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In statistical hypothesis testing, the alternative hypothesis (or maintained hypothesis or research hypothesis) is the proposed result of a failed null hypothesis. Whereas the null hypothesis is the assumption of no difference between populations, the alternative hypothesis acts as the negation of this assumption. The alternative hypothesis is the hypothesis that sample observations are influenced by some non-random cause.

In the domain of science two rival hypotheses can be compared by explanatory power and predictive power.

Notation
Typical notation for the alternative hypothesis is

$$\text{H}_1$$ or $$\text{H}_a $$

and typical notation for null hypothesis is

$$\text{H}_0$$

Example
An example of an alternative hypothesis might be that a coin is not evenly weighted such that it lands on heads more than tails (or on tails more than heads). The null hypothesis is that the coin is evenly weighted and will land on heads and tails with equal proportions in the long run.

$$\text{H}_0$$ : The coin is equally weighted

$$\text{H}_1$$ : The null hypothesis is false; the coin is not equally weighted

Suppose we flipped the coin 50 times, resulting in 40 Heads and 10 Tails. Given this result, we would be inclined to reject the null hypothesis, and therefore accept the alternative hypothesis. That is, we would conclude that the coin is probably not fair and balanced.

More details can be found in statistical hypothesis testing.

History
The concept of an alternative hypothesis in testing was devised by Jerzy Neyman and Egon Pearson, and it is used in the Neyman–Pearson lemma. It forms a major component in modern statistical hypothesis testing. However it was not part of Ronald Fisher's formulation of statistical hypothesis testing, and he opposed its use. In Fisher's approach to testing, the central idea is to assess whether the observed dataset could have resulted from chance if the null hypothesis were assumed to hold, notionally without preconceptions about what other model might hold. Modern statistical hypothesis testing accommodates this type of test since the alternative hypothesis can be just the negation of the null hypothesis.

Types of alternative hypothesis
In the case of a scalar parameter, there are four principal types of alternative hypothesis:
 * Point. Point alternative hypotheses occur when the hypothesis test is framed so that the population distribution under the alternative hypothesis is a fully defined distribution, with no unknown parameters. Such hypotheses are usually of no practical interest but are fundamental to theoretical considerations of statistical inference and are the basis of the Neyman–Pearson lemma.
 * One-tailed directional. A one-tailed directional alternative hypothesis is concerned with the region of rejection for only one tail of the sampling distribution.
 * Two-tailed directional. A two-tailed directional alternative hypothesis is concerned with both regions of rejection of the sampling distribution.
 * Non-directional. A non-directional alternative hypothesis is not concerned with either region of rejection, but, rather, it is only concerned that the null hypothesis is not true. Non-directional alternative hypotheses are the most common when attempting to strictly reject the null hypothesis. For a alternative hypothesis that a coin is not balanced, observers need only find that one side appears proportionally more to reject the null hypothesis (and accept the alternative). Wether it is the heads or tails that the coin is unbalanced for does not matter.