Hypothesis testing and parametric test

Hypothesis Testing
Dr. Keerti Jain
NIIT University, Neemrana

Some Important Terms
• Population
• Sample
• Population Parameter
• Sample Statistic
• Point Estimation
• Interval Estimation
• Confidence Interval
14-08-2019Dr.Keerti Jain, NIIT University 2

Hypothesis
A statistical hypothesis is a claim
(assertion, statement, belief or
assumption) about an unknown
population parameter value.

Hypothesis Testing
The process that enables a decision maker
to test the validity (or significance) of his
claim by analysis the difference between
the value of sample statistics and the
corresponding hypothesized population
parameter value, is called hypothesis
testing.

Steps of Hypothesis Testing
• Step1: State the Null Hypothesis and Alternate
Hypothesis
• Step II: State the level of significance.
• Step III: Select the suitable test of significance or Test
Statistic.
• Step IV: Interpretation (Decision).
0H
1H

Errors in Hypothesis Testing
Decision State of Nature
Type I error (α) Correct decision with
confidence (1-β)
Correct decision with
confidence (1-α)
Type II error (β)
0 is TrueH 0 is FalseH
0Accept H
0Reject H

Elements of a Hypothesis Test
• Null hypothesis – The null hypothesis represents the
claim or statement made about the value or the
population parameter. It is denoted by , where H
stands for hypothesis and zero stands for no
difference between sample statistic and parameter
value 𝐻0: 𝜇 = 𝜇0
• Alternative hypothesis - Statement contradictory to
the null hypothesis (will always contain an inequality).
It is denoted by
0H
1H
1 0 1 0 1 0: , : , :H H H       

• Directional Hypothesis (One tailed test)
Example:
:There is no difference between the average pulse rates
of men and women.
: Men have lower average pulse rates than women do.
• Non Directional Hypothesis (Two tailed test)
Example:
: There is no difference between the average pulse rates
of men and women.
:There is difference between the average pulse rates of
men and women.
1H
0H
0H
1H

One Tailed Test (Directional)
• Left tailed test
𝐻0: 𝜇 = 𝜇0; 𝐻1: 𝜇 > 𝜇0

Contd…
Right tailed test
𝐻0: 𝜇 = 𝜇0; 𝐻1: 𝜇 < 𝜇0

Two Tailed Test (Non Directional)
0 0 1 0: , :H H    

Certain Critical Values for Sample
Statistic Z
Rejection
Region
Level of Significance, α per cent
α = 0.10 α =0.05 α =0.01 α =0.005
One-tailed
region
Two-tailed
region
1.28 1.645 2.33 2.58
1.645 1.96 2.58 2.81

Contd…
Test statistic =
Value of sample statistic- value of hypothesized population paramter
standard error of the sample statistic
,
/ /
x x
z t
n s n
 

 
 

Formulate a Decision Rule to Accept
Null Hypothesis
The decision rules falls within the area of acceptance:
• If calculated absolute value of test statistic is less
than or equal to its critical (tabulated) value, then
accept the null hypothesis
• Otherwise reject null hypothesis.

P - Value ………….????
• It provides an alternative way to decide whether a null
hypothesis is to be accepted.
• Probability Value or p - value is the probability of
observing a sample outcome even more extreme than
the observed value when the null hypothesis is true.
• The smaller the p - value, the smaller are the chances
that variations are caused by chance/random factors.
• It is also called observed level of significance.

P-value….??? Contd..
• It has following advantages and that’s the reason mostly
statistical softwares are giving printouts with p - values:
• It allows a decision maker to use his/her own level of
significance and make decision accordingly once sample
results are available with necessary statistic
• t provides very precise information about the highest level
of significance at which the null hypothesis must be
accepted.

Example
An auto company decided to introduce a new six
cylinder car whose mean petrol consumption is
claimed to be lower than that of the existing auto
engine. It was found that the mean petrol
consumption for 50 cars was 10 km per litre with a
standard deviation of 3.5 km/ litre. Test for the
company at 5 percent level of significance, the
claim that in the new car petrol consumption is 9.5
km per litre on the average.

