00 - Lecture - 04_MVA - Applications and Assumptions of MVA.pdf

Application and Assumptions of Multivariate Data Analysis

1. Multivariate Data Analysis Multivariate Data Analysis
Dr. J D Chandrapal
MBA – marketin
g

,
PGDHRM
,
P HD
,
CII
(
Award
)
– London
g, ,
,( )
Development Officer - LIC of India –Ahmedabad - 9825070933

2. MVA A
pp
lications MVA A
pp
lications
pp pp
Data reduction or structural simplification. Several multivariate methods, such
as principal components analysis
, allow the summary of multiple variables through a
comparatively
smaller
set
of
'synthetic
'
variables
generated
by
the
analyses
themselves
.
comparatively
smaller
set
of
synthetic
variables
generated
by
the
analyses
themselves
.
Thus, high-dimensional patterns arepresented in a lower-dimensional space, aiding
interpretation.
Sorting
and
grouping
Many
ecological
questions
are
concerned
with
the
similarity
or
Sorting
and
grouping
.
Many
ecological
questions
are
concerned
with
the
similarity
or
dissimilarity
of a collection of entities and their assignment to groups. Several multiv ariate
methods, such as cluster analysis
and non-metric dimensional scaling
, allow detection of
potential groups in the data. Active classification based on multivariat e data may also be
performed by methods such as linea
r
discriminant analysis
.
Investigation of the dependence among variables. Dependence among response
variables, among response and explanatory variables, or among explanatory variables is
of key interest. Methods that detect dependence, such as redundancy analysis
,are
valuable in detecting influence or covariation.
Prediction. Once the de
p
endence amon
g
variables has been detected and
p
g
characterised, multivariate models may be constructed to allow predicti on.
Hypothesis construction and testing. Exploratory techniques can reveal patterns in
data from which h
yp
otheses ma
y
be constructed. Several methods
,
such
yp
y
,
as MANOVA test
, allow the testing of statistical hypotheses on multivariate data.
Appropriately constructed assertions may thus be tested.

3. Data Reduction and Sim
p
lification Data Reduction and Sim
p
lification
•Several multivariate methods, such as principal components
analysis
allow
the
summary
of
multiple
variables
through
a
p p
analysis
,
allow
the
summary
of
multiple
variables
through
a
comparatively smaller set of 'synthetic' variables generated by the
analyses
themselves
.
the
analyses
themselves
.
•This is a statistical approach which is useful in data reduction
by
reducing
variables
that
is
capable
of
accounting
for
a
large
by
reducing
variables
that
is
capable
of
accounting
for
a
large
portion of the total variability in the items. It is also useful in constructs
validity
constructs
validity
.
•Principal component analysis is the most widely used method
i
tti
li
t
(F t )
It
dt ii
t
f
i
nex
t
rac
ti
ng
li
near componen
t
s
(F
ac
t
or
)
.
It
d
e
t
erm
i
n
i
ng a se
t
o
f
loadings (Values); leads to the estimation of the total communality
Communalities
are
the
proportion
of
common
communality
.
Communalities
are
the
proportion
of
common
variance within a variable.

4. Data Reduction and Sim
p
lification Data Reduction and Sim
p
lification
Factor Item
Major changes in insurance products
p p
1
Major

changes

in

insurance

products
Product Innovation
Com
p
etitive
p
remium rates
pp
Alternative Distribution Channel Sales promotional activities
2
Satisfactory work culture
Improved Customer services
Rifiitk R
espons
i
ve o
f
serv
i
c
i
ng ne
t
wor
k
Customer centric Insurance Market Public
’s awareness of the need for insurance
3
Public s

awareness

of

the

need

for

insurance

Education on financial planning
Easy market access
Increased Competition

5. Sortin
g
and Grou
p
in
g
Sortin
g
and Grou
p
in
g
gpggpg
•Sortin
g
and
g
rou
p
in
g
.Man
y
ecolo
g
ical
q
uestions are concerned with
g
gpg
y
g
q
the similarity or dissimilarity
of a collection of entities and their
assi
g
nment to
g
rou
p
s.
g
gp
•When we have multiple variables, Groups of “similar” objects or
variables are created, based upon measured characteristics.
•
Several
multivariate
methods
such
as
cluster
analysis
and
non
metric
•
Several
multivariate
methods
,
such
as
cluster
analysis
and
non
-
metric
dimensional scaling
, allow detection of potential groups in the data.
•Active classification based on multivariate data may also be performed
by methods such as linear discriminant analysis
.

