Topic of Research Methodology

1. Presentation
on
Testing of Hypothesis
By:
Ankit Dubey
Chintan H.Trivedi

2. • Basics of Hypothesis
• Procedure of Hypothesis Testing
• Testing Of Hypothesis for Comparing two
related samples
• Hypothesis testing of Proportion
• Limitations of Test of Hypothesis

3. HYPOTHESIS
• For a researcher hypothesis is a formal question
that he intends to resolve.
• Hypothesis may be defined as a proposition or a
set of proposition set forth as an explanation for
the occurrence of some specified group of
phenomena.
• Example: vehicle A performs better than vehicle B

4. NULL HYPOTHESIS AND ALTERNATIVE
HYPOTHESIS
• Null hypothesis: when population mean is
equal to hypothesis mean
• Alternative hypothesis : the hypothesis which
is accepted when null hypothesis fails
• A set of alternatives to null hypothesis is
called alternative hypothesis
• Null hypothesis always include = to sign
• Alternative hypothesis may include any of the
inequality signs

5. ALTERNATIVE HYPOTHESIS
ALTERNATIVE
HYPOTHESIS
MEANING
H0: μ ≠ μ0 Population mean is not equal to hypothesis
mean
H1 :μ>μ0 Population mean is greater than hypothesis
mean
H1 :μ<μ0 Population mean is greater than hypothesis
mean

6. TYPE I AND TYPE II ERROR
POSSIBLE HYPOTHESIS TEST OUTCOMES
ACTUAL SITUATION
DECISION H0 True H0 False
ACCEPT H0 No error Type II error
Probability =1-α Probability =β
REJECT H0 Type I Error No Error
Probability =α Probability = 1-β

7. CRITICAL VALUE
It divides area under probability curve into two
regions critical and acceptance regions
In two tailed test we wish to test H0 : μ =μ0
against H1 : μ≠ μ0 we have two critical values
which divides probability curve into two
regions

8. Level of significance
• Denoted by α is the probability of Type 1 error.
• It is usually 5% which should be chosen with
great care thought and reason.
• If level of significance is 5% then researcher is
willing to take as much as 5% percent risk of
rejecting null hypothesis if it is true.
• Maximum Value of probability of rejecting H0
when it is true.
• Determined usually in advance prior to testing
of hypothesis.

9. Two Tailed And One Tailed Test
• Testing three types of Hypothesis given by:
• H0 :μ = μ0 Against H1 :μ ≠ μ0 (two tailed test)
• H0 :μ ≤ μ0 Against H1 :μ > μ0 (Right tailed test)
• H0 :μ ≥ μ0 Against H1 :μ < μ0 (Left tailed test)

12. PROCEDURE FOR HYPOTHESIS TESTING
• It is used to test validity of hypothesis
• Setting up the hypothesis: Formal statement
of null and alternative hypothesis
• Selecting significance level:pre determined
level of significance based on sample size,
variability of measurements with in sample
• Test statistic: to obtain value of mean
,variance etc on basis of distribution selected

13. PROCEDURE FOR HYPOTHESIS TESTING
• Critical value: depends on type of
distribution, level of significance, type of test
• Decision: compare the value of test statistic
and critical value
• we reject null hypothesis when
• Test statistic value< lower than critical
value>upper critical value
• Value of test statistic > critical value
• Value of test statistic < critical value

14. HYPOTHESIS TESTING FOR
COMPARING TWO RELATED SAMPLES
• Paired t-test is a way to test for comparing two
related samples,
• For a paired t-test, observations in the two
samples collected in the form matched pairs
• “each observation in the one sample must be
paired with an observation in the other sample in
such a manner that these observations are
somehow “matched” or related, in an attempt to
eliminate extraneous factors which are not of
interest in test.”

15. • Such a test is generally considered appropriate in
a before-and-after-treatment study.
• Testing a group of certain students before and
after training in order to know whether the
training is effective, in which situation we may
use paired t-test.
• To apply this test, :
a. Work out the difference score for each
matched pair
b. Find out the average of such differences, D
c. Sample variance of the difference score.

16. • If the values from the two matched samples
are denoted as Xi and Yi and the differences
by Di (Di = Xi – Yi), then the mean of the
differences i.e.,
• And Variance of the differences:

17. • To work out the test statistic (t):
Where

18. Example

19. • Formulation of Null And Alternate Hypothesis
• Work out Test Statistic

20. • Standard deviation of differences:

21. HYPOTHESIS TESTING OF PROPORTION
• Population is divided into exclusive and
exhaustive tests based on certain attribute
• One class having attribute and other not
• Important parameter is proportion of
population having that attribute
H0 :π≤ π0, H1 : π> π0 (Right or Upper tail test)
H0 : π≥ π0 , H1 π< π0 (left or lower tail test)
H0 : π= π0 , H1 : π≠ π0 (two upper tail test)

22. • Test statistic for this test
Z = p - π0
√ π0 (1- π0 )/n
p=sample proportion

23. HYPOTHESIS TESTING OF TWO
PROPORTION
• Two population proportions are compared
• Two population proportion are obtained from
two different samples
• Hence following hypothesis is tested
H0 : π1 = π2 against H1 : π1 ≠ π2
H0 : π1 = π2 against H1 : π1 > π2 or H0 π1 ≤ π2
Against H1 : π1 > π2
H0 : π1 = π2 Against H1 π1 < π2 or π1 ≥ π2 Against H1
π1 < π2

24. EXAMPLE
Q) A sample survey indicates that out of 3232 births,
1705 were boys and the rest were girls. Do these
figures confirm the hypothesis that the sex ratio is
50 : 50? Test at 5 per cent level of significance.
Solution: Starting from the null hypothesis that the sex
ratio is 50 : 50 we may write:

27. Hypothesis testing for Variance
1. H0 :σ2 = σ0
2 , H1 : σ2 ≠ σ0
2 (two tailed test)
2. H0 :σ2 = σ0
2 , H1 : σ2 > σ0
2 (right tailed test)
3. H0 :σ2 = σ0
2 , H1 : σ2 < σ0
2 (left tailed test)

28. Limitations of Test of Hypothesis
• Important limitations are as follows:
• Testing is not decision-making itself; the tests are only
useful aids for decision-making.
• “proper interpretation of statistical evidence is
important to intelligent decisions.”
• Test do not explain the reasons as to why does the
difference exist, viz. between the means of the two
samples.
• They simply indicate whether the difference is due to
fluctuations of sampling or because of other reasons
• The tests do not tell us the reasons causing the
difference.

29. • Results of significance tests are based on 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 hypotheses.
• This is specially so in case of small samples where the
probability of drawing erring inferences happens to be
generally higher.
• For greater reliability, the size of samples be sufficiently
enlarged.

30. • All these limitations suggest that in problems
of statistical significance, the inference
techniques (or the tests) must be combined
with adequate knowledge of the subject-
matter along with the ability of good
judgement.

31. Thank You!