Hypothesis testing; z test, t-test. f-test
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About This Presentation
Hypothesis is usually considered as the principal instrument in research and quality control. Its main function is to suggest new experiments and observations. In fact, many experiments are carried out with the deliberate object of testing hypothesis. Decision makers often face situations wherein th...
Slide Content
Hypothesis Testing;
Z-Test, T-Test, F-Test
BY NARENDER SHARMA
Z-Test, T-Test, F-Test
BY NARENDER SHARMA
Shakehandwith Life
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FutureManagersandEngineersinMBAandB.E./B.Tech.,
StudentsinGraduationandPost-Graduation,Researchers,Academicians.
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What is Hypothesis?
Hypothesis is a predictive statement, capable of
being tested by scientific methods, that relates an
independent variables to some dependent
variable.
A hypothesis states what we are looking for and it is
a proportion which can be put to a test to
determine its validity
e.g.
Students who receive counseling will show a greater
increase in creativity than students not receiving
counseling
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Hypothesis is a predictive statement, capable of
being tested by scientific methods, that relates an
independent variables to some dependent
variable.
A hypothesis states what we are looking for and it is
a proportion which can be put to a test to
determine its validity
e.g.
Students who receive counseling will show a greater
increase in creativity than students not receiving
counseling
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3
Characteristics of Hypothesis
Clear and precise.
Capable of being tested.
Stated relationship between variables.
limited in scope and must be specific.
Stated as far as possible in most simple terms so that the same is
easily understand by all concerned. But one must remember that
simplicity of hypothesis has nothing to do with its significance.
Consistent with most known facts.
Responsive to testing with in a reasonable time. One can’t spend a
life time collecting data to test it.
Explain what it claims to explain; it should have empirical reference.
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Clear and precise.
Capable of being tested.
Stated relationship between variables.
limited in scope and must be specific.
Stated as far as possible in most simple terms so that the same is
easily understand by all concerned. But one must remember that
simplicity of hypothesis has nothing to do with its significance.
Consistent with most known facts.
Responsive to testing with in a reasonable time. One can’t spend a
life time collecting data to test it.
Explain what it claims to explain; it should have empirical reference.
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Null Hypothesis
It is an assertion that we hold as true unless we have
sufficient statistical evidence to conclude otherwise.
Null Hypothesis is denoted by ??????
0
If a population mean is equal to hypothesised mean
then Null Hypothesis can be written as
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5
??????
0:??????=??????
0
It is an assertion that we hold as true unless we have
sufficient statistical evidence to conclude otherwise.
Null Hypothesis is denoted by ??????
0
If a population mean is equal to hypothesised mean
then Null Hypothesis can be written as
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??????
0:??????=??????
0
Alternative Hypothesis
The Alternativehypothesis is negation of null
hypothesis and is denoted by ??????
??????
If Null is given as
Then alternative Hypothesis can be written as
??????
0:??????=??????
0
??????
??????:??????≠??????
0
??????
??????:??????>??????
0
??????
??????:??????<??????
0
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The Alternativehypothesis is negation of null
hypothesis and is denoted by ??????
??????
If Null is given as
Then alternative Hypothesis can be written as
??????
0:??????=??????
0
??????
??????:??????≠??????
0
??????
??????:??????>??????
0
??????
??????:??????<??????
0
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Level of significance and
confidence
Significance means the percentage risk to reject a
null hypothesis when it is true and it is denoted by �.
Generally taken as 1%, 5%, 10%
(1−�)is the confidence interval in which the null
hypothesis will exist when it is true.
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confidence
Significance means the percentage risk to reject a
null hypothesis when it is true and it is denoted by �.
Generally taken as 1%, 5%, 10%
(1−�)is the confidence interval in which the null
hypothesis will exist when it is true.
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Risk of rejecting a Null Hypothesis
when it is true
Designation
Risk
??????
Confidence
??????−??????
Description
Supercritical
0.001
0.1%
0.999
99.9%
More than $100 million
(Large loss of life, e.g.nuclear
disaster
Critical
0.01
1%
0.99
99%
Lessthan $100 million
(A few lives lost)
Important
0.05
5%
0.95
95%
Less than $100 thousand
(Nolives lost, injuries occur)
Moderate
0.10
10%
0.90
90%
Less than $500
(Noinjuries occur)
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when it is true
Designation
Risk
??????
Confidence
??????−??????