6. Sortin
g
and Grou
p
in
g
Sortin
g
and Grou
p
in
g
gpggpg

7. Investi
g
ation of De
p
endence Investi
g
ation of De
p
endence
gp gp
•Investigation of the dependence among variables is an investigation
regarding the nature o
f
the relationships among variables is o
f
interest.
•
Are
all
the
variables
mutually
independent
or
are
one
or
more
variables
•
Are
all
the
variables
mutually
independent
or
are
one
or
more
variables
dependent on the others?
•Researcher is interested in investigate....
Dependence
among
response
variables,
Dependence
among
response
variables,
Dependence among response and explanatory variables,
De
p
endence amon
g
ex
p
lanator
y
variables
p
g
py
•Methods that detect dependence, such as redundancy analysis
,are
valuable
in
detecting
influence
or
Covariaces
valuable
in
detecting
influence
or
Covariaces
.

8. 9.0 7
0
8.0 6.07
.
0
es
405.0
Price
---
HNI
3.04
.
0
Fuel
---
Middle Class
102.0 1
.
0
1.0 2.0 3.0 4.0 5.0 6.0 7.0 8.0 9.0 10.0
Sale of Petrol Vehicles

9. Prediction Prediction
•Once the dependence among variables has been detected and
htid
lti i t
dl
b
ttd
t
ll
c
h
arac
t
er
i
se
d
,mu
lti
var
i
a
t
emo
d
e
l
smay
b
e cons
t
ruc
t
e
d
t
oa
ll
ow
prediction.
•Prediction techniques rely on a predictive equation; multiple regression
is, indeed, a prime multivariate analysis prediction technique.
•With predictive analysis, you can unfold anddevelop initiatives
that will
not only enhance your various operational processes but also help you
gain an all-important edge on the competition.
•If you understand why a trend, pattern, or event happened through
data
you
will
be
able
to
develop
an
informed
projection
of
how
things
data
,
you
will
be
able
to
develop
an
informed
projection
of
how
things
may unfold in particular areas of the business.

10. General Equation for Prediction in CLRM
•In another word it can be said that the prediction is a kind of estimate
that contains true value +
/
- Error
•so the general equation in regression analysis is
Outcome
i
=
(Model)
+
Error
Outcome
i
=
(Model)
+
Error
Model = a+bx Where a = constant =
b = Parameter =
x
i
= Predictor = Value of IV or Explanatory Variable
Standard Error =

11. Anatomy of CLRM
•In regression the model we fit is linear, which means that we
summarize
a
trend
in
data
set
with
a
strait
line
summarize
a
trend
in
data
set
with
a
strait
line
.
500 450
--
--
Residual = 100
Actual Data
400 350
--
--
es (Car)
300 250 200
--
--
--
lized Sale
Line of “Best” Fit
Actual Y
275
Predicted
200 150 100
--
--
Y = Rea
Y=a +bX
Intercept
Coefficient
Dependent
Independent
Predicted

Y
175
50 0
--
--
5
I
10
15
20
25
IIII
30
0
I
I
35
I
Intercept
Coefficient
(-) Residual
X = Ad Budget
̷ Linear (Actual Sales)
■ Actual Sale

12. H
yp
othesis H
yp
othesis
yp yp
•Hypothesis construction and testing.Once the dependence among
ibl
h
b
dt td
d
htid
lti i t
dl
var
i
a
bl
es
h
as
b
een
d
e
t
ec
t
e
d
an
d
c
h
arac
t
er
i
se
d
,mu
lti
var
i
a
t
emo
d
e
l
s
may be constructed to allow prediction.
•Prediction techniques rely on a predictive equation; multiple regression
is, indeed, a prime multivariate analysis prediction technique.
•With predictive analysis, you can unfold anddevelop initiatives
that will
not only enhance your various operational processes but also help you
gain an all-important edge on the competition.
•If you understand why a trend, pattern, or event happened through
data
you
will
be
able
to
develop
an
informed
projection
of
how
things
data
,
you
will
be
able
to
develop
an
informed
projection
of
how
things
may unfold in particular areas of the business.

13. What is Hypothesis What is Hypothesis
Predictions about Research findings Research

findings

Predictions
Proposition based
Ri
Proposition on
R
eason
i
ng
Hypothesis
Tentative answer to
Research
q
uestion
q
Answer
Claim - Property of
Wh l P l ti
Claim
Wh
o
l
e
P
opu
l
a
ti
on
Simply, a hypothesis is a specific, testable prediction means you can
support or refute it through scientific
research methods
support

or

refute

it

through

scientific
research

methods
It should be based on existing theories and knowledge

14. W
hat is H
yp
othesis Test

Hypothesis tests are normally done for one and two samples. A sample is
taken
out
from
the
population
and
analysed
yp
is
taken
out
from
the
population
and
analysed
.

For one sample, researchers are often interested in whether a
lti
htiti
h
th
i
ilt
t
ti
popu
l
a
ti
on c
h
arac
t
er
i
s
ti
csuc
h
as
th
e mean
i
sequ
i
va
l
en
t
t
o a cer
t
a
i
n
value.

For two samples, they may be interested in i
f
there is a difference
between two means from two different populations.

Statistical hypothesis tests depend on a statistic designed to measure the degree of evidence for various alternative hypotheses.

Basically, hypothesis testing involves on examination based on sample evidence and probability theory to determine whether hypothesis is reasonable statement.

15. H
yp
othesis Testin
g
yp g
The general goal of a hypothesis test is to rule out chance (sampling error)
as
a
plausible
explanation
for
the
results
from
a
research
study
error)
as
a
plausible
explanation
for
the
results
from
a
research
study
.
Hypothesis testing is a technique to help determine whether a specific
treatment has an effect on the individuals in a population.

Hypothesis testing
can be used to determine whether a statement about
the value of a population parameter should or shouldn’t be rejected.

A statistical hypothesis is a statement about the probability distribution of
a random variable.

A
hypothesis
test
is
a
procedure
for
testing
a
claim
about
a
property
of
a

A
hypothesis
test
is
a
procedure
for
testing
a
claim
about
a
property
of
a
population
uses data from a sample to test the two competing statements
indicated byH
0
andH
a
.

The null hypothesis
,denoted by
H
0
,is a tentative assumption about a
population parameter.

The alternative h
yp
othesis
,
denoted b
y
H
a
,
is the o
pp
osite o
f
what is stated
yp
,
y
a
,
pp
in the null hypothesis.

16. Core Set of Terms
All hypothesis tests use the same core set of terms and concepts. The following descriptions
of
common
terms
and
concepts
refer
to
a
hypothesis
test
in
which
descriptions
of
common
terms
and
concepts
refer
to
a
hypothesis
test
in
which
the means of two populations are being compared.
•
Null Hypothesis (H
) and Alternate Hypothesis (
H
)
11
•
Null

Hypothesis

(H
0
)

and
Alternate

Hypothesis

(
H
a
)
22
•Test Statistic
33
•Significance and Power •
Critical Value and p
Value
44
•
Critical

Value

and

p
-
Value
55
•Decision
66
•Type I (also known as ‘α’) Errors and Type II (also known as ‘β ’) Errors
Z
Vl
77
•
Z
-
V
a
l
ue

17. H
0
and
H
a
H
0
and
H
a
The H
0
is a hypothesis which the researcher tries to disprove, reject or nullify
. The
'null'
often
refers
to
the
common
view
of
something
while
The
H
is
what
the

The
word
“null”
in
this
context
means
that
it
's
a
commonly
accepted
fact
'null'
often
refers
to
the
common
view
of
something
,
while
The
H
a
,
is
what
the
researcher really thinks is the cause of a phenomenon. 
The
word
“null”
in
this
context
means
that
it s
a
commonly
accepted
fact
that researchers work to nullify. It doesn't mean that the statement is null itself
! (Perhaps the term should be called “nullifiable hypothesis” as
that might cause less confusion).

Purpose: A H
0
is a hypothesis that says there is no statistical significance
between
the
two
variables
It
is
usually
the
hypothesis
a
researcher
will
try
between
the
two
variables
.
It
is
usually
the
hypothesis
a
researcher
will
try
to disprove or discredit
.AnH
a
is one that states there is a statistically
significant relationship between two variables.

