# 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 ${H_o}$.