1. Hypothesis Testing I

2. Hypothesis Testing

3. Learning Objectives Procedure of hypothesis testing
Formulation of the Hypothesis
Selection of statistical test to be used
The level of significance
Calculation of sample statistics
Determination of the critical values
Comparison of the value of sample statistics with the critical value
Conclusion of business research.

4. What is a Hypothesis?
A hypothesis is an assumption about the population parameter.
A parameter is a characteristic of the population, like its mean or variance.
The parameter must be identified before analysis.
I assume the mean GPA of this class is 3.5!
© T/Maker Co.

5. Hypothesis
Hypothesis is an assumption /statement to be proved or disproved.
It is a statement that is considered to be true till it is proved to be false. It is an assumption or guess.
Researcher hypothesis is a formal question that he intends to resolve.
Researcher hypothesis is predictive statement, capable to being tested by scientific methods that relates an independent variable to some dependent variable.

6. Hypothesis
Examples:
Student who receive counseling will show a greater increase in creativity than students not receiving counseling.
The automobile ‘A’ is performing as well as automobile ‘B’

7. Characteristics of Hypothesis
Hypothesis should be clear and precise.
Hypothesis should be capable of being tested.
Hypothesis should state the relationship between variables.
Hypothesis should be limited in scope and must be specific.
Hypothesis should be stated in most simple terms.

8. Why Hypothesis Testing ?
Hypothesis Testing enables us to make probablity statements about population parameter .
The hypothesis may not proved absolutely, but in practices it is accepted if it has withstood a critical testing.

9. Testing of Hypothesis
It is a process of testing the significance of a parameter of the population on the basis of a sample.

10. Testing of Hypothesis-main features
In Testing of Hypothesis we compute a statistic (It is characteristic of a sample) drawn out of a population to verify whether the drawn sample belongs to the same population.

11. General Idea of Hypothesis Testing
Make an initial assumption.
Collect evidence (data).
Based on the available evidence, decide whether or not the initial assumption is reasonable.

12. Hypothesis Testing Process
Assume the
population
mean age is 50.
(Null Hypothesis)
Population
The Sample
Mean Is 20
No, not likely!
REJECT
Null Hypothesis
Sample

13. Steps in Hypothesis Testing
1. Formulation of Hypothesis (determine the Null Hypothesis H0 and Alternative Hypothesis Ha)
2. Selection of statistical test to be used.
3. The level of significance.
4. Calculation of sample statistics (x , Z , SE)
5. Determination of the critical values
6. Comparison of the value of sample statistics with the critical value (check whether x falls within the acceptance region)
7. Conclusion of business research

14. Basic concept & Steps in Hypothesis Testing
1. Formulation of Hypothesis :
Null & Alternative Hypothesis :
We must state the assumed or hypothesized value of the population parameter before we begin sampling.
The null and alternative hypothesis are chosen before the sample is drawn.

15. Basic concept & Steps in Hypothesis Testing
Formulation of Hypothesis :
Null Hypothesis :-
The assumption of test is called the null hypothesis.
The null hypothesis is that the population parametric value is equal to hypothesized mean ( Statistics)
The hypothesis that we put to the test is called the null hypothesis, symbolized Ho.
The null hypothesis usually states the situation in which there is no difference (the difference is “null”) between populations.
Null hypothesis is that hypothesized value of the population mean is equal to population mean (5.5 ft)
. H0 : µ = µHo = 5.5 ft.

16. Steps in Hypothesis Testing
1. Formulation of Hypothesis :
Alternative hypothesis:
If our statistic do not support this null hypothesis . We should conclude that something else is true.
The set of alternatives to the null hypothesis is called alternative hypothesis.
Reject the null hypothesis is known as Alternative hypothesis.
H0 : µ = µHo Null hypothesis
Ha : µ ≠ µHo Alternative hypothesis

17. Steps in Hypothesis Testing
Formulation of Hypothesis :
Alternative hypothesis:
The set of alternatives to the null hypothesis is called alternative hypothesis.
Ha : µ ≠ µHo (Two tail test - less than or more than)
Ha : µ > µHo (One tail test - Right tail test)
Ha : µ < µHo (One tail test - Left tail test)

