Procedure : Large sample theory and test of significants for single mean
The following are the
steps involved in hypothesis testing problems
1. Null hypothesis: Set
up the null hypothesis H0
hypothesis: Set up the alternative hypothesis . This will enable us to decide
whether we have to use two tailed test or single tailed test (right or left
3. Level of
significance: Choose the appropriate level of significant (a) depending on the
reliability of the estimates and permissible risk. This is to be fixed before
sample is drawn. i.e., a is fixed in advance.
4. Test statistic :
Compute the test statistic
5. Conclusion: We
compare the computed value of Z in step 4 with the significant value or
critical value or table value Za at the given level of significance .
If | Z |< Za i.e., if the calculated value of is less
than critical value we say it is not significant. This may due to fluctuations
of sampling and sample data do not provide us sufficient evidence against the
null hypothesis which may therefore be accepted.
If | Z |> Za i.e., if the calculated value of is greater than critical value Za then we say it is
significant and the null hypothesis is rejected at level of significance α.
Let xi , (i = 1,2,3,...,n) is a random sample of size from a normal population with mean μ and variance σ2 then the sample mean is distributed normally with mean and variance σ2/n, .
Thus for large samples,
the standard normal variate corresponding to is :
Under the null
hypothesis that the sample has been drawn from a population with mean and
variance σ2 , i.e., there is no
significant difference between the sample mean () and
the population mean (α) , the test statistic (for large samples) is:
If the population
standard deviation s is unknown then we use its estimate provided by the sample
variance given by σˆ2 = s2 = > σˆ = s