Description
Supercritical
0.001
0.1%
0.999
99.9%
More than $100 million
(Large loss of life, e.g.nuclear
disaster
Critical
0.01
1%
0.99
99%
Lessthan $100 million
(A few lives lost)
Important
0.05
5%
0.95
95%
Less than $100 thousand
(Nolives lost, injuries occur)
Moderate
0.10
10%
0.90
90%
Less than $500
(Noinjuries occur)
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Type I and Type II Error
Situation
Decision
Accept Null Reject Null
Nullis true Correct Type I error
(������)
Nullis false Type II error
(������)
Correct
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Situation
Decision
Accept Null Reject Null
Nullis true Correct Type I error
(������)
Nullis false Type II error
(������)
Correct
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Two tailed test @
5%Significancelevel
Acceptance and Rejection
regions in case of a Two
tailed test
���������������
/�����������������
(�=0.025��2.5%)
���������������
/�����������������
(�=0.025��2.5%)
Suitable When ??????
0:??????=??????
0
??????
??????:??????≠??????
0
??????
0:??????=??????
0
�����??????���������������
�����������������
(1−�)=95%
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5%Significancelevel
Acceptance and Rejection
regions in case of a Two
tailed test
���������������
/�����������������
(�=0.025��2.5%)
���������������
/�����������������
(�=0.025��2.5%)
Suitable When ??????
0:??????=??????
0
??????
??????:??????≠??????
0
??????
0:??????=??????
0
�����??????���������������
�����������������
(1−�)=95%
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Left tailed test @
5%Significancelevel
Acceptance and Rejection
regions in case of a left tailed
test
??????
0:??????=??????
0
�����??????���������������
�����������������
(1−�)=95%
���������������
/�����������������
(�=0.05��5%)
Suitable When ??????
0:??????=??????
0
??????
??????:??????<??????
0
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5%Significancelevel
Acceptance and Rejection
regions in case of a left tailed
test
??????
0:??????=??????
0
�����??????���������������
�����������������
(1−�)=95%
���������������
/�����������������
(�=0.05��5%)
Suitable When ??????
0:??????=??????
0
??????
??????:??????<??????
0
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Right tailed test @
5%Significancelevel
Acceptance and Rejection
regions in case of a Right
tailed test
Suitable When ??????
0:??????=??????
0
??????
??????:??????>??????
0
??????
0:??????=??????
0
�����??????���������������
�����������������
(1−�)=95%
���������������
/�����������������
(�=0.05��5%)
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5%Significancelevel
Acceptance and Rejection
regions in case of a Right
tailed test
Suitable When ??????
0:??????=??????
0
??????
??????:??????>??????
0
??????
0:??????=??????
0
�����??????���������������
�����������������
(1−�)=95%
���������������
/�����������������
(�=0.05��5%)
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Procedure for Hypothesis
Testing
State the null
(Ho)and alternate
(Ha) Hypothesis
State a
significance level;
1%, 5%, 10% etc.
Decide a test
statistics; z-test, t-
test, F-test.
Calculate the
value of test
statistics
Calculate the p-
value at given
significance level
from the table
Compare
the p-value
with
calculated
value
P-value >
Calculated
value
P-value <
Calculated
value
Accept Ho
Reject Ho
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Testing
State the null
(Ho)and alternate
(Ha) Hypothesis
State a
significance level;
1%, 5%, 10% etc.
Decide a test
statistics; z-test, t-
test, F-test.
Calculate the
value of test
statistics
Calculate the p-
value at given
significance level
from the table
Compare
the p-value
with
calculated
value
P-value >
Calculated
value
P-value <
Calculated
value
Accept Ho
Reject Ho
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Hypothesis
Testing of
Means
Z-TEST AND T-TEST
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Testing of
Means
Z-TEST AND T-TEST
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Z-Test for testing means
Test Condition
Population normal and
infinite
Sample size large or small,
Population variance is
known
Ha may be one-sided or
two sided
Test Statistics
??????=
??????−????????????0
????????????
??????
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Test Condition
Population normal and
infinite
Sample size large or small,
Population variance is
known
Ha may be one-sided or
two sided
Test Statistics
??????=
??????−????????????0
????????????
??????
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Z-Test for testing means
Test Condition
Population normal and
finite,
Sample size large or small,
Population variance is
known
Ha may be one-sided or
two sided
Test Statistics
??????=
??????−??????
??????
0
??????
??????
�
× �−��−1
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Test Condition
Population normal and
finite,
Sample size large or small,
Population variance is
known
Ha may be one-sided or
two sided
Test Statistics
??????=
??????−??????
??????
0
??????
??????
�
× �−��−1
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Z-Test for testing means
Test Condition
Population is infinite and
may not be normal,
Sample size is large,
Population variance is
unknown
Ha may be one-sided or
two sided
Test Statistics
??????=
??????−????????????0
????????????