"The statement being tested in a test of statistical significance is called
the null hypothesis. The test of significance is designed to assess the
strength
of
the
evidence
against
the
H
0
The
statement
that
is
strength
of
the
evidence
against
the
H
0
..
The
statement
that
is
being tested against the null hypothesis is the alternative hypothesis.

18. Test Statistic and Si
g
nificance
g

Test Statistic
: The test statistic is the tool to decide whether or not to
reject
the
H
It
is
obtained
by
taking
observed
value
(sample
statistic)
reject
the
H
0
.
It
is
obtained
by
taking
observed
value
(sample
statistic)
and converting it into a standard score under the assumption that the H
is
true
The
test
statistic
depends
fundamentally
on
the
number
of
H
0
is
true
.
The
test
statistic
depends
fundamentally
on
the
number
of
observations that are being evaluated. It differs from situation to situation
The
whole
notion
of
hypothesis
rests
on
the
ability
to
specify
situation
.
The
whole
notion
of
hypothesis
rests
on
the
ability
to
specify
(exactly or approximately) the distribution that the test statistic foll ows

Significance
(- Alpha): It is a measure of the statistical strength of the
h
yp
othesis test. It is often characterized as the
p
robabilit
y
o
f
incorrectl
y
yp
py
y
concluding that the H
0
is false. Theshould be specified up front. The
is typically one of three values: 10%, 5%, or 1%. A 1%represents the
strongest test of the three. For this reason, 1% is a higher than 10%.

19. Power and Critical Value

Power
: Related to significance, the powerof a test measures the probability
of
correctly
concluding
that
the
H
is
true
Power
is
not
something
that
of
correctly
concluding
that
the
H
0
is
true
.
Power
is
not
something
that
researcher can choose. It is determined by several factors, including the
significance level selected and the size of the difference between the thi ngs
researcher is trying to compare. Unfortunately, significance and power are inversely related. Increasing significance decreases power.
This makes it
difficult
to
design
experiments
that
have
both
very
high
significance
and
difficult
to
design
experiments
that
have
both
very
high
significance
and
power.

Critical Value
: The critical value is the standard score
that separates the
rejection region (

) from the rest of a given curve. The critical value in a
hypothesis
test
is
based
on
two
things
:
the
distribution
of
the
test
statistic
hypothesis
test
is
based
on
two
things
:
the
distribution
of
the
test
statistic
and the significance level.
The critical value(s) refer to the point in the test
statistic distribution that give thetails of the distribution an area (meaning
probability) exactly equal to the significance level that was chosen.

20. Decision and
p
-Value
p

Decision
: Your decision to reject or accept the null hypothesis is based on
comparing the test statistic to the critical value. If the test statistic e xceeds
the critical value, you should rejectthe null hypothesis. In this case, you
would say that the difference between the two population means is
significant. Otherwise, you accept the null hypothesis.

p-Value
: It is the area to the left or right of the test statistic. The p-value of a
hypothesis test gives another way to evaluate the null hypothesis. The p-
value represents the highest significance level at which particular test
f
f
statistic would justi
f
y rejecting the null hypothesis. For example, i
f
the
significance level of 5% is chosen, and the p-value turns out to be .03 (or
3
%
)
it
ld
b
jtifid
i
jti
th
ll
hthi
3
%
)
,
it
wou
ld
b
e
j
us
tifi
e
d
i
nre
j
ec
ti
ng
th
enu
ll
h
ypo
th
es
i
s.

21. Type I and Type II Errors
Because hypothesis tests are based on sample data, there must
be possibility of errors.
•The probability of Type I error (α) is usually determined
Type I error
in advance. when the null hypothesis is true as an
equality is called the level of significance
•
Applications
of
hypothesis
testing
that
only
control
the
(α)
rejecting H
0
•
Applications
of
hypothesis
testing
that
only
control
the
Type I error are often called significance tests
,
0
when it is true
•Difficult to control for the probability of making a Type II
error when we try to reduce type I error, the probability
Type II error
(
β
)
of committing type II error increases.
•S
tatisticians avoid the risk of making a Type II error by
“
“
(
β
)
accepting H
0
when it is false
using
“
do not rejectH
0
” and not
“
acceptH
0
,
when

it

is

false

22. Type I and Type II Errors
Population Condition
H
0
True
(The drug doesn’t work)
H
0
False
(The drug works)
Conclusion
Correct
Decision
Type II Error
Accept
H
0
Decision
1 -α
Type II Error
Correct
Accept
H
0
Correct
Decision
1 -β
Type I Error
Reject
H
0
False Negative
False Positive
Goal: Keep