Solution
Step 1: 𝐻0: 𝜇 = 9.5 km/litre
𝐻1: 𝜇 ≠ 9.5 km/litre
Step 2: At 5% level of significance.
Step 3: Given 𝑥 = 10, 𝑛 = 50, 𝑠 = 3.5, 𝑧 𝛼 2 =
1.96 at
α =0.05 level of significance.
Thus using z-test statistics
𝑧 =
𝑥−𝜇
𝜎 𝑥
=
𝑥−𝜇
𝑠/ 𝑛
=
10−9.5
3.5/ 50
= 1.01
Step 4: 𝑧 𝑐𝑎𝑙 = 1.01 < 𝑧𝑡𝑎𝑏 = 𝑧 𝛼 2(𝑐𝑟𝑖𝑡𝑖𝑐𝑎𝑙 𝑣𝑎𝑙𝑢𝑒) =
1.96 at α =0.05 level of significance.

Contd..
Therefore null hypothesis is accepted. Hence the new car’s petrol
consumption is 9.5 km/litre.
The p-value approach :
The probability of finding 𝑧 𝑐𝑎𝑙 = 1.01 is 0.3437 (from normal table).
The p-value is the area to the right as well as left of the calculated value
of z-test statistic (for two-tailed test).
Since 𝑧 𝑐𝑎𝑙 = 1.01, then the area to its right is 0.5 − 0.3437 = 0.1563.
Therefore p-value = 2(0.1563)= 0.3126 > α =0.05, the null hypothesis
is accepted.

For every hypothesis-testing problem, we require a test which may
be …
• PARAMETRIC
• NON PARAMETRIC

Types of Variables
• Nominal Variable
• Ordinal Variable
• Interval Variable
• Ratio Variable

PARAMETRIC TEST ...
• A PARAMETRIC test is a test whose model requires and
specifies certain conditions about the parameters of the
population from which the sample is drawn.
• Such tests makes certain assumptions about the nature of the
underlying population like Normal Probability Distribution and
their validity rests upon the validity of these assumptions.
• These test are more powerful and strong in their assertions and
are usually applicable when data is interval scale or Ratio Scale.
• These tests are very much rich and developed.

NON PARAMETRIC TESTS...
• These are the tests whose model does not specify conditions and
assumptions about the parameters of the population; they lack
parameters.
• These are widely used for nominal or ordinal data where no
parametric tests is applicable.
• These tests are not very powerful and strong in their assertions.
• Non-parametric statistical tests are typically much easier to learn and
apply than are parametric tests.
• These tests usually convert data into ranks or signs and thereby may
loose some important information.

TESTS RELATED TO INTERVAL/RATIO
SCALE DATA – ONE SAMPLE
ONE SAMPLE
INTERVAL/RATIO SCALE DATA
VARIATION TESTS
CENTRAL TENDENCY
TESTS
c2
USE Z-TEST
or
t-TEST

Example of One Sample
• Has a visitor of the site placed an order. (Proportion test)-
t-test or z-test.
• A study of incidence of heart diseases in the middle-age
workers in Indian Managers. (Mean test)-t-test or z-test.
• It is claimed the Indian stock markets are not very risky as
compared to other emerging markets. (Variability test)-
Chi-Square test.

TEST RELATED TO INTERVAL/RATIO
SCALE – TWO SAMPLES
TWO SAMPLES
RELATED
SAMPLES
UNRELATED
SAMPLES
PAIRED t-TEST
USE Z-TEST
OR
t-TEST FOR DIFFERENCES
IN MEANS & PROPORTIONS
VARIATION
TEST
F-TEST

Examples of Two Samples
• Rozana is a retail chain. They have launched a special
incentive point scheme in NCR region which run for last
6 months. To know whether such an incentive programme
has any impact on sales. Related samples-(Difference in
central tendency)-Paired t-test.
• The social conditions of Textile workers in India- A
comparative study Delhi and Mumbai- Unrelated
Samples- (Mean test)-z-test or t-test.
• Which stock exchange has more fluctuations in prices –
BSE or NSE. Unrelated Samples-(Variability)- F-test.

TEST RELATED TO INTERVAL/RATIO
SCALE – MORE THAN TWO SAMPLES
MORE THAN 2 SAMPLES
ANALYSIS OF
VARIANCE

Reference
• J.K Sharma, Business Statistics, Pearson education.
• Srivastava, Rego, Statistics for Management, Tata Mcgrawhill.
• Johna S. Croucher, Statistics: Making Business Decisions,
McGrawhill.
• Levin, Rubin, Statistics for Management, Pearson Prentice hall.
• Render, Stair, Hanna, Badri, Quantitative Analysis for
Management, Pearson Prentice hall.
• http://www.graphpad.com/support/faqid/1089/

Thank
you
14-08-2019 31Dr.Keerti Jain, NIIT University

Hypothesis testing and parametric test

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