples
Simple random sampling
 Systematic sampling
 Stratified sampling
 Multi-stage sampling
 Cluster sampling

Simple random sampling
Table of random numbers
684257954125632140
582032154785962024
362333254789120325
985263017424503686
Systematic sampling
Sampling fraction
Ratio between sample size and population
size
Systematic sampling
Cluster sampling
Cluster: a group of sampling units close to each
other i.e. crowding together in the same area or
neighborhood
Cluster sampling
Section 1
Section 2
Section 3
Section 5
Section 4
Stratified sampling

Stratified sampling: Stratified sampling is a probability sampling
technique wherein the researcher divides the entire population into
different subgroups or strata, then randomly selects the final
subjects proportionally from the different strata. For example, by
gender, social class, education level, religion, etc. Then the population is
randomly sampled within each category or stratum. If 38% of the population
is college-educated, then 38% of the sample is randomly selected from the
college-educated population
Stratified sampling

It is important to note that the strata must be
non-overlapping. Having overlapping subgroups
will grant some individuals higher chances of
being selected as subject. This completely
negates the concept of stratified sampling as a
type of probability sampling.

Stratified samples are as good as or better than
random samples, but they require a fairly
detailed advance knowledge of the population
characteristics, and therefore are more difficult
to construct.
Uses of Stratified Random Sampling

Stratified random sampling is used when the
researcher wants to highlight a specific
subgroup within the population. This technique is
useful in such researches because it ensures
the presence of the key subgroup within the
sample.

Researchers also employ stratified random
sampling when they want to observe existing
relationships between two or more subgroups.
With a simple random sampling technique, the
researcher is not sure whether the subgroups
that he wants to observe are represented
equally or proportionately within the sample.
Uses of Stratified Random
Sampling

With stratified sampling, the researcher can
representatively sample even the smallest and most
inaccessible subgroups in the population. This allows the
researcher to sample the rare extremes of the given
population.

With this technique, you have a higher statistical
precision compared to simple random sampling. This is
because the variability within the subgroups is lower
compared to the variations when dealing with the entire
population. Because this technique has high statistical
precision, it also means that it requires a small sample
size which can save a lot of time, money and effort of the
researchers.
Multi-Stage Sampling

The four methods we've covered so far -- simple,
stratified, systematic and cluster -- are the simplest
random sampling strategies.

In most real applied social research, we would use
sampling methods that are considerably more complex
than these simple variations.

The most important
combine the simple
variety of useful ways
needs in the most
possible.

When we combine sampling methods, we call this With
this technique, you have a higher statistical precision
compared to simple random sampling.
principle here is that we can
methods described earlier in a
that help us address our sampling
efficient and effective manner
Multi-Stage Sampling

This is because the variability within the subgroups is lower
compared to the variations when dealing with the entire
population. Because this technique has high statistical
precision, it also means that it requires a small sample size
which can save a lot of time, money and effort of the
researchers.

Consider a national sample of school districts stratified by
economics and educational level. Within selected districts, we
might do a simple random sample of schools. Within schools,
we might do a simple random sample of classes or grades.
And, within classes, we might even do a simple random
sample of students. In this case, we have three or four stages
in the sampling process and we use both stratified and simple
random sampling. By combining different sampling methods
we are able to achieve a rich variety of probabilistic sampling
methods that can be used in a wide range of social research
contexts.
Probability

Suppose that an event E can happen in h ways out of a
total of n possible equally likely ways. Then the
probability of occurrence of the event ( called its
success) is denoted by

The probability of nonoccurrence of the event (called its
failure) is denoted by

Thus
E” is sometimes denoted by
The event “not
Probability

Example. When a die is tossed, there are 6 equally
possible ways in which the die can fall
The event E, that a 3 or 4 turns up is
and the probability of E is Pr(E)=2/6 or 1/3. The
probability of not getting a 3 or 4 (i.e. getting a 1, 2, 5, or
6) is Pr{E} = 1-Pr{E}=2/3

Note that the probability of an event is a number
between 0 and 1. If the event cannot occur, its probability
is 0. If it must occur(i.e. Its occurrence is certain), its
probability is 1
Properties of Probability

Properties
Sample Space

The sample space is a set S comprised of all the
possible outcomes of the experiment.

The elements of a sample space are called elementary
outcomes, or simply outcomes.

The sample space may be finite or infinate e.g.
S={1,2,3,4,5,6}, S = N(set of natural numbers).

The elements of S must be mutually exclusive and
exhaustive, in the sense that once the experiment is
carried out, there is exactly one element of S that occurs.

