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 $${H_o}$$.