mb0050 slides unit 10

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C o n f i d e n t i a l

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Program : MBA

Semester : III

Subject Code : MB0050

Subject Name : Research Methodology

Unit Number : 10

Unit Title : Testing of Hypothesis

Lecture Number : 10

Lecture Title : Testing of Hypothesis

Book Id : B1700

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Testing of Hypothesis

Objectives :

Discuss the concepts used in the testing of hypothesis exercise.

Explain the steps used in testing of hypothesis exercise.

Explain the test of the significance of the mean of a single

population using both t and Z test.

Explain the test of the significance of difference between two

population means using t and Z tests.

Discuss the test of the significance of a single population

proportion.

Explain the test of the significance of the difference between two

population proportions using a Z test.

In this unit we will study the concepts and steps in the testing of

hypothesis exercise. 2

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Unit-10 Testing of Hypothesis

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Lecture Outline

Introduction

Concepts in Testing of Hypothesis

Tests Concerning Means-the Case of Single Population

Tests for Difference between two Population Means

Tests Concerning Population Proportion- the Case ofSingle Population

Tests for Difference between two PopulationProportions

Summary

Check Your Learning

Activity

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Unit-10 Testing of Hypotheses

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Introduction

A hypothesis is an assumption or a statement that may or may

not be true.

The hypothesis is tested on the basis of information obtained from

a sample.

Instead of asking, for example, what the mean assessed value of

an apartment in a multistoried building is, one may be interested

in knowing whether or not the assessed value equals some

particular value, say `80 lakh.

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Null hypothesis

The hypotheses that are proposed with the intent of receiving a rejection forthem are called null hypotheses.

Alternative hypotheses

Rejection of null hypotheses leads to the acceptance of alternative hypotheses.The alternative hypotheses are denoted by H1.

One-tailed and two-tailed tests

A test is called one-sided (or one-tailed) only if the null hypothesis getsrejected when a value of the test statistic falls in one specified tail of thedistribution. The test is called two tailed if null hypothesis gets rejected when avalue of the test statistic falls in either one or the other of the two tails of itssampling distribution.

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Steps in Testing of Hypothesis Exercise:

1. Setting up of a hypothesis

2. Setting up of a suitable significance level

3. Determination of critical region

4. Computing the value of test-statistic

5. Making decision

In case a hypothesis is rejected, the difference between the

sample statistic and the hypothesized population parameter is

considered to be significant.

If the hypothesis is accepted, the difference between the sample

statistic and the hypothesized population parameter is not

regarded as significant and can be attributed to chance.

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Test Statistic for Testing Hypothesis about Population Mean

One of the important things that have to be kept in mind is the use of

an appropriate test statistic.

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Tests Concerning Means-the Caseof Single Population

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Case 1: Large Sample

Irrespective of the sample sizen, if the value of the population standard deviation is known,a Z test is appropriate. There can be alternate cases of two-tailed and one-tailed tests ofhypotheses.

The test statistic is given by:

Where,

= Sample mean

= Population standard deviation

H0= Value of under the assumptionthat the null hypothesis is true

n = Size of sample

Corresponding to the null hypothesis H0 : = 0, the criteria shownin the table below can beused.

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Tests Concerning Means-the Caseof Single Population

Case 2: Small Sample

If the sample size is small (n 30.

Shape of t and NormalDistribution

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The procedure for testing the hypothesis of a mean is similar to that explained for a large

sample. The test statistic used in this case is as follows:

Where, (s = Sample standard deviation and n 1 = Degrees of freedom)

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Tests for Difference between twoPopulation Means

Case of large sample:

In case both the sample sizes are greater than 30, a Z test is used. The

hypothesis to be tested may be written as:

H0 : 1 = 2

H1 : 1 2

Where, 1 = mean of population 1

2 = mean of population 2

Case of small sample:

If the size of both the samples is less than 30 and the population

standard deviation is unknown, the procedure described above to discuss

the equality of two population means is not applicable in the sense that a

t test would be applicable under the assumptions:

Two population variances are equal.

Two population variances are not equal.10

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Case of large sample:

In case both the sample sizes are greater than 30, a Z test is used. The

hypothesis to be tested may be written as:

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Case of small sample: If the size of both the samples is less than 30 and the population

standard deviation is unknown, the procedure described above to discuss

the equality of two population means is not applicable in the sense that a

t test would be applicable under the assumptions:

Two population variances are equal.

Two population variances are not equal.

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7/24/2019 MB0050 Slides Unit 10


Tests Concerning Population Proportionthe Case of Single Population

In the tests about proportion, one is interested in examiningwhether the respondents possess a particular attribute or not.

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Tests for Difference between twoPopulation Proportions

Here, the interest is to test whether the two populationproportions are equal or not. The hypothesis under investigation

is:

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Summary

A hypothesis is a statement or an assumption regarding a population, which

may or may not be true.

The sequences of steps that need to be followed for the testing of

hypothesis are: setting up of a hypothesis, setting up of a suitable

significance level, determination of a test statistic, determination of critical

region, computing the value of test-statistic and making decision

In the test procedure for a single population mean or for examining the

equality of two population means, for large samples, a Z test is appropriate

whereas for the small samples, a t test is used under the two cases where:

population variances are equal and population variances are not equal.

In the testing procedures concerning the proportion of a single population

and the difference between two population proportions the hypotheses

concerning them are carried out using a Z test under the assumption that

the normal distribution could be used as an approximation to the binomial

distribution for a large sample.15

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Check Your Learning

1. The company XYZ manufacturing bulbs hypothesizes that the life of its

bulbs is 145 hours with a known standard deviation of 210 hours. A random

sample of 25 bulbs gave a mean life of 130 hours. Using a 0.05 level ofsignificance, can the company conclude that the mean life of bulbs is less

than the 145 hours?

Ans: A Z test will be used.

2. Average annual income of the employees of a company has been reported to

be `18,750. A random sample of 100 employees was taken. Then average

annual income was found to be `19,240 with a standard deviation of `2,610.

Test at 5 per cent level of significance whether the sample results are

representative of population results.


3. If 54 out of a random sample of 150 boys smoke, while 31 out of random

sample of 100 girls smoke, can we conclude at the 0.05 level of significance

that the proportion of male smokers is higher than that of female smokers? Use

the 0.05 level of significance to test the null hypothesis that the prescribed

programme of exercise is not effective in reducing weight.


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Activity

It is believed that the proportion of male smokers is higher than

that of female smokers. To verify this, you may visit a co-

educational college with a large number of students. Ask them

whether they smoke or not. Carry out an appropriate test to

examine the belief at 5% level of significance.

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