Example: If the experiment consists of a single roll of
ordinary die, the natural sample space is the set S =
{1,2,3,4,5,6}, consisting of six elements. The outcome 2
indicates that the result of the roll was 2.
Event Space

Is any subset of sample space.
Conditional Probability
If
are two events, the probability that
occurs
given that
has occurred its denoted by
, and is called the conditional probability of
given
has
occurred.
If the occurrence or non-occurrence of
does not affect the
probability of occurrence of
, then
and we
say that
are independent events; otherwise, they are
dependent events.
If we denote by
the event that “both
occur ,”
sometimes called a compound event, then

In particular,
For three events
for independent events
, we have
Conditional Probability(Cont…)

That is, the probability of occurrence of
is equal to
(the probability of )X(the probability of
given that
has
occurred) X (the probability of
given that both and
have occurred). In particular,
Example:
Let
and
be the events ‘‘heads on fifth toss’’ and ‘‘heads on
sixth toss’’ of a coin, respectively. Then
and
are
independent events, and thus the probability of heads on both
the fifth and sixth tosses is (assuming the coin to be fair)
Conditional Probability(Cont…)
Example1:
If the probability that A will be alive in 20 years is 0.7 and the
probability that B will be alive in 20 years is 0.5, then the
probability that they will both be alive in 20 years is
(0.7)(0.5)=0.35.
Example2:
Suppose that a box contains 3 white balls and 2 black balls. Let
E1 be the event ‘‘?first ball drawn is black’’ and E2 the event
‘‘second ball drawn is black,’’ where the balls are not replaced
after being drawn. Here E1 and E2 are dependent events.
The probability that the first ball drawn is black is
.The probability that the second ball drawn is black, given that
the first ball drawn was black, is
. Thus the
probability that both balls drawn are black is
Conditional Probability(Cont…)
MUTUALLY EXCLUSIVE EVENTS:
Two or more events are called mutually exclusive if the
occurrence of any one of them excludes the occurrence of the
others. Thus if
and
are mutually exclusive events, then
If
denotes the event that ‘‘either
or
or
both occur,’’ then
In particular
for mutually exclusive events
Conditional Probability(Cont…)
Example
If is the event ‘‘drawing an ace from a deck of cards’’ and
is
the event ‘‘drawing a king,’’ then
. The
probability of drawing either an ace or a king in a single draw is
Since both an ace and a king cannot be drawn in a single draw
and are thus mutually exclusive events(shown below).
Conditional Probability(Cont…)
Example
If is the event ‘‘drawing an ace from a deck of cards’’ and
is
the event ‘‘drawing a spade,’’ then
and
are not mutually
exclusive since the ace of spades can be drawn(shown below).
. Thus probability of drawing either an ace or a spade or both is
Note that the event ‘‘ and
’’ consisting of those outcomes in
both events is the ace of spades.).
Probability Distribution
Random Variable: A random variable is a function
defined at each point of the sample space.
Discrete:
If a variable X can assume a discrete set of values X1,
X2...,XK with respective probabilities p1,p2,...,pK,
where p1+p2 +…+pK=1, we say that a discrete
probability distribution for X has been defined. The
function p(X), which has the respective values p1,
p2,...,pK for X =X1, X2,...,XK, is called the probability
function, or frequency function, of X. Because X can
assume certain values with given probabilities, it is
often called a discrete random variable. A random
variable is also known as a chance variable or
stochastic variable.
Probability Distribution(Cont…)
Example
Let a pair of fair dice be tossed and let X denote the sum of the
points obtained. Then the probability distribution is as shown in
Table below. For example, the probability of getting sum 5 is 4/
36 =1/ 9; thus in 900 tosses of the dice we would expect 100
tosses to give the sum 5.
Probability Distribution(Cont…)
Continuous
The above ideas can be extended to the case where the variable
X may assume a continuous set of values. The relativefrequency polygon of a sample becomes, in the theoretical or
limiting case of a population, a continuous curve (such as
shown in figure below) whose equation is Y=p(X). The total
area under this curve bounded by the X axis is equal to 1, and
the area under the curve between lines X = a and X = b
(shaded in Fig. below) gives the probability that X lies between
a and b, which can be denoted by Pr(a < X < b):
Probability Distribution(Cont…)
We call p(X) a probability density function, or briefly a density
function, and when such a function is given we say that a
continuous probability distribution for X has been defined. The
variable X is then often called a continuous random variable. As
in the discrete case, we can define cumulative probability
distributions and the associated distribution functions.
MATHEMATICAL EXPECTATION
If p is the probability that a person will receive a sum of money S,
the mathematical expectation (or simply the expectation) is
defined as pS.
Example
Find E(X) for the distribution of the sum of the dice given in Table
below. The distribution is given in the following EXCEL printout.
The distribution is given in A2:B12 where the p(X) values have
been converted to their decimal equivalents. In C2, the
expression =A2*B2 is entered and a click-and-drag is
performed from C2 to C12. In C13, the expression
=Sum(C2:C12) gives the mathematical expectation which
equals 7
MATHEMATICAL EXPECTATION
MATHEMATICAL EXPECTATION
Properties of Mathemical Expectation
Suppose X and Y are two random variables. Then





E(X+Y) = E(X) + E(Y)
E(X-Y) = E(X) - E(Y)
E(XY) = E(X)E(Y)
E(cX) = cE(X) where c any constant
E(c) = 0
MOMENTS
MOMENTS
The expected values E(X), E(X2), E(X3), ..., and E(Xr) are called
moments.
The first moment is the mean and measures the center of the
distribution
MOMENTS
Some functions of moments are sometimes difficult to
find. Therefore, special functions, called moment-generating
functions can sometimes make finding the mean and
variance of a random variable simpler
MOMENTS