18. Null and Alternative Hypothesis
If H0 : µ = µHo = 5.5 ft., The Alternative hypothesis could be
Alternative Hypothesis
To be read as follows
Ha : µ ≠ µHo
The alternative hypothesis is that the population mean is not equal to 5.5 ft i.e., it may be more or less than 5.5 ft
Ha : µ > µHo
The alternative hypothesis is that the population mean is greater than 5.5 ft
Ha : µ < µHo
The alternative hypothesis is that the population mean is less than 5.5 ft

19. Steps in Hypothesis Testing
Null and Alternative Hypotheses
Both Ho and Ha are statements about population parameters, not sample statistics.
A decision to retain the null hypothesis implies a lack of support for the alternative hypothesis.
A decision to reject the null hypothesis implies support for the alternative hypothesis.

20. Steps in Hypothesis Testing
Considerations to set the Null Hypothesis:
H0 is one which one whishes to disprove.
Type I error involve great risk ( level of significance) which is chosen very small.
H0 should always be specific.
Generally in testing hypothesis we proceed on the basis of null hypothesis , keeping alternative hypothesis in view. Because the assumption is that null hypothesis is true.

21. Steps in Hypothesis Testing
Selection of statistical test to be used:
Consideration of following three factors while selecting of appropriate statistical test:
1.Type of research question (Means or Proportions)
2.Number of samples.
3.Measurement scale used.

22. Types of statistical test and its characteristics
Hypo.
testing
Number of samples
Measurement scale
Test
Requirement
Hypo
thesis about
Mean
One (Large sample)
Interval or ratio
Z - test
n ≥ 30 when σp is known
One (Small sample)
t - test
n < 30 when σp is unknown
Two (Large sample)
Two (Small sample)
One way
ANOVA

23. Types of statistical test and its characteristics
Hypothesis testing
Number of samples
Measurement scale
Test
Requirement
Hypothesis about Proportions
One (Large sample)
Interval or ratio
Z - test
n ≥ 30 when σp is known
One (Small sample)
t - test
n < 30 when σp is unknown
Two (Large sample)
Nominal
Two (Small sample)

24. Types of statistical test and its characteristics
Hypothesis testing
Number of samples
Measurement scale
Test
Requirement
Hypothesis about frequency distribution
one
Nominal x2
Two or more
Variance
Two or more samples
Interval or ratio
F – test or
(ANOVA test)

25. Types of statistical test and its characteristics Summary
σp is known
Z - test
σp is unknown
t - test

26. t - distribution Normal distribution df = inf
t- distribution for sample size n=15 (df = 15-1 = 14)
t- distribution for sample size n=2 (df = 2-1 = 1)

27. Characteristics of the t - distribution
Both t-distribution and Normal distribution are symmetrical.
t-distribution is flatter than the normal distribution.
A t- distribution is lower at the mean and higher at the tails than a normal distribution.
When sample size gets larger, the shape of the t-distribution loses its flatness & become approximately equal to the normal distribution.
In fact sample size is more than 30, the t-distribution is so close to normal distribution that we will use the normal to approximate the t.

28. t - distribution & Degree of Freedom
There is separate t- distribution for each sample size.
There is different t- distribution for each of the possible degree of freedom.
What is degree of freedom? Degree of Freedom is the number of values we can choose freely. (n-1)
When there are two elements in our sample we have (2-1)= 1 degree of freedom, and with seven elements in our sample we have (7-1)= 6 degree of freedom.

29. Steps in Hypothesis Testing
3. The Significance Level :
Level of significance measure of degree of risk that researcher (test) might reject the null hypothesis when null hypothesis is true.
It is the criterion used for rejecting the null hypothesis.
It is always some percentage (usually 5%).

30. Steps in Hypothesis Testing
3. The Significance Level :
A 5% level of significance implies that there is 5% probability that we may conclude that there is a difference between the sample statistics & hypothesized population parameter, when there is no difference between them.
In another words out of 100, 5 or less than 5 sample mean that display a difference between the sample statistics & hypothesized population parameter, when there exists no difference.
It means researcher is willing to take as much as 5% risk of rejecting H0 .

