**PROCEDURE FOR HYPOTHESIS TESTING **

To test a hypothesis means to tell (on the basis of the data the researcher has collected) whether or not the hypothesis seems to be valid. In hypothesis testing the main question is: whether to accept the null hypothesis or not to accept the null hypothesis? Procedure for hypothesis testing refers to all those steps that we undertake for making a choice between the two actions i.e., rejection and acceptance of a null hypothesis. The various steps involved in hypothesis testing are stated below:

**(i) ***Making a formal statement: *The step consists in making a formal statement of the null hypothesis (*H *) and also of the alternative hypothesis (*H *). This means that hypotheses should be clearly stated, considering the nature of the research problem. For instance, Mr. Mohan of the Civil Engineering Department wants to test the load bearing capacity of an old bridge which must be more than 10 tons, in that case he can state his hypotheses as under:

Null hypothesis *H*0 : μ = 10 tons

Alternative Hypothesis Ha : μ > 10 tons

Take another example. The average score in an aptitude test administered at the national level is 80. To evaluate a state’s education system, the average score of 100 of the state’s students selected on random basis was 75. The state wants to know if there is a significant difference between the local scores and the national scores. In such a situation the hypotheses may be stated as under:

Null hypothesis H0 : μ = 80

Alternative Hypothesis Ha : μ ≠ 80

The formulation of hypotheses is an important step which must be accomplished with due care in accordance with the object and nature of the problem under consideration. It also indicates whether we should use a one-tailed test or a two-tailed test. If *H*_{a}* *is of the type greater than (or of the type *a *lesser than), we use a one-tailed test, but when *H*_{a}* *is of the type “whether greater or smaller” then *a *we use a two-tailed test.

**(ii) ****Selecting a significance level:** The hypotheses are tested on a pre-determined level of significance and as such the same should be specified. Generally, in practice, either 5% level or 1% level is adopted for the purpose. The factors that affect the level of significance are: (a) the magnitude of the difference between sample means; (b) the size of the samples; (c) the variability of measurements within samples; and (d) whether the hypothesis is directional or non-directional (A directional hypothesis is one which predicts the direction of the difference between, say, means). In brief, the level of significance must be adequate in the context of the purpose and nature of enquiry.

**(iii) ***Deciding the distribution to use: *After deciding the level of significance, the next step in hypothesis testing is to determine the appropriate sampling distribution. The choice generally remains between normal distribution and the *t*-distribution. The rules for selecting the correct distribution are similar to those which we have stated earlier in the context of estimation.

**(iv) ***Selecting a random sample and computing an appropriate value: *Another step is to select a random sample(s) and compute an appropriate value from the sample data concerning the test statistic utilising the relevant distribution. In other words, draw a sample to furnish empirical data.

**(v) ***Calculation of the probability: *One has then to calculate the probability that the sample result would diverge as widely as it has from expectations, if the null hypothesis were in fact true.

**(vi) ***Comparing the probability: *Yet another step consists in comparing the probability thus calculated with the specified value for α , the significance level. If the calculated probability is equal to or smaller than the α value in case of one-tailed test (and α/2 in case of two-tailed test), then reject the null hypothesis (i.e., accept the alternative hypothesis), but if the calculated probability is greater, then accept the null hypothesis. In case we reject *H*_{o }we run a risk of(at most the level of significance), committing an error of Type I, but if we accept *H*_{o}, then we run some risk (the size of which cannot be specified as long as the *H*_{o}* *happens to be vague rather than specific) of committing an error of Type II.

## Hypothesis Testing [Procedure]

Mathematics

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PROCEDURE FOR HYPOTHESIS TESTINGTo test a hypothesis means to tell (on the basis of the data the researcher has collected) whether or not the hypothesis seems to be valid. In hypothesis testing the main question is: whether to accept the null hypothesis or not to accept the null hypothesis? Procedure for hypothesis testing refers to all those steps that we undertake for making a choice between the two actions i.e., rejection and acceptance of a null hypothesis. The various steps involved in hypothesis testing are stated below:

(i)The step consists in making a formal statement of the null hypothesis (Making a formal statement:H) and also of the alternative hypothesis (H). This means that hypotheses should be clearly stated, considering the nature of the research problem. For instance, Mr. Mohan of the Civil Engineering Department wants to test the load bearing capacity of an old bridge which must be more than 10 tons, in that case he can state his hypotheses as under:Null hypothesis

H0 : μ = 10 tonsAlternative Hypothesis Ha : μ > 10 tons

Take another example. The average score in an aptitude test administered at the national level is 80. To evaluate a state’s education system, the average score of 100 of the state’s students selected on random basis was 75. The state wants to know if there is a significant difference between the local scores and the national scores. In such a situation the hypotheses may be stated as under:

Null hypothesis H0 : μ = 80

Alternative Hypothesis Ha : μ ≠ 80

The formulation of hypotheses is an important step which must be accomplished with due care in accordance with the object and nature of the problem under consideration. It also indicates whether we should use a one-tailed test or a two-tailed test. Ifis of the type greater than (or of the typeis of the type “whether greater or smaller” then

H_{a}alesser than), we use a one-tailed test, but whenH_{a}awe use a two-tailed test.(ii)The hypotheses are tested on a pre-determined level of significance and as such the same should be specified. Generally, in practice, either 5% level or 1% level is adopted for the purpose. The factors that affect the level of significance are: (a) the magnitude of the difference between sample means; (b) the size of the samples; (c) the variability of measurements within samples; and (d) whether the hypothesis is directional or non-directional (A directional hypothesis is one which predicts the direction of the difference between, say, means). In brief, the level of significance must be adequate in the context of the purpose and nature of enquiry.Selecting a significance level:(iii)After deciding the level of significance, the next step in hypothesis testing is to determine the appropriate sampling distribution. The choice generally remains between normal distribution and theDeciding the distribution to use:t-distribution. The rules for selecting the correct distribution are similar to those which we have stated earlier in the context of estimation.(iv)Another step is to select a random sample(s) and compute an appropriate value from the sample data concerning the test statistic utilising the relevant distribution. In other words, draw a sample to furnish empirical data.Selecting a random sample and computing an appropriate value:(v)One has then to calculate the probability that the sample result would diverge as widely as it has from expectations, if the null hypothesis were in fact true.Calculation of the probability:(vi)Yet another step consists in comparing the probability thus calculated with the specified value for α , the significance level. If the calculated probability is equal to or smaller than the α value in case of one-tailed test (and α/2 in case of two-tailed test), then reject the null hypothesis (i.e., accept the alternative hypothesis), but if the calculated probability is greater, then accept the null hypothesis. In case we rejectComparing the probability:Hwe run a risk of(at most the level of significance), committing an error of Type I, but if we accept_{o }H, then we run some risk (the size of which cannot be specified as long as the_{o}H_{o}