1 hypothesis testing i 2 hypothesis testing 3 learning objectives 1)procedure of hypothesis testing...

1

Hypothesis Testing I

2

Hypothesis Testing

3

Learning Objectives

1) Procedure of hypothesis testing2) Formulation of the Hypothesis 3) Selection of statistical test to be used 4) The level of significance5) Calculation of sample statistics6) Determination of the critical values7) Comparison of the value of sample

statistics with the critical value8) Conclusion of business research.

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!

© 1984-1994 T/Maker Co.

What is a Hypothesis?

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

featuresIn 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 TestingMake an initial assumption.

Collect evidence (data).

Based on the available evidence, decide whether or not the initial assumption is reasonable.

Population

Assume thepopulationmean age is 50.(Null Hypothesis)

REJECT

The SampleMean Is 20

SampleNull Hypothesis

50?20 XIs

Hypothesis Testing Process

No, not likely!

13

Steps in Hypothesis Testing1. 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 values6. 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 Testing1. 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

1. 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.

Basic concept & Steps in

Hypothesis Testing

16

Steps in Hypothesis Testing


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


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


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

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 Testing2. 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 characteristicsHypo.

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)

Interval or ratio t - test n < 30 when σp is unknown

Two (Large sample)

Interval or ratio Z - test n ≥ 30 when σp is known

Two (Small sample)

Interval or ratio t - test n < 30 when σp is unknown

Two (Small sample)

Interval or ratio One way

ANOVA

23

Types of statistical test and its characteristicsHypothesis 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)

Interval or ratio

t - test n < 30 when σp is unknown

Two (Large sample)

Nominal Z - test n ≥ 30 when σp is known

Two (Small sample)

Interval or ratio

t - test n < 30 when σp is unknown

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 Nominal x2

Variance Two or more samples

Interval or ratio F – test or

(ANOVA test)

25

Types of statistical test and its characteristics Summary

n > 30 n < 30

σp is known Z - test Z - test

σp is unknown Z - test t - test

26

t - distribution

Normal distributiondf = inf

t- distribution for sample size n=2 (df = 2-1 = 1)

t- distribution for sample size n=15 (df = 15-1 = 14)

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


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


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 .

Steps in Hypothesis

Testing

31



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 TestingType 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

Decision

Wrong Decision

(Type I error) =

(Serious)

- significance Level

H0 is false

Wrong Decision

(Type II error) = β dangerous

Correct

Decision

35


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 TestingHypothesis 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 TestingHypothesis 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 levelType II: generally unspecified and unknownBoth 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


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 Testing5. 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


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


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) > 1.96It 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 TestingOne 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


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


One tail test:Left tail Test /Lower-tailed Test: Symbolically Ha : µ < µHo

Assume significance level is 5%Acceptance Region - A : (Z) > - 1.645

48


One tail test:Left tail Test /

Lower-tailed Test: The critical region

is located only in

Left end of the

sampling distribution.

49


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


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


One tail test:Right tail Test /

Upper-tailed Test The critical region

is located only in

right end of the

sampling distribution.

Level of Significance, a and the Rejection

RegionH0: 3

H1: < 30

0

0

H0: 3

H1: > 3

H0: 3

H1: 3

/2

Critical Value(s)

Rejection Regions

53

Table of Normal distributionCritical 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

+ 2.326

+ 2.054

+ 1.751

+1.645

+ 1.282

Left –tail test - 2.326

- 2.054

- 1.751

- 1.645

- 1.282

54

Criteria for judging significance at various important levelsSignificance 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 ± 2.5758 S.E. 99.005

56


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 distributionCritical 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

+ 2.326

+ 2.054

+ 1.751

+1.645

+ 1.282

Left –tail test - 2.326

- 2.054

- 1.751

- 1.645

- 1.282

58


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


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

known,/

0

n

Xz

30,unknown,/

0

nns

Xz

61


B. The test of significance of a Mean for

small sample:

t - tests for a population mean

30,/

0

nns

Xt

62


C. The test of significance of difference between two

mean : 1. Large scale sample, n > 30

1 2

1 2 1 2

1 2 1 2

2 21 2

1 2

( ) ( )

( ) ( )

X X

X Xz

X Xz

N N

63


C. The test of significance of difference between two

mean : 2. Small sample size, n < 30

1 2

1 2 1 2 1 2

2 22

1 21 2

( ) ( ) ( )

1 1X X p pp

X X X X X Xt

s s ss

N NN N

2 2

2 1 1 2 2

1 2

( 1) ( 1)

2p

N s N ss

N N

64


D. The test of significance for Proportions:

1. Large sample test for proportions:

1 hypothesis testing i 2 hypothesis testing 3 learning objectives 1)procedure of hypothesis testing...

Documents

testing of hypothesis

hypothesis testing slide

formulation of hypothesis

researcher hypothesis

procedure of hypothesis

hypothesis examples

null alternative hypothesis

alternative hypothesis