In scientific research, the null hypothesis (often denoted H0) is the claim that no relationship exists between two sets of data or variables being analyzed. The null hypothesis is that any experimentally observed difference is due to chance alone, and an underlying causative relationship does not exist, hence the term "null." In addition to the null hypothesis, an alternative hypothesis is also developed, which claims that a relationship does exist between two variables.
Basic definitions
The null hypothesis and the alternative hypothesis are types of conjectures used in statistical tests, which are formal methods of reaching conclusions or making decisions on the basis of data. The hypotheses are conjectures about a statistical model of the population, which are based on a sample of the population. The tests are core elements of statistical inference, heavily used in the interpretation of scientific experimental data, to separate scientific claims from statistical noise.
"The statement being tested in a test of statistical significance is called the null hypothesis. The test of significance is designed to assess the strength of the evidence against the null hypothesis. Usually, the null hypothesis is a statement of 'no effect' or 'no difference'." It is often symbolized as H0.
The statement that is being tested against the null hypothesis is the alternative hypothesis. Symbols include H1 and Ha.
Statistical significance test: "Very roughly, the procedure for deciding goes like this: Take a random sample from the population. If the sample data are consistent with the null hypothesis, then do not reject the null hypothesis; if the sample data are inconsistent with the null hypothesis, then reject the null hypothesis and conclude that the alternative hypothesis is true."
The following adds context and nuance to the basic definitions.
Given the test scores of two random samples, one of men and one of women, does one group differ from the other? A possible null hypothesis is that the mean male score is the same as the mean female score:
H0: μ1 = μ2
where
H0 = the null hypothesis, μ1 = the mean of population 1, and μ2 = the mean of population 2.
A stronger null hypothesis is that the two samples are drawn from the same population, such that the variances and shapes of the distributions are also equal.
Terminology
Simple hypothesis Any hypothesis which specifies the population distribution completely. For such a hypothesis the sampling distribution of any statistic is a function of the sample size alone. Composite hypothesis Any hypothesis which does not specify the population distribution completely. Example: A hypothesis specifying a normal distribution with a specified mean and an unspecified variance.
The simple/composite distinction was made by Neyman and Pearson.
Exact hypothesis Any hypothesis that specifies an exact parameter value. Example: μ = 100. Synonym: point hypothesis. Inexact hypothesis Those specifying a parameter range or interval. Examples: μ ≤ 100; 95 ≤ μ ≤ 105.
Fisher required an exact null hypothesis for testing (see the quotations below).
A one-tailed hypothesis (tested using a one-sided test) is an inexact hypothesis in which the value of a parameter is specified as being either: above or equal to a certain value, or below or equal to a certain value. A one-tailed hypothesis is said to have directionality.
Fisher's original (lady tasting tea) example was a one-tailed test. The null hypothesis was asymmetric. The probability of guessing all cups correctly was the same as guessing all cups incorrectly, but Fisher noted that only guessing correctly was compatible with the lady's claim.
Technical description
The null hypothesis is a default hypothesis that a quantity to be measured is zero (null). Typically, the quantity to be measured is the difference between two situations. For instance, trying to determine if there is a positive proof that an effect has occurred or that samples derive from different batches.
The null hypothesis states that a quantity (of interest) is larger or equal to zero and smaller or equal to zero. If either requirement can be positively overturned, the null hypothesis is "excluded from the realm of possibilities".