Hypothesis Testing

Hypothesis Testing

Concept of hypothesis • A hypothesis is a proposition which the researcher wants to verify • In a problem research it is necessary to formulate a hypothesis.

Procedure of hypothesis testing • • • • •

Formulate a hypothesis Set up a suitable significance level Choose a test criterion Compute Make decisions

Formulate a hypothesis • The conventional approach to hypothesis testing is to set up 2 hypotheses instead of one in such a way that if one is false or rejected, the other is true or accepted. • Let us consider that the average income of population is Rs.9000 • H : µ = Rs. 9000 where H is the null hypothesis • The alternate hypothesis is H :µ ≠ Rs.9000 0

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Set up a suitable significance level • Having formulated the hypothesis , the next step is to test its validity at a certain level of significance. • The confidence with which a hypothesis is accepted or rejected depends upon the significance level used for this purpose • A significance level of say 5% means that means that in long run, the researcher is likely to be wrong in accepting a false hypothesis and rejecting a true hypothesis in 5 out of 100 occasions.

Select test criterion • The next step in testing the hypothesis is selecting an appropriate statistical technique as the test criterion • When the hypothesis pertains to a large sample Z- test implying the normal distribution is used • When the sample size is small, t-test will become more suitable

Compute • After having selected the statistical technique, the next is performance of different computations • These computations include the testing statistic as also its standard error

Make decisions • The last step includes drawing a statistical decision involving the acceptance or rejection of the null hypothesis • Statement rejecting the hypothesis is much stronger than that accepting it

Two types errors in hypothesis testing • When a hypothesis is tested, there are four possibilities – The hypothesis is true but our test leads to its rejection – The hypothesis is false but the test leads to its acceptance – The hypothesis is true and the test leads to its acceptance – The hypothesis is false and the test leads to its rejection

Decision

H0 is true (S1)

H0 is false (S2)

Accept H0 (A1)

Correct decision

Type II error(β)

Reject H0 Type I error Correct (A2) (α) decision • When (α) =0.10 it means that the true hypothesis will be accepted in 90 out of every 100 occasions.

Example • A business firm wants to introduce another product in the market. Thus it has to choose one of the two decisions: whether to introduce the product or not. • Now the states of nature are two : the product fails, the product succeeds. • The firm thus runs the risk of wrong decision making in two ways: – The company doesn’t launch the product though it would have succeeded if it was launched. This is type II error (β) – The company launches the product and then it fails. This is type I error(α)

Parametric and non-parametric tests • The parametric tests assume that parameters such as mean , standard deviation etc. exist and are used in testing the hypothesis. • The underlying assumption in such tests the source of data is normally distributed. • The parametric tests which are commonly used are: • Z-Test, t-test, F-test

One tail test

• A characteristic of this test is that alternative hypothesis is one-sided. • For example if H : µ = 50 • It can be H : µ > 50 , called the right tail test • H : µ < 50, it is called the left tail test • Another characteristic of one sided test is that the alternative hypothesis includes a series of hypothesis H1: µ = 51 H1: µ = 52 H1: µ = 53…….. • Such an alternative hypothesis is called the composite hypothesis. O

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Interpreting the significance level • The purpose of hypothesis testing is not to question the computed value of sample statistic , but to make a judgment about difference between the sample statistic and a hypothesized population parameter • Next step is to decide what criterion to use for deciding whether to accept or reject the null hypothesis.

• If we assume that the hypothesis is correct the significance level will indicate the percentage of sample mean that is outside the certain limits • The confidence levels indicate the percentage of sample mean that is outside certain limits

0.025 of the area

0.95 of the area

0.025 of the area

In these two regions there is a significant difference between the sample statistic and the hypothesized population parameter