??????
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Test Condition
Population is infinite and
may not be normal,
Sample size is large,
Population variance is
unknown
Ha may be one-sided or
two sided
Test Statistics
??????=
??????−????????????0
????????????
??????
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Z-Test for testing means
Test Condition
Population is finite and may
not be normal,
Sample size is large,
Population variance is
unknown
Ha may be one-sided or
two sided
Test Statistics
??????=
??????−??????
??????
0
??????
??????
�
× �−��−1
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Test Condition
Population is finite and may
not be normal,
Sample size is large,
Population variance is
unknown
Ha may be one-sided or
two sided
Test Statistics
??????=
??????−??????
??????
0
??????
??????
�
× �−��−1
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T-Test for testing means
Test Condition
Population is infinite and
normal,
Sample size is small,
Population variance is
unknown
Ha may be one-sided or
two sided
Test Statistics
�=
??????−????????????0
????????????
??????
���ℎ�.�.=�−1
??????
??????=
??????
??????− ??????
2
(�−1)
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Test Condition
Population is infinite and
normal,
Sample size is small,
Population variance is
unknown
Ha may be one-sided or
two sided
Test Statistics
�=
??????−????????????0
????????????
??????
���ℎ�.�.=�−1
??????
??????=
??????
??????− ??????
2
(�−1)
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T-Test for testing means
Test Condition
Population is finite and
normal,
Sample size is small,
Population variance is
unknown
Ha may be one-sided or
two sided
Test Statistics
���ℎ�.�.=�−1
??????
??????=
??????
??????− ??????
2
(�−1)
�=
??????−??????
??????0
??????
??????
�
× �−��−1
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Test Condition
Population is finite and
normal,
Sample size is small,
Population variance is
unknown
Ha may be one-sided or
two sided
Test Statistics
���ℎ�.�.=�−1
??????
??????=
??????
??????− ??????
2
(�−1)
�=
??????−??????
??????0
??????
??????
�
× �−��−1
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Hypothesis
testing for
difference
between
means
Z-TEST, T-TEST
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testing for
difference
between
means
Z-TEST, T-TEST
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Z-Test for testing difference
between means
Test Condition
Populations are normal
Samples happen to be
large,
Population variances are
known
Ha may be one-sided or
two sided
Test Statistics
??????=
??????
1− ??????
2
??????
??????1
2
�
1
+
??????
??????2
2
�
2
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between means
Test Condition
Populations are normal
Samples happen to be
large,
Population variances are
known
Ha may be one-sided or
two sided
Test Statistics
??????=
??????
1− ??????
2
??????
??????1
2
�
1
+
??????
??????2
2
�
2
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Z-Test for testing difference
between means
Test Condition
Populations are normal
Samples happen to be large,
Presumed to have been
drawn from the same
population
Population variances are
known
Ha may be one-sided or two
sided
Test Statistics
??????=
??????
1− ??????
2
??????
??????
21
�
1
+
1
�
2
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between means
Test Condition
Populations are normal
Samples happen to be large,
Presumed to have been
drawn from the same
population
Population variances are
known
Ha may be one-sided or two
sided
Test Statistics
??????=
??????
1− ??????
2
??????
??????
21
�
1
+
1
�
2
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T-Test for testing difference
between means
Test Condition
Samples happen to be small,
Presumed to have been
drawn from the same
population
Population variances are
unknown but assumed to be
equal
Ha may be one-sided or two
sided
Test Statistics
�=
??????
1− ??????
2
�
1−1??????
??????1
2
+�
2−1??????
??????2
2
�
1+�
2−2
×
1
�
1
+
1
�
2
���ℎ�.�.=(�
1+�
2−2)
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between means
Test Condition
Samples happen to be small,
Presumed to have been
drawn from the same
population
Population variances are
unknown but assumed to be
equal
Ha may be one-sided or two
sided
Test Statistics
�=
??????
1− ??????
2
�
1−1??????
??????1
2
+�
2−1??????
??????2
2
�
1+�
2−2
×
1
�
1
+
1
�
2
���ℎ�.�.=(�
1+�
2−2)
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Hypothesis
Testing for
comparing
two related
samples
PAIRED T-TEST
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Testing for
comparing
two related
samples
PAIRED T-TEST
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Paired T-Test for comparing
two related samples
Test Condition
Samples happens to be
small
Variances of the two
populations need not be
equal
Populations are normal
Ha may be one sided or
two sided
Test Statistics
�=
??????−0
??????
????????????????????????.
�
���ℎ(�−1)�.�.
??????= Mean of differences
??????
