Testing of Hypothesis

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Business Research: Major steps for testing a hypothesis

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CHAPTER-12:

TESTING OF HYPOTHESES CHAPTER-12 RESEARCH METHODOLOGY CONCEPTS AND CASES DR DEEPAK CHAWLA DR NEENA SONDHI

What is a Hypothesis?:

What is a Hypothesis? 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. Hypothesis tests are widely used in business and industry for making decisions. 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 Rs 80 lakh . Some other examples could be whether a new drug is more effective than the existing drug based on the sample data, and whether the proportion of smokers in a class is different from 0.30. SLIDE 7-1 RESEARCH METHODOLOGY CONCEPTS AND CASES DR DEEPAK CHAWLA DR NEENA SONDHI SLIDE 12-1

Concepts in Testing of Hypothesis:

Concepts in Testing of Hypothesis Null hypothesis: The hypotheses that are proposed with the intent of receiving a rejection for them are called null hypotheses. This requires that we hypothesize the opposite of what is desired to be proved. For example, if we want to show that sales and advertisement expenditure are related, we formulate the null hypothesis that they are not related. Null hypothesis is denoted by H 0 . Alternative hypothesis : Rejection of null hypotheses leads to the acceptance of alternative hypotheses. The rejection of null hypothesis indicates that the relationship between variables (e.g., sales and advertisement expenditure) or the difference between means (e.g., wages of skilled workers in town 1 and town 2) or the difference between proportions have statistical significance and the acceptance of the null hypotheses indicates that these differences are due to chance. Alternative hypothesis is denoted by H 1 . SLIDE 7-1 RESEARCH METHODOLOGY CONCEPTS AND CASES DR DEEPAK CHAWLA DR NEENA SONDHI SLIDE 12-2

Concepts in Testing of Hypothesis:

Concepts in Testing of Hypothesis One-tailed and two-tailed tests: A test is called one-sided (or one-tailed) only if the null hypothesis gets rejected when a value of the test statistic falls in one specified tail of the distribution. Further, the test is called two-sided (or two-tailed) if null hypothesis gets rejected when a value of the test statistic falls in either one or the other of the two tails of its sampling distribution. Type I and type II error: if the hypothesis H 0 is rejected when it is actually true, the researcher is committing what is called a type I error. The probability of committing a type I error is denoted by alpha (α). This is termed as the level of significance. Similarly, if the null hypothesis H 0 when false is accepted, the researcher is committing an error called Type II error. The probability of committing a type II error is denoted by beta (β). The expression 1 – β is called power of test. SLIDE 7-1 RESEARCH METHODOLOGY CONCEPTS AND CASES DR DEEPAK CHAWLA DR NEENA SONDHI SLIDE 12-3

Steps in Testing of Hypothesis Exercise:

Steps in Testing of Hypothesis Exercise 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 Making decision SLIDE 7-1 RESEARCH METHODOLOGY CONCEPTS AND CASES DR DEEPAK CHAWLA DR NEENA SONDHI SLIDE 12-4

Test statistic for testing hypothesis about population mean:

Test statistic for testing hypothesis about population mean The table below summarizes the test statistic for testing hypothesis about population mean. SLIDE 7-1 RESEARCH METHODOLOGY CONCEPTS AND CASES DR DEEPAK CHAWLA DR NEENA SONDHI SLIDE 12-5

Test Concerning Means – Case of Single Population:

Test Concerning Means – Case of Single Population Case of large sample - I n case the sample size n is large or small but the value of the population standard deviation is known, a Z test is appropriate. The test statistic is given by, SLIDE 7-1 RESEARCH METHODOLOGY CONCEPTS AND CASES DR DEEPAK CHAWLA DR NEENA SONDHI SLIDE 12-6

Test Concerning Means – Case of Single Population:

Test Concerning Means – Case of Single Population If the population standard deviation σ is unknown, the sample standard deviation SLIDE 7-1 RESEARCH METHODOLOGY CONCEPTS AND CASES DR DEEPAK CHAWLA DR NEENA SONDHI SLIDE 12-7 is used as an estimate of σ. There can be alternate cases of two-tailed and one-tailed tests of hypotheses. Corresponding to the null hypothesis H 0 : μ = μ 0 , the following criteria could be formulated as shown in the table below:

Test Concerning Means – Case of Single Population:

Test Concerning Means – Case of Single Population It may be noted that Z α and Z α /2 are Z values such that the area to the right under the standard normal distribution is α and α/2 respectively. Case of small sample: In case the sample size is small (n ≤ 30) and is drawn from a population having a normal population with unknown standard deviation σ, a t test is used to conduct the hypothesis for the test of mean. The t distribution is a symmetrical distribution just like the normal one. However, t distribution is higher at the tail and lower at the peak. The t distribution is flatter than the normal distribution. With an increase in the sample size (and hence degrees of freedom), t distribution loses its flatness and approaches the normal distribution whenever n > 30. SLIDE 7-1 RESEARCH METHODOLOGY CONCEPTS AND CASES DR DEEPAK CHAWLA DR NEENA SONDHI SLIDE 12-8

Test Concerning Means – Case of Single Population:

Test Concerning Means – Case of Single Population A comparative shape of t and normal distribution is given in the figure below: SLIDE 7-1 RESEARCH METHODOLOGY CONCEPTS AND CASES DR DEEPAK CHAWLA DR NEENA SONDHI SLIDE 12-9

Test Concerning Means – Case of Single Population:

Test Concerning Means – Case of Single Population The null hypothesis to be tested is: H 0 : μ = μ 0 The alternative hypothesis could be one-tailed or two-tailed test. The test statistics used in this case is: SLIDE 7-1 RESEARCH METHODOLOGY CONCEPTS AND CASES DR DEEPAK CHAWLA DR NEENA SONDHI SLIDE 12-10 The procedure for testing the hypothesis of a mean is identical to the case of large sample.

Tests for Difference Between Two Population Means:

Tests for Difference Between Two Population 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: H 0 : μ 1 = μ 2 H 1 : μ 1 ≠ μ 2 Where, μ 1 = mean of population 1 μ 2 = mean of population 2 The above is a case of two-tailed test. The test statistic used is: SLIDE 7-1 RESEARCH METHODOLOGY CONCEPTS AND CASES DR DEEPAK CHAWLA DR NEENA SONDHI SLIDE 12-11

Tests for Difference Between Two Population Means:

Tests for Difference Between Two Population Means SLIDE 7-1 RESEARCH METHODOLOGY CONCEPTS AND CASES DR DEEPAK CHAWLA DR NEENA SONDHI SLIDE 12-12 The Z value for the problem can be computed using the above formula and compared with the table value to either accept or reject the hypothesis.

Tests for Difference Between Two Population Means:

Tests for Difference Between Two Population Means 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: a) Two population variances are equal. b) Two population variances are not equal. SLIDE 7-1 RESEARCH METHODOLOGY CONCEPTS AND CASES DR DEEPAK CHAWLA DR NEENA SONDHI SLIDE 12-13

Tests for Difference Between Two Population Means:

Tests for Difference Between Two Population Means Population variances are equal - If the two population variances are equal, it implies that their respective unbiased estimates are also equal. In such a case, the expression becomes: SLIDE 7-1 RESEARCH METHODOLOGY CONCEPTS AND CASES DR DEEPAK CHAWLA DR NEENA SONDHI SLIDE 12-14 To get an estimate of σˆ 2 , a weighted average of s 1 2 and s 2 2 is used, where the weights are the number of degrees of freedom of each sample. The weighted average is called a ‘pooled estimate’ of σ 2 . This pooled estimate is given by the expression:

Tests for Difference Between Two Population Means:

Tests for Difference Between Two Population Means The testing procedure could be explained as under: H0 : μ1 = μ2 ⇒ μ1 – μ2 = 0 H1 : μ1 ≠ μ2 ⇒ μ1 – μ2 ≠ 0 In this case, the test statistic t is given by the expression: SLIDE 7-1 RESEARCH METHODOLOGY CONCEPTS AND CASES DR DEEPAK CHAWLA DR NEENA SONDHI SLIDE 12-15 Once the value of t statistic is computed from the sample data, it is compared with the tabulated value at a level of significance α to arrive at a decision regarding the acceptance or rejection of hypothesis.

Tests for Difference Between Two Population Means:

Tests for Difference Between Two Population Means Population variances are not equal - In case population variances are not equal, the test statistic for testing the equality of two population means when the size of samples are small is given by: SLIDE 7-1 RESEARCH METHODOLOGY CONCEPTS AND CASES DR DEEPAK CHAWLA DR NEENA SONDHI SLIDE 12-16 The degrees of freedom in such a case is given by the expression: The procedure for testing of hypothesis remains the same as was discussed when the variances of two populations were assumed to be same.

Tests Concerning Population Proportion:

Tests Concerning Population Proportion The case of single population proportion - Suppose we want to test the hypothesis, H 0 : p = p 0 H 1 : p ≠ p 0 For large sample, the appropriate test statistic would be: SLIDE 7-1 RESEARCH METHODOLOGY CONCEPTS AND CASES DR DEEPAK CHAWLA DR NEENA SONDHI SLIDE 12-17

Tests Concerning Population Proportion:

Tests Concerning Population Proportion For a given level of significance α, the computed value of Z is compared with the corresponding critical values, i.e. Z α /2 or – Z α /2 to accept or reject the null hypothesis. SLIDE 7-1 RESEARCH METHODOLOGY CONCEPTS AND CASES DR DEEPAK CHAWLA DR NEENA SONDHI SLIDE 12-18

Tests Concerning Population Proportion:

Tests Concerning Population Proportion Two Population Proportions - Here, the interest is to test whether the two population proportions are equal or not. The hypothesis under investigation is: H 0 : p 1 = p 2 ⇒ p 1 – p 2 = 0 H 1 : p 1 ≠ p 2 ⇒ p 1 – p 2 ≠ 0 The alternative hypothesis assumed is two sided. It could as well have been one sided. The test statistic is given by: SLIDE 7-1 RESEARCH METHODOLOGY CONCEPTS AND CASES DR DEEPAK CHAWLA DR NEENA SONDHI SLIDE 12-19

Tests Concerning Population Proportion:

Tests Concerning Population Proportion SLIDE 7-1 RESEARCH METHODOLOGY CONCEPTS AND CASES DR DEEPAK CHAWLA DR NEENA SONDHI SLIDE 12-21

Tests Concerning Population Proportion:

Tests Concerning Population Proportion SLIDE 7-1 RESEARCH METHODOLOGY CONCEPTS AND CASES DR DEEPAK CHAWLA DR NEENA SONDHI SLIDE 12-22

Tests Concerning Population Proportion:

Tests Concerning Population Proportion Therefore, the estimate of standard error of difference between the two proportions is given by: SLIDE 7-1 RESEARCH METHODOLOGY CONCEPTS AND CASES DR DEEPAK CHAWLA DR NEENA SONDHI SLIDE 12-23

Tests Concerning Population Proportion:

Tests Concerning Population Proportion Now, for a given level of significance α, the sample Z value is compared with the critical Z value to accept or reject the null hypothesis. SLIDE 7-1 RESEARCH METHODOLOGY CONCEPTS AND CASES DR DEEPAK CHAWLA DR NEENA SONDHI SLIDE 12-24

END OF CHAPTER:

END OF CHAPTER RESEARCH METHODOLOGY CONCEPTS AND CASES DR DEEPAK CHAWLA DR NEENA SONDHI