hypothesis testing

this chapter is very important and helps you understand the basics of hypothesis. type 1 and type 2 error should be remembered

1. HYPOTHESIS TESTING By: Dr Ilona Asst Prof

2. A statistical hypothesis is an assumption about a population parameter. This assumption may or may not be true. The best way to determine whether a statistical hypothesis is true would be to examine the entire population. Since that is often impractical, researchers typically examine a random sample from the population

3. If sample data are not consistent with the statistical hypothesis, the hypothesis is rejected There are two types of statistical hypotheses . Null hypothesis: The null hypothesis, denoted by H0 , is usually the hypothesis that sample observations result purely from chance. Alternative hypothesis: The alternative hypothesis, denoted by H1 or Ha , is the hypothesis that sample observations are influenced by some non-random cause. It is not due to chance.

4. Statisticians follow a formal process to determine whether to reject a null hypothesis, based on sample data. This process, called hypothesis testing, consists of four steps

5. Characteristics of hypothesis Hypothesis should be clear and precise. Hypothesis should be capable of being tested. Hypothesis should state relationship between variables, if it happens to be a relational hypothesis Hypothesis should be limited in scope and must be specific Hypothesis should be stated as far as possible in most simple terms so that the same is easily understandable by all concerned

6. 6. Hypothesis should be amenable to testing within a reasonable time 7. Hypothesis must explain the facts that gave rise to the need for explanation

7. FOUR STEPS OF HYPOTHESIS TESTING The researcher states A hypothesis to be tested Formulates an analysis plan Analyzes sample data according to the plan Accepts or rejects the null hypothesis, based on results of the analysis.

8. STEP 1:State the hypotheses Every hypothesis test requires the analyst to state a null hypothesis and an alternative hypothesis. The hypotheses are stated in such a way that they are mutually exclusive. That is, if one is true, the other must be false; and vice versa

9. STEP 2: Formulate an analysis plan The analysis plan describes how to use sample data to accept or reject the null hypothesis. It should specify the following elements. The Level Of Significance : Often, researchers choose significance levels equal to 0.01, 0.05, or 0.10; but any value between 0 and 1 can be used. - Thus the significance level is the maximum value of the probability of rejecting H0 when it is true and is usually determined in advance before testing the hypothesis

10. Test method: t-score, chi-square, etc. If the test statistic probability is less than the significance level, the null hypothesis is rejected (or) if the calculated test statistic is more than the critical value, the null hypothesis is rejected.

11. STEP 3: Analyse the sample data Test statistic is selected based on the type of data, normality assumption, and homogeneity of variance

12. STEP 4: Interpret Results Apply the decision rule described in the analysis plan. If the value of the test statistic is unlikely, based on the null hypothesis, reject the null hypothesis.

13. DECISION ERRORS Two types of errors can result from a hypothesis test. Type I error: A Type I error occurs when the researcher rejects a null hypothesis when it is true. The probability of committing a Type I error is called the significance level. This probability is also called alpha, and is often denoted by α Reject H0 Do not reject H0 H0 is true H0 is false Type 1 Error α correct Correct Type II Error ß

14. Type II error: A Type II error occurs when the researcher fails to reject a null hypothesis that is false. The probability of committing a Type II error is called Beta, and is often denoted by β. The probability of not committing a Type II error is called the Power of the test.

15. The analysis plan includes decision rules for rejecting the null hypothesis. In practice, statisticians describe these decision rules in two ways - with reference to a P-value or with reference to a region of acceptance.

16. P-value: The strength of evidence in support of a null hypothesis is measured by the P-value. Suppose the test statistic is equal to S. The P-value is the probability of observing a test statistic as extreme as S, assuming the null hypothesis is true. If the P-value is less than the significance level, we reject the null hypothesis.

17. Region of acceptance: The region of acceptance is a range of values. If the test statistic falls within the region of acceptance, the null hypothesis is not rejected. The region of acceptance is defined so that the chance of making a Type I error is equal to the significance level.

18. The set of values outside the region of acceptance is called the region of rejection. If the test statistic falls within the region of rejection, the null hypothesis is rejected. In such cases, we say that the hypothesis has been rejected at the α level of significance

19. One-Tailed and Two-Tailed Tests: One-Tailed Tests A test of a statistical hypothesis, where the region of rejection is on only one side of the sampling distribution, is called a one-tailed test. For example, suppose the null hypothesis states that the mean is less than or equal to 10. The alternative hypothesis would be that the mean is greater than 10. The region of rejection would consist of a range of numbers located on the right side of sampling distribution; that is, a set of numbers greater than 10.

20. Two-Tailed Tests A test of a statistical hypothesis, where the region of rejection is on both sides of the sampling distribution, is called a two-tailed test. For example, suppose the null hypothesis states that the mean is equal to 10. The alternative hypothesis would be that the mean is less than 10 or greater than 10. The region of rejection would consist of a range of numbers located on both sides of sampling distribution; that is, the region of rejection would consist partly of numbers that were less than 10 and partly of numbers that were greater than 10

21. Note: When null hypothesis is rejected– researcher concludes that unlikely chance alone produce observed difference– thus called significant effect(not produced by chance) When Null hypothesis is not rejected—researcher concludes its due to chance– not significant

22. TESTS OF HYPOTHESES Parametric tests or standard tests of hypotheses; and Non-parametric tests or distribution-free test of hypotheses Parametric tests are: Test include (1) z-test; (2) t-test 3) χ2 -test (4) F-test The sample should be normally distributed

23. LIMITATIONS OF THE TESTS OF HYPOTHESES The tests should not be used in a mechanical fashion. Testing is not decision-making itself; the tests are only useful aids for decision-making. Test do not explain the reasons as to why does the difference exist, say between the means of the two samples When a test shows that a difference is statistically significant, then it simply suggests that the difference is probably not due to chance Statistical inferences based on the significance tests cannot be said to be entirely correct evidences concerning the truth of the hypotheses

24. Thus, the inference techniques (or the tests) must be combined with adequate knowledge of the subject-matter along with the ability of good judgement

25. References: C.R. Kothari. Research methodology Portney . Research methodology

26. For detail online explanation contact Dr.Ilona watsapp numb-8722369286