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Test of Hypothesis

Topics coveredMeaning of hypothesisCharacteristics of hypothesisBasic concepts concerning testing of hypothesis Null and Alternative Hypothesis Type I and Type II errors Level of significance Decision rule or test of hypothesis

Test of HypothesisHypothesis- Hypothesis is generally considered the most important instrument in research. Its main function is to suggest new functions and ideas. In social sciences where direct knowledge of population parameters is rare ,hypothesis testing is the often used for deciding whether sample data supports our purpose or not.

Meaning of hypothesisIn ordinary context:Hypothesis means mere assumptions or supposition which are to be proved or disproved.In research context:Hypothesis is a formal question that is intended to resolve.

A research hypothesis is a predictive statement capable of being tested by scientific methods that relate an independent variable to some dependent variable.

For example, Consider statements like the following ones:Students who receive counseling will show a greater increase in creativity than students not receiving counselingthe automobile A is performing as well as automobile B. These are hypotheses capable of being objectively verified and tested. Thus, we may conclude that a hypothesis states what we are looking for and it is a proposition which can be put to a test to determine its validity.

Characteristics of hypothesis State relation

Clear & PreciseRelated to problemTestableAmenable with timeSpecific & Simple

Characteristics of a hypothesisHypothesis should be able to relate to a variable.Hypothesis should be clear and preciseHypothesis must be consistent with most known factsHypothesis should be capable of being testedHypothesis must be stated in very simple terms.Hypothesis must be limited in scope and must be specific

Basic concepts concerning testing of hypothesis

Null Hypothesis and Alternative hypothesis

The null hypothesis is generally symbolized as H0 and the alternative hypothesis as Ha. Suppose we want to test the hypothesis that the population mean is equal to the hypothesized mean m(H0) = 100.Then we would say that the null hypothesis is that the population mean is equal to the hypothesized mean 100 and symbolically we can express as:H0: m =m (H0)=100

The null hypothesis, H0 represents a theory that has been put forward, either because it is believed to be true or because it is to be used as a basis for argument, but has not been proved. For example, in a clinical trial of a new drug, the null hypothesis might be that the new drug is no better, on average, than the current drug. We would write H0: there is no difference between the two drugs on average.

Alternative hypothesisIf our sample results do not support this then we assume something else is true. The alternative that we will accept is known as alternative hypothesis HaAlternative Hypothesis would beHo: Ho i.e population mean is not equal to hypothized mean

Alternative hypothesisThe alternative hypothesis, Ha, is a statement of what a statistical hypothesis test is set up to establish For example, in a clinical trial of a new drug, the alternative hypothesis might be that the new drug has a different effect, on average, compared to that of the current drug. We would write Ha: the two drugs have different effects, on average..

Alternative hypothesisThe alternative hypothesis might also be that the new drug is better, on average, than the current drug. In this case we would write Ha: the new drug is better than the current drug, on average

Following considerations are kept in viewThe alternative hypothesis is one which one wants to prove We give special consideration to the null hypothesis. This is due to the fact that the null hypothesis relates to the statement being tested, whereas the alternative hypothesis relates to the statement to be accepted if the null is rejected. The final conclusion once the test has been carried out is always given in terms of the null hypothesis. We either 'reject H0 in favour of Ha' or 'do not reject H0'; we never conclude 'reject Ha', or even 'accept Ha'

Following considerations are kept in viewIf we conclude 'do not reject H0', this does not necessarily mean that the null hypothesis is true, it only suggests that there is no sufficient evidence against H0 in favour of Ha; rejecting the null hypothesis then, suggests that the alternative hypothesis may be true.

2. Type I and type II errors in a Hypothesis Yest

Type I Error Type I error : In a hypothesis test, a type I error occurs when the null hypothesis is rejected when it is in fact true; that is, H0 is wrongly rejected. For example H0: there is no difference between the two drugs on average.Type I error will occur if we conclude that the two drugs produce different effects when actually there isnt a difference.

A type I error is often considered to be more serious, and therefore more important to avoid, than a type II error. The hypothesis test procedure is therefore adjusted so that there is a guaranteed 'low' probability of rejecting the null hypothesis wrongly; this probability is never 0.

Type II Error 2. Type II error refers to the situation when we accept the null hypothesis when it is false. H0: there is no difference between the two drugs on average. Type II error will occur if we conclude that the two drugs produce the same effect when actually there is a difference.

Type I and Type II Errors Example

Your null hypothesis is that the battery for a heart pacemaker has an average life of 300 days, with the alternative hypothesis that the average life is more than 300 days. You are the quality control manager for the battery manufacturer.Would you rather make a Type I error or a Type II error? Based on your answer to part (a), should you use a high or low significance level?

Type I and Type II Errors ExampleGiven H0 : average life of pacemaker = 300 days, and HA: Average life of pacemaker > 300 daysIt is better to make a Type II error (where H0 is false i.e average life is actually more than 300 days but we accept H0 and assume that the average life is equal to 300 days)As we increase the significance level () we increase the chances of making a type I error. Since here it is better to make a type II error we shall choose a low .

3. Level of significanceThis is a very important concept in the context of hypothesis testing.The significance level of a statistical hypothesis test is a fixed probability of wrongly rejecting the null hypothesis H0, if it is in fact true. It is the probability of a type I error and is set by the investigator in relation to the consequences of such an error. That is, we want to make the significance level as small as possible in order to protect the null hypothesis and to prevent, as far as possible, the investigator from inadvertently making false claims.

Significance levelThe significance level is usually denoted by Significance Level = P(type I error) = Usually, the significance level is chosen to be = 0.05 = 5%.One-sided Test A one-sided test is a statistical hypothesis test in which the values for which we can reject the null hypothesis, H0 are located entirely in one tail of the probability distribution.

ContinuedIn other words, the critical region for a one-sided test is the set of values less than the critical value of the test, or the set of values greater than the critical value of the test. A one-sided test is also referred to as a one-tailed test of significance

ExampleSuppose we wanted to test a manufacturers claim that there are, on average, 50 matches in a box. We could set up the following hypotheses Ho: =50 as againstHa: 50

ExampleEither of these two alternative hypotheses would lead to a one-sided test. Presumably, we would want to test the null hypothesis against the first alternative hypothesis since it would be useful to know if there is likely to be less than 50 matches, on average, in a box (no one would complain if they get the correct number of matches in a box or more).

ContinuedYet another alternative hypothesis could be tested against the same null, leading this time to a two-sided test: Ho: =50 as againstHa: 50

ContinuedThat is, nothing specific can be said about the average number of matches in a box; only that, if we could reject the null hypothesis in our test, we would know that the average number of matches in a box is likely to be less than or greater than 50.

Two-Sided Test A two-sided test is a statistical hypothesis test in which the values for which we can reject the null hypothesis, H0 are located in both tails of the probability distribution. A two-sided test is also referred to as a two-tailed test of significance.

4.Decision rule or test of hypothesisGiven a hypothesis H0 and an alternative hypothesis Ha, we make a rule which is known as decision rule according to which we accept H0 (i.e., reject Ha) or reject H0 (i.e., accept Ha). For instance, if (H0 is that a certain lot is good (there are very few defective items in it) against Ha) that the lot is not good (there are too many defective items in it), then we must decide the number of items to be tested and the criterion for accepting or rejecting the hypothesis. We might test 10 items in the lot and plan our decision saying that if there are none or only 1 defective item among the 10, we will accept H0 otherwise we will reject H0 (or accept Ha). This sort of basis is known as decision rule.