Definition:Alternative Hypothesis

Definition

In the context of a hypothesis test, the alternative hypothesis, usually denoted $H_1$ or $\mathrm H_1$, is the hypothesis which is to be accepted if the null hypothesis is demonstrated not to hold.

Examples

Arbitrary Example

Let $\delta$ be a hypothesis test.

Let $H_0$ and $H_1$ be the null hypothesis and alternative hypothesis of $\delta$ respectively.

Let $\theta$ be the test statistic which is being used to determine whether $H_0$ or $H_1$ holds.

Let the null hypothesis state that $T = 2$:

$H_0: \theta = 2$

The alternative hypothesis may be, for example:

$H_1: \theta \ne 2$

or that:

$H_1: \theta > 2$

Also see

• Definition:Null Hypothesis

• Results about the alternative hypothesis can be found here.

Sources

• 1989: Ephraim J. Borowski and Jonathan M. Borwein: Dictionary of Mathematics ... (previous) ... (next): hypothesis testing
• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): hypothesis testing
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): alternative hypothesis
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): hypothesis testing
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): alternative hypothesis
• 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): hypothesis testing
• 2021: Richard Earl and James Nicholson: The Concise Oxford Dictionary of Mathematics (6th ed.) ... (previous) ... (next): alternative hypothesis