hypothesis testing

Hypothesis testingMathworldPlanetmath is a statistical inferencial procedure in which a statement based on some experimental or observational study is formulated, tested, then put through a decision process. The decