Testing the hypothesis - Acceptance or Rejection of Hypothesis for PhD Research

Given a hypothesis H0 and an alternative hypothesis H1,we make a rule which is known as a decision rule according to which we accept H0 or reject H0. For example, suppose we want to examine that the mean age of the people in a city is 40 years .In order to conduct the hypothesis testing,we need to be a bit more specific if we wish to examine that 

(i) the mean age of the people in a city is 40 years or not ;or
(ii)the mean age of the people in a city is 40 years or higher ;or
(iii)the mean age of the people inacity is 40 years or lower .
For testing the above claims,we first set up the hypothesis which are given as below.

(i)Ho:μ=40 Against Hο:μ≠40 (for claim(i))
(ii)Ho:μ=40 Against H1:μ> 40 (for claim (ii))
(iii)Ho:μ =40 Against Against H1 :μ< 40 (for claim (iii))

Now we draw a probabilistic sample of a size from the aforesaid population. Size of the sample is already known. Since we are testing the claim about population mean, we obtain sample mean as sample mean is a good estimate or a population mean .Suppose sample means comes to be twenty years.This is significantly lower than the claimed mean population age 40 years, if claim is true ,the probability of getting such a different sample mean would be very small. So when we get sample mean as 20, we do not believe on the claim.If the sample mean is close to assumed population mean is close to the assumed population mean, Ho is accepted .If the sample mean is far-off from the assumed population mean Ho is rejected.How far is far enough to reject Ho? The concept of critical value is used to decide on this.

Also, in a hypothesis test, we initially assume that the null hypothesis is true and we proceed to try to reject the null hypothesis using the sample. In case, when we cannot reject the null hypothesis it only means that sample has sufficient information to reject null hypothesis it only means sample has insufficient information to reject null hypothesis at given level of significance.It does not mean that the parametric statement under alternative hypothesis is true. Therefore,whenever we say that null hypothesis is accepted, it only means that null hypothesis cannot be rejected as there is no statistical evidence against it.

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