,

reasonably small
22
Error
)
I
I
P(Typ
e
β
Error
)
I
P(Typ
e
α



23. Z-Value

Z value is a measure of standard deviation (σ) i.e. how many SD(σ) away from mean is the observed value.

z-score is a very useful statistic because it allows us to calculate the
probability of a score occurring within our normal distribution.

Z-scores range from -3 SD(σ) (which would fall to the far left o
f
the normal
distribution curve) up to +3 SD(σ) (which would fall to the far right of the
normal
distribution
curve)
normal
distribution
curve)
.

In order to use a z-score, you need to know the mean μ and also the population std deviation σ.

Technically, z-scores are a conversion of individual scores into a standard
form. The conversion allows you to more easily compare different data; it is
bd
kld
bt
th
lti ’
tdd
diti
d
b
ase
d
on your
k
now
l
e
d
ge a
b
ou
t
th
e popu
l
a
ti
on
’ss
t
an
d
ar
d
d
ev
i
a
ti
on an
d
mean. A z-score tells you how many SD(σ) from the mean your result is..

The
z
-
score
formula
doesn
’
t
say
anything
about
sample
size
;
The
rule
of

The
z
score
formula
doesn t
say
anything
about
sample
size
;
The
rule
of
thumb applies that your sample size should be above 30 to use it.

24. Z-Value and t-Value

The t Statistic is used in a t test when you are deciding if you should supportor reject
the
null
hypothesis
It
’s
very
similar
to
a
Z
-
score
reject
the
null
hypothesis
.
It s
very
similar
to
a
Z
-
score

The Z score is scaled down by thepopulationstd deviation (σ). The t score is
scaled down by thesamplestd deviation (s). You usually have the latter, not so
much
the
former
However
due
to
the
central
limit
theorem
in
most
cases
with
a
much
the
former
.
However
,
due
to
the
central
limit
theorem
in
most
cases
with
a
very large sample of means you can assume normality and use the Z score.

In order to use Z, we must know four things:
•
The
population
mean
•
The
population
mean
.
•The population standard deviation.
•The sample mean.
The
sample
size
•
The
sample
size
.
Usually in stats, you don’t know anything about a population, so instead ofaZ
score you use a t Test with a t Statistic. The major difference between usingaZ
d
t
ttiti
i
th t
h
t
ti t
th
lti
tdd
score an
d
a
t
s
t
a
ti
s
ti
c
i
s
th
a
t
you
h
ave
t
oes
ti
ma
t
e
th
e popu
l
a
ti
on s
t
an
d
ar
d
deviation. The t test is also used if you have a small sample size (less than 3 0).

The greater the t, the more evidence you have that your observations are
iifi l
diff
f
A
ll
l
i
id
h
s
i
gn
ifi
cant
l
y
diff
erent
f
rom average.
A
sma
ll
e
r
tva
l
ue
i
sev
id
ence t
h
at you
r
observations arenotsignificantly different from average. :

25. W
hen to use a t score
I
l
i
Yes
Use the Z-Score
Do you know the
population Std
Dii
Yes N
I
s samp
l
es
i
ze
above 30?
Yes No
Use the t-Score
D
ev
i
at
i
on σ
N
o
Use the t Score
Z-Value for Proportion Z-Value for Mean t-Value for Mean
σ
known and normally
σ
unknown & normally
np ≥ 5 and nq
≥5
σ
known

and

normally

distributed population
or
σ
unknown

&

normally

distributed population
or
nq
≥

5

σ known and n>30 σ known and n<30

26. Characteristics of H
yp
othesis
yp
11
•Hypothesis must be conceptually clear
22
•Hypothesis should be empirically testable
33
•Hypothesis must be specific
44
•Hypothesis should be closest to things observable
55
•Hypothesis should be related to body of theory •
Hypothesis should be related to available techniques
66
•
Hypothesis

