The Null and the Alternative

In statistical hypothesis testing, the alternative hypothesis and the null hypothesis are the two rival hypotheses which are compared by a statistical hypothesis test. An example might be where water quality in a stream has been observed over many years. A test can be made of the null hypothesis (that there is no change in quality between the first and second halves of the data) against the alternative hypothesis (that the quality is poorer in the second half of the record).

The Null Hypothesis

The null hypothesis refers to a general or default position: that there is no relationship between two measured phenomena, or that a potential medical treatment has no effect. Rejecting or disproving the null hypothesis (and thus concluding that there are grounds for believing that there is a relationship between two phenomena or that a potential treatment has a measurable effect) is a central task in the modern practice of science and gives a precise sense in which a claim is capable of being proven false.

The concept of a null hypothesis is used differently in two approaches to statistical inference, though the same term is used, a problem shared with statistical significance. In the significance testing approach of Ronald Fisher, a null hypothesis is potentially rejected or disproved on the basis of data that is significantly under its assumption, but never accepted or proved. In the hypothesis testing approach of Jerzy Neyman and Egon Pearson, a null hypothesis is contrasted with an alternative hypothesis, and these are decided between on the basis of data, with certain error rates.

Sir Ronald Fisher

Sir Ronald Fisher, pictured here, was the first to coin the term null hypothesis.

The Alternative Hypothesis

In the case of a scalar parameter, there are four principal types of alternative hypothesis:

1. 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.
2. One-tailed directional. A one-tailed directional alternative hypothesis is concerned with the region of rejection for only one tail of the sampling distribution.
3. Two-tailed directional. A two-tailed directional alternative hypothesis is concerned with both regions of rejection of the sampling distribution.
4. Non-directional. A non-directional alternative hypothesis is not concerned with either region of rejection, but, rather, only that the null hypothesis is not true.

The concept of an alternative hypothesis forms a major component in modern statistical hypothesis testing; however, it was not part of Ronald Fisher's formulation of statistical hypothesis testing. 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.

The Test

A hypothesis test begins by consider the null and alternate hypotheses, each containing an opposing viewpoint.

$H_0$: The null hypothesis: It is a statement about the population that will be assumed to be true unless it can be shown to be incorrect beyond a reasonable doubt.

$H_a$: The alternate hypothesis: It is a claim about the population that is contradictory to $H_0$ and what we conclude when we reject $H_0$.

Since the null and alternate hypotheses are contradictory, we must examine evidence to decide if there is enough evidence to reject the null hypothesis or not. The evidence is in the form of sample data.

We can make a decision after determining which hypothesis the sample supports (there are two options for a decision). They are "reject $H_0$" if the sample information favors the alternate hypothesis or "do not reject $H_0$" or "fail to reject $H_0$" if the sample information is insufficient to reject the null hypothesis.

Example 1

$H_0$: No more than 30% of the registered voters in Santa Clara County voted in the primary election.

$H_a$: More than 30% of the registered voters in Santa Clara County voted in the primary election.

Example 2

We want to test whether the mean grade point average in American colleges is different from 2.0 (out of 4.0).

$H_0: \mu = 2.0 | H_a: \mu \neq 2.0$

Example 3

We want to test if college students take less than five years to graduate from college, on the average.

$H_0: \mu \geq 5 | H_a: \mu <5$