31. Steps in Hypothesis Testing
3. The Significance Level :
There was no proper mechanism to decided the level of significance.
The level of significance is set by researcher based on various factors.
Sensitivity of the study
Cost involved for each type of research

32. Steps in Hypothesis Testing
Type I and Type II errors :
Type I error : Rejecting a null hypothesis when it is true is called type I error and probability of type I error is also called as significance level of the test. (ά)
Type I error means rejecting H0 which should have been accepted.
Type I error 5% it means that there are about 5 chances in 100 that we will reject H0 when it is true , it means we are accepting 5% error in decision making.

33. Steps in Hypothesis Testing
Type I and Type II errors :
Type II error : Accepting a null hypothesis when it is false is called as type II error the probability of type II error is β.
Both types of error cannot be reduced simultaneously.

34. Type I and Type II errors
Actual
Decision
Accept H0
Reject H0
H0 is true
Correct
Wrong Decision
(Type I error) =
(Serious)
 - significance Level
H0 is false
(Type II error) = β dangerous

35. Steps in Hypothesis Testing
Example . Suppose we want to test the
null hypothesis that an anti-pollution
device for cars is effective. Explain under
what conditions we would commit a Type
I error and under what conditions we
would commit a Type II error.

36. Steps in Hypothesis Testing
Solution. We would commit a Type I error
if we rejected the device when it was
indeed effective. We would commit a
Type II error if we failed to reject the
device when it was ineffective.

37. Steps in Hypothesis Testing
We call the probability of type I error, or
the probability of rejecting the null
hypothesis when it is true, .
We call the probability of type II error, or
the probability of not rejecting the null
hypothesis when it is false, .

38. Type I and Type II errors
Type I:  is the specified significance level
Type II:  generally unspecified and unknown
Both types of errors may be reduced simultaneously by increasing n.
Type one error involves the time and trouble of reworking a batch of chemicals that should have been accepted. Where as Type II error taking a chance that an entire group of user of this chemical compound will be poisoned, then in such situation one should prefer a Type I error to Type II error.

39. Steps in Hypothesis Testing
4. Calculation of sample statistics:
It involve two steps:
1. compute the SE of sample statistics
2. compute the standardize value (z, t) by using SE.

40. Steps in Hypothesis Testing
5. Determination of the critical values:
Convert Sample Statistic to test statistic, for example Z, t or F-statistic
Compare to Critical value obtained from a table.
Critical values associated with the standardized values are determined so as to evaluate whether the standardized value will fall into the acceptance region or rejection region.
Critical values depend on two tail test or one tail test
The selection of test depends on the formulation of hypothesis.

41. Steps in Hypothesis Testing
5. Determination of the critical values:
Critical Values:
We can use the normal
curve table to calculate
the Z values, called
critical values, that
separate the upper
2.5% and lower 2.5%
of sample means from
the remainder.

42. Steps in Hypothesis Testing
Two tail test or one tail test:
Two tail test:
Two tail test rejects the null hypothesis if the sample mean is significantly higher or lower than the hypothesized value of the population parameter.
In this case the null hypothesis is some specified value and alternative hypothesis is a value is not equal to the specified value of null hypothesis.
Reject the null hypothesis if the sample statistics is either too far below or above population parameter.

43. Steps in Hypothesis Testing
Two tail test or one tail test:
Two tail test: Symbolically
Ho : µ = µHo
Ha : µ ≠ µH ( Ha : µ > µHo , Ha : µ < µHo )
Assume significance level is 5%
Acceptance Region- A : (Z) ≤ 1.96
Rejection Region- R : (Z) > 1.96
It means probability of the rejection will be 0.95%

44. Steps in Hypothesis Testing
Two tail test or one tail test:
Two tail test: The alternative hypothesis states that the population parameter may be either less than or greater than the value stated in H0.
The critical region
is divided between
both tails of the
sampling
distribution.

45. Steps in Hypothesis Testing
One tail test:
A one test would be used when we are to test whether the population mean is either lower than or higher than some hypothesized value.
In this case rejection lies entirely on one extreme of the curve.