should

be

related

to

available

techniques
77
•I
t

s
h
ou
l
d

be
r
e
l
e
v
a
n
t

to

e
xi
st
in
g

e
nvir
o
nm
e
n
ta
l
co
n
d
i
t
i
o
n
s

7
ts oudbe ee a ttoe st ge o e ta co dto s

27. Ste
p
s in H
yp
othesis Testin
g
1
•State the null hypothesis, H
0
and the alternative hypothesis, H
1
pyp g
2
•Specify the level of significance, , (sample size, Type I & II errors)
3
•Determine the appropriate test statistic and sampling distribution.
compute the value of the test statistic.
•p-Value Approach
•Critical Value Approach
4
•Use the value of test statistic
to compute p-Value
4
•Use level of significance to
determine the critical value
and the rejection rule.
5
•Reject H
0
if
p-value <

5
•Use value of test statistic &
rejection rule to determine
wh
et
h
e
r
to
r
eject

H
0
et e to eject
0

28. 1
•State the H
0
and the H
1
0
1
Begin with the assumption that the null hypothesis is true

Similar to the notion of innocent until
proven guilty
Similar

to

the

notion

of

innocent

until
proven

guilty
H
0
=Null hypothesis (innocent)
Held on to unless there is sufficient evidence to the contrary H
a
=
Alternative hypothesis (guilty) We reject H
0
in favor of H
a
if there
H
a
Alternative

hypothesis

(guilty)

We

reject

H
0
in

favor

of

H
a
if

there

is enough evidence favoring H
a

Example
New sales force bonus plan is developed in an attempt to increase sales New

sales

force

bonus

plan

is

developed

in

an

attempt

to

increase

sales
H
a
= The new bonus plan increase sales.
H
0
= The new bonus plan does not increase sales.

Sfff
H
dH
bt lti t

S
ummary o
f

f
orms
f
or
H
0
an
d

H
a
a
b
ou
t
a popu
l
a
ti
on parame
t
ers
One
-
tailed
One
-
tailed
Two
-
tailed
One
tailed
(lower-tail)
One
tailed
(upper-tail)
Two
tailed

29. 2
•S
p
ecif
y
in
g
the level of Si
g
nificance
pyg g
A major west coast city provides one of the most comprehensive
di l
i
i
th
ld
Oti
i
lti l
emergency me
di
ca
l
serv
i
ces
i
n
th
ewor
ld
.
O
pera
ti
ng
i
namu
lti
p
l
e
hospital system with approximately 20 mobile medical units, the
service goal is to respond to medical emergencies with a mean time of
12
minutes
or
less
12
minutes
or
less
.
The director of medical services wants to formulate a hypothesis test
that
could
use
a
sample
of
emergency
response
times
to
determine
that
could
use
a
sample
of
emergency
response
times
to
determine
whether or not the service goal of 12 minutes or less is being achieved.
The emergency service is
meeting the
response
HH




The

emergency

service

is

meeting

the

response

goal; no follow-up action is necessary.
Th i i
tti
th
HH
00
:

:

 

 
 
Th
e emergency serv
i
ce
i
s no
t
mee
ti
ng
th
e response
goal; appropriate follow-up action is necessary.
H
a
:



where:

= mean response time for the
population
of medical
where:

= mean response time for the
population
of medical
emergency requests

30. Si
g
nificance Level
g
The significance level (denoted byα)is the probability that the test statistic will fall in the critical region when the null hypothesis is
actually true (making the mistake of rejecting the null hypothesis when
it
is
true)
Common
choices
for
α
are
0
05
0
01
and
0
10
Suggested Guidelines for Interpreting
p
-Values
it
is
true)
..
Common
choices
for
α
are
0
.
05
,
0
.
01
,
and
0
.
10
.
Less than .01
Overwhelming evidence to conclude H
a
is true
Between .01 & .05
Strong evidence to conclude H
a
is true
Between .05 & .10
Weak evidence to conclude H
a
is true
Greater than .10
Insufficient evidence to concludeH
a
is true

