Fundamentals, Standard Error, Estimation, Interval Estimation, Hypothesis, Characteristics of Hypothesis, Testing The Hypothesis, Type I & Type II error, One tailed & Two tailed test, Tabulated Values, Chi-square (2) Test, Analysis of variance (ANOVA)Introduction, The Sign Test, The rank sum test or The Mann-Whitney U test, Determination of Sample Size

1. Research Methods in
Management
Module V : Estimation Theory & Hypothesis Testing
Module VI :Techniques of Association of Attributes &
Testing

2. Some Fundamentals
 Total No of Units Population (N) &
Sample (n)
 Parameter (θ) : The Statistical Measures of Population
are called Parameters.
 Population Mean (µ) = ∑xα/N
 Population Proportion (P) = X/N
Where X is the no of units possessing
some attribute
 Population Variance (σp
2
) = ∑(Xα- µ)2
/N

3. Some Fundamentals
 Statistic (T) : The Statistical Measures of Sample are
called Statistics.
 Sample Mean (X) = ∑x/n
 Sample Proportion (p) = x/n
Where x is the no of units possessing
some attribute
 Sample Variance(σs
2
) = ∑(X - X)2
/n-1
* SD of population & sample can be calculated by taking the
square root of variance.

4. Standard Error
 A measure of precision achieved by sampling (for
mean & proportion) is called sampling error.
S.E. (X) = σ/√n
S.E. (p) = √pq/n

5. Estimation
 Inferring about population parameter on
the basis of statistic is called estimation.
 Point Estimation
 Interval Estimation

6. Interval Estimation
Interval Estimation for Mean
95% Limits =X ± 1.96 σ/√n
99% Limits =X ± 2.58 σ/√n

7. Interval Estimation
Interval Estimation for Proportion
95% Limits= p ± 1.96 √pq/n
99% Limits= p ± 2.58√pq/n

8. Hypothesis
 Hypothesis means a tentative result about any
situation or a assumption to be proved or
disapproved. For a researcher it is a question
that he intends to solve
 Example: The performance of trained employees
is better than the performance of untrained
employees

9. Characteristics of Hypothesis
 It should be clear & precise
 It should be capable of being tested
 It should state relationship between variables.
 It should be stated in a simple terms
 It must be able to tasted in a reasonable time

10. Testing The Hypothesis
1. State the null hypothesis (H0) & alternate
hypothesis (Ha or H1).
Example : In a bulb production company, manager
assumed out of 100 bulbs on an average 5
bulbs having some defect.
Null hypothesis H0 : µ = 5
Alternative hypothesis Ha : µ ≠ 5

11. Testing The Hypothesis
2. Establish a level of significance
5% or 1%
3. Choosing a suitable test (calculate the value)
4. Conclude the result
If calculated value < tabulated value then H0
accepted
If calculated value >tabulated value then H0 rejected

12. Type I & Type II error
H0 is True
H0 is False
H0 Accepted H0 Rejected
Type I error
Type II error
Correct
Decision
Correct
Decision

13. One tailed & Two tailed test
 If Ha is the type of the greater than or the type of lesser
than, we use one tailed test.
Null hypothesis H0 : µ < 5
Alternative hypothesis Ha : µ > 5
 If Ha is of the type “whether greater or smaller” then
we use two tailed test
Null hypothesis H0 : µ = 5
Alternative hypothesis Ha : µ ≠ 5

14. One sample test
 When a single sample is taken from the
population and researcher estimate the values
of parameters on the basis of single sample
statistics then one sample test is used to test the
significance of characteristics of sample i.e.
statistics
 z-test if n > 30
 t-test if n < 30

15. Tabulated Values
Type of z – test 1% 5%
One tailed 2.33 1.645
Two tailed 2.58 1.96
For t – test tabulated valued are referred from statistical table at
desired degree of freedom

16. Two sample test
 When two samples are taken from the
population then two sample test is used to test
the significance of difference of sample means
 z-test if n > 30
 t-test if n < 30

17. Chi-square (χ2) Test
 Chi-square test is used either to determine the
association between the two variables or to test
the difference b/w the attributes of two sample
 Example:
 Whether quinine is affective in controlling fever?
 Whether the consumption pattern of two ice
creams is different from each other?

18. Process
 Formulate the Hypothesis
 Calculate the value of χ2
 Compare the calculated value with
tabulated value at the desired degree
of freedom
 Conclude the result
If calculated value < tabulated value then
H0 accepted
If calculated value >tabulated value then
H0 rejected

19. Analysis of variance (ANOVA)
Introduction
 When we have two sample then to determine
the significant difference between sample
means we apply z-test & t-test, but in the case of
more than two samples we cant apply these
tests.
 ANOVA is used when multiple sample cases are
involved, using this technique one can draw
interferences about whether the samples have
been drawn from populations having same
mean

20. The Sign Test
 The sign test is the simplest of the non-parametric
tests. Its name comes from the fact that it is
based on the directions (i.e. signs of pluses or
minuses) of a pair of observations and not on
their numerical magnitude.
 In any problem where sign test is used, we count:
 No. of + signs
 No. of – Signs
 No. of 0’s

21. Process
 Formulate the Hypothesis
 Population proportion is always taken 0.5
 Calculate the sample proportion which is the
ratio b/w +ive signs & total no. of signs.
 Apply z-test formula for proportion & conclude
the results

22. The rank sum test or
The Mann-Whitney U test
 This is a very popular test to determine whether
the two independent samples have been drawn
from the same populations or it is used to
determine whether there is a significant
difference between two sets of population

23. Process
 Formulate the Hypothesis
 Combine two samples and rank them from
maximum to minimum
 Calculate the value of U, μ & σ by using
appropriate formula
 Apply z-test formula [ z = (U – μ)/σ ]& conclude
the results

24. The one sample run test
 The one sample run test is used to judge the
randomness of a sample on the basis of the order
in which the observations are taken.
 A run is a succession of identical letters (or other
kinds of symbols) which is followed and preceded
by different letters or no letters at all.

25. Process
 Formulate the Hypothesis
 H0 :There is a randomness in series
 Ha :There is no randomness in series
 Count the no. of symbols or letters
 n1 = number of occurrences of type 1
 n2 = number of occurrences of type 2
 r = number of runs ( no. of sets)
 Calculate Mean (µr) & S.D. (σr) by using formulas
 Calculate value of z and conclude the results.

26. Determination of Sample Size

27. presented by
Dr. Dharmesh Motwani.
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