46. Steps in Hypothesis Testing
One tail test:
Left tail Test /Lower-tailed Test: Ex. We are manufacturing 6 volt batteries and we claim that our batteries last on an average (µ) 100 hours. If somebody wants to test the accuracy of your claim – he can take a random sample of our batteries and find the average (x) of a sample. He will reject our claim only if the value of (x) so calculated considerably lower than 100 hours, but will not reject our claim if the value of (x) is considerably higher than 100 hours.

47. Steps in Hypothesis Testing
One tail test:
Left tail Test /Lower-tailed Test:
Symbolically
Ha : µ < µHo
Assume significance level is 5%
Acceptance Region - A : (Z) >

48. Steps in Hypothesis Testing
One tail test:
Left tail Test /
Lower-tailed Test:
The critical region
is located only in
Left end of the
sampling distribution.

49. Steps in Hypothesis Testing
One tail test:
Right tail Test / Upper-tailed Test :
Ex. If we are making low calories ice-cream and claim that it has on an average only 500 calories per kg. and a investigator wants to test our claim, he can take a sample and compute (x) if the value of (x) is much higher than 500 calories then he will reject our claim . But will not reject our claim if the value of (x) is much lower than 500 calories.
Hence the rejection region in this case will be only on right end tail of the curve.

50. Steps in Hypothesis Testing
One tail test:
Right tail Test / Upper-tailed Test:
Symbolically
Ha : µ > µHo
Assume significance level is 5%
Acceptance Region - A : (Z) ≤ 1.645

51. Steps in Hypothesis Testing
One tail test:
Right tail Test /
Upper-tailed Test
The critical region
is located only in
right end of the
sampling distribution.

52. Level of Significance, a and the Rejection Region
Critical Value(s) a
H0: m ³ 3 H1: m < 3
Rejection Regions a
H0: m £ 3 H1: m > 3
a/2
H0: m = 3 H1: m ¹ 3

53. Table of Normal distribution
Critical value (Z)
Level of Significance
1 %
2 %
4 %
5 % 10
Two –tail test
± 2.56
± 2.326
± 2.054
± 1.96
± 1.645
Right –tail test
+1.645
Left –tail test

54. Criteria for judging significance at various important levels
Significance level
Confidence level
Critical Value
Confidence Limit
5.0%
95.055
1.96
± 1.96SE
1.0%
99.0%
2.57
± 2.57SE
2.7%
99.73% 3
± 3SE

55. Critical region (acceptance region)
Range
Percent Values
µ ± 1 S.E.
68.27%
µ ± 2 S.E.
95.45%
µ ± 3 S.E.
99.73%
µ ± 1.96 S.E.
95.00%
µ ± S.E.
99.005

56. Steps in Hypothesis Testing
6. Comparison of the value of sample statistics with the critical value:
Once the boundaries are defined, the next step is to compare the standardized value with the critical value and check whether it falls within acceptance region.
Acceptance region (critical value) is depend on the test .

57. Table of Normal distribution
Critical value (Z)
Level of Significance
1 %
2 %
4 %
5 % 10
Two –tail test
± 2.56
± 2.326
± 2.054
± 1.96
± 1.645
Right –tail test
+1.645
Left –tail test

58. Steps in Hypothesis Testing
7. Conclusion of business research:
We reject null hypothesis if the value of sample statistics fall in the rejection region and accept the null hypothesis if the sample statistics fall within the accepted region.

59. Steps in Hypothesis Testing
Sample Statistics - Hypothesized Parameter
Test Statistic =
Standard error of statistic

60. Measuring the power of hypothesis testing
A. The test of significance of a Mean for large sample:
z - tests for a population mean

61. Measuring the power of hypothesis testing
B. The test of significance of a Mean for small sample:
t - tests for a population mean

62. Measuring the power of hypothesis testing
C. The test of significance of difference between two mean : 1. Large scale sample, n > 30

63. Measuring the power of hypothesis testing
C. The test of significance of difference between two mean : 2. Small sample size, n < 30

64. Measuring the power of hypothesis testing
D. The test of significance for Proportions:
1. Large sample test for proportions:

65. Thank You