31. 3
•Identif
y
the Test Statistic
y

32. Types of Hypothesis Tests Types of Hypothesis Tests
•A two-tailed test rejects the null hypothesis if, say, the
sample
mean
is
significantly
higher
or
lower
than
the
sample
mean
is
significantly
higher
or
lower
than
the
hypothesised value of the mean of the population. Such a
test is appropriate when the null hypothesis is some
specified value and the alternative hypothesis is a value
t
l
t
th
ifi d
l
f
th
ll
hthi
Two-tailed
no
t
equa
l
t
o
th
espec
ifi
e
d
va
l
ue o
f
th
enu
ll
h
ypo
th
es
i
s,
•
A
Left
-
tailed
test
would
be
used,
whether
the
population
A
Left
tailed
test
would
be
used,
whether
the
population
mean is lower than some hypothesised value. For instance,ifour H0:µ=andHa:µ<H0,thenweare
interested in what is known as left-tailed test (wherein
there
is
one
rejection
region
only
on
the
left
tail)
Left-tailed
there
is
one
rejection
region
only
on
the
left
tail)
A
iht
tild
tt
ld
b
d
hth
th
lti
•
A
r
i
g
ht
-
t
a
il
e
d
t
es
t
wou
ld
b
euse
d
,w
h
e
th
e
r
th
e popu
l
a
ti
on
mean is higher than some hypothesised value. For
instance,ifour H0:µ=andHa:µ>H0,thenweare
interested in what is known as left-tailed test
(
wherein
Right-tailed
(
there is one rejection region only on the left tail)

33. p
-Value Approach -Lower-Tailed Test
Ab t
Plti
M
K
Ab
ou
t
a
P
opu
l
a
ti
on

M
ean:
σ
K
nown
pp- -Value Value <<
 
, so , so reject reject H H
0 0
..

= 10

= 10
Sampling Sampling

=
.
10

=
.
10
Sampling
distribution of
Sampling
distribution of
p-value 

p-value 

of of








z z
0 0
z z

34. p
-Value Approach -Upper-Tailed Test Ab t
Plti
M
K
Ab
ou
t
a
P
opu
l
a
ti
on

M
ean:
σ
K
nown
pp- -Value Value <<
 
, so , so reject reject H H
0 0
..
Sampling Sampling

= .04

= .04
distribution
o
f
distribution
o
f
p-Value p-Value 







0 0
z z

35. Critical Value Approach
One-Tailed Hypothesis Testing

The
test statistic
z
has a standard normal
probability

The

test

statistic

z
has

a

standard

normal

probability

distribution.

We can use the standard normal probability distribution table to

We

can

use

the

standard

normal

probability

distribution

table

to

find the z-value with an area of ain the lower (or upper) tail of
the distribution. the

distribution.

The value of the test statistic that established the boundary o f
the rejection region is called the
critical value
for the test
the

rejection

region

is

called

the

critical

value
for

the

test
.
 
The re
j
ection rule is: The re
j
ection rule is:
j j
• •
Lower tail: Reject H
0
if z<
-z

•
U
pp
er tail: Re
j
ect
H
0
if z>
z

pp j
0


36. Critical Value Approach -Lower-Tailed Test
Ab t
Plti
M
K
Ab
ou
t
a
P
opu
l
a
ti
on

M
ean

: 
K
nown
Sampling
Di t ib ti
Sampling
Di t ib ti Di
s
t
r
ib
u
ti
on
ofDi
s
t
r
ib
u
ti
on
ofofof

37. Critical Value Approach
Upper-Tailed Test :

Known
Sampling Sampling
distribution
of
distribution
of

38. Here are some common values
Confidence
Areabetween
Areain one
z
-
score
Here are some common values
Confidence
Level
Area

between
0 and z-score
Area

in

one
tail (alpha/2)
z
score
50% 0.2500 0.2500 0.674 80%
0.4000
0.1000
1.282
80%
0.4000
0.1000
1.282
90% 0.4500 0.0500 1.645 95% 0.4750 0.0250 1.960 98%
0 4900
0 0100
2 326
98%
0
.
4900
0
.
0100
2
.
326
99% 0.4950 0.0050 2.576

39. Two Tailed Tests Two Tailed Tests

40. Two Tailed Tests Two Tailed Tests

41. Two Tailed Tests Two Tailed Tests

42. Thank You Thank You