????????????????????????.= Standard deviation of differences
�=������������ℎ�������
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two related samples
Test Condition
Samples happens to be
small
Variances of the two
populations need not be
equal
Populations are normal
Ha may be one sided or
two sided
Test Statistics
�=
??????−0
??????
????????????????????????.
�
���ℎ(�−1)�.�.
??????= Mean of differences
??????
????????????????????????.= Standard deviation of differences
�=������������ℎ�������
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Hypothesis
Testing of
proportions
Z-TEST
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Testing of
proportions
Z-TEST
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Z-test for testing of proportions
Test Condition
Use in case of qualitative
data
Sampling distribution may
take the form of binomial
probability distribution
Ha may be one sided or two
sided
����=�.�
�����������������=�.�.�
Test statistics
??????=
�−�
�.�
�
�=������������������
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Test Condition
Use in case of qualitative
data
Sampling distribution may
take the form of binomial
probability distribution
Ha may be one sided or two
sided
����=�.�
�����������������=�.�.�
Test statistics
??????=
�−�
�.�
�
�=������������������
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Hypothesis
Testing for
difference
between
proportions
Z-TEST
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Testing for
difference
between
proportions
Z-TEST
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Z-test for testing difference
between proportions
Test Condition
Sample drawn from two
different populations
Test confirm, whether the
difference between the
proportion of success is
significant
Ha may be one sided or
two sided
Test statistics
??????=
�
1− �
2
�
1 �
1
�
1
+
�
2 �
2
�
2
�
1=proportionofsuccessinsampleone
�
2=proportionofsuccessinsampletwo
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between proportions
Test Condition
Sample drawn from two
different populations
Test confirm, whether the
difference between the
proportion of success is
significant
Ha may be one sided or
two sided
Test statistics
??????=
�
1− �
2
�
1 �
1
�
1
+
�
2 �
2
�
2
�
1=proportionofsuccessinsampleone
�
2=proportionofsuccessinsampletwo
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Hypothesis
testing of
equality of
variances of
two normal
populations
F-TEST
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testing of
equality of
variances of
two normal
populations
F-TEST
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F-Test for testing equality of
variances of two normal
populations
Test conditions
The populations are normal
Samples have been drawn
randomly
Observations are
independent; and
There is no measurement
error
Ha may be one sided or two
sided
Test statistics
??????=
??????
??????1
2
??????
??????2
2
���ℎ�
1−1and�
2−1d.f.
??????
??????1
2
is the sample estimate for ??????
??????1
2
??????
??????2
2
is the sample estimate for ??????
??????2
2
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variances of two normal
populations
Test conditions
The populations are normal
Samples have been drawn
randomly
Observations are
independent; and
There is no measurement
error
Ha may be one sided or two
sided
Test statistics
??????=
??????
??????1
2
??????
??????2
2
���ℎ�
1−1and�
2−1d.f.
??????
??????1
2
is the sample estimate for ??????
??????1
2
??????
??????2
2
is the sample estimate for ??????
??????2
2
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Limitations of the test of
Hypothesis
Testing of hypothesis is not decision making itself; but help
for decision making
Test does not explain the reasons as why the difference
exist, it only indicate that the difference is due to
fluctuations of sampling or because of other reasons but
the tests do not tell about the reason causing the
difference.
Tests are based on the probabilities and as such cannot be
expressed with full certainty.
Statistical inferences based on the significance tests
cannot be said to be entirely correct evidences
concerning the truth of the hypothesis.
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Hypothesis
Testing of hypothesis is not decision making itself; but help
for decision making
Test does not explain the reasons as why the difference
exist, it only indicate that the difference is due to
fluctuations of sampling or because of other reasons but
the tests do not tell about the reason causing the
difference.
Tests are based on the probabilities and as such cannot be
expressed with full certainty.
Statistical inferences based on the significance tests
cannot be said to be entirely correct evidences
concerning the truth of the hypothesis.
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Thank You
34
34
Shakehandwith Life
LeadingTraining,Coaching,ConsultingservicesinDelhiNCRforManagersatalllevels,
FutureManagersandEngineersinMBAandB.E./B.Tech.,
StudentsinGraduationandPost-Graduation,Researchers,Academicians.
TrainingwithMS-Excelformanagerialdecisionmakingskills,
WorkingwithMS-Exceltosolveallmathematicalandstatisticalproblem.
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Corporate Training and Management Education
Call Now
9468267324, 8684861131
WhatsApp
9468267324
www.shakehandwithlife.in, www.shakehandwithlife.puzl.com
E-mail: [email protected]
LeadingTraining,Coaching,ConsultingservicesinDelhiNCRforManagersatalllevels,
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