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. For example, stating “More than 25% people are literate in our country” means we are hoping to get this result from the sample. It will be taken as an alternate hypothesis {H_1} and the null hypothesis {H_o} will be that 25% or less are literate. To be more specific, {H_o} will be 25% or less 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 the proper direction of inequality. Thus we write {H_1}:p > 0.25.

In this case the {H_1} contains the inequality of more than (>). We shall explain later that {H_1} may be written with the 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.

For

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

This is the simple approach, which is allowed for 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}.