Hypothesis testing compares the data being studied to an observed characteristic of the population from which the data are sampled. This is a method of inferential statistics, and the data must be properly sampled in order for the results of testing to be valid.
The researcher has a proposed hypothesis about a population characteristic and conducts a study to discover if it is reasonable, or, acceptable. The proposed hypothesis is called the alternative hypothesis and is labelled Ha.
The observed characteristic is a value such as a mean, or a proportion, or a variance, that is already known as "true." This value is called a parameter. The null hypothesis states what the parameter is, and is labelled Ho.
The alternative hypothesis claims that the population characteristic is different than the observed parameter. This difference is either that the characteristic has increased, decreased, or, possibly either increased or decreased.
The standard notation for these hypotheses are:
Ho: ε ≤ #
Ha: ε > # (an increase)
Ho: ε ≥ #
Ha: ε < # (a decrease)
Ho: ε = #
Ha: ε ≠ # (either increase or decrease)
- where ε represents the symbol for the parameter.
For example, a study of the mean value would show a μ symbol:
Ho: μ = #
Ha: μ ≠ #
The researcher will measure the sample's characteristic and use it to calculate a test statistic. There are a number of different test statistic formulas, that depend upon what data is used, and which parameter is being tested.
The test is based upon an assumed distribution of the population. The null is made upon this assumption. The test statistic will have a certain likelihood for occuring, according to the distribution being used. When this likelihood is small, this indicates that the sample data are either from an unusual sample, or, that the distribution of the population actually is different than assumed. If the sample is properly drawn, there is small risk that the sample is unusual, and, so, it is safe to draw a conclusion that the distribution may be changed. This allows the conclusion that the null hypothesis may have changed, and that the alternative hypothesis might be accepted instead. This conclusion leads the researcher to "reject" the null hypothesis.
The likelihood that is small "enough" to reject the null is a subjective rule that is determined by the researcher before the test is conducted. This likelihood is called alpha α. Common practice sets alpha to either .01, .05, .10. This is also called a rejection region - referring to the graphed area of the distribution.
An alternative approach to using alpha is to calculate a p-value, which is thought to bring more flexibilty to the conclusion.
The case that the sample data are unusual, and the underlying population actually would have fit the distribution, so, the null hypothesis is rejected in error, is called Type I error. The probability for making this error is equal to the value of alpha, or, to the p-value, whichever has been used to draw the erroneous conclusion.
