Alternative Hypothesis

The hypothesis which is accepted when the null hypothesis has been rejected is called the alternative hypothesis. It is denoted by {H_1} or {H_A}. Whatever we are expecting from the sample data is taken as the alternate hypothesis. “More than 25% people are literate in our country”. We are hoping to get this result from the sample. It will be taken as an alternate hypothesis {H_1} and null hypothesis {H_o} will be that 25 % or less than that is literate. To be more specific, {H_o} will be 25 % or 1ess are literate and{H_1} will be more than 25 % are literate. It is written as:

{H_o}:p \leqslant 0.25   (25 % or less)       {H_1}:p  > 0.25 (More than 25 %)

To keep the things simple, we can write H0 in the form of equality as {H_o}:p =  0.25
but it is important to write H1 with proper direction of inequality. Thus we write{H_1}:p > 0.25.

In this case the{H_1}contains the inequality more than (>). We shall explain later that{H_1} may be written with inequality less than (<) or not equal ( \ne  ). In general, if the hypothesis about the population parameter 8 is 00, then H can be written in three different ways.

{H_o}:\theta = {\theta _o}{\text{ }}{H_1}:\theta \ne {\theta _o}{\text{ }}{H_1}:\theta > {\theta _o}{\text{ }}{H_1}:\theta < {\theta _o}

But this is the simple approach which is allowed for the students. Another way of writing the above hypotheses{H_o} and{H_1} is

\begin{gathered} {\text{(a) }}{H_o}:\theta = {\theta _o}{\text{, }}{H_1}:\theta \ne {\theta _o} \\ {\text{(b) }}{H_o}:\theta \leqslant {\theta _o}{\text{, }}{H_1}:\theta > {\theta _o} \\ {\text{(c) }}{H_o}:\theta \geqslant {\theta _o}{\text{,}}{H_1}:\theta  < {\theta _o} \\ \end{gathered}

The alternative hypothesis {H_1} never contains the sign of equality. Thus {H_1} will not contain ‘=‘, ‘ \leqslant ‘or ‘ \geqslant ‘signs. The equality sign ‘=‘and inequalities like ‘ \leqslant ‘and ‘ \geqslant ‘are used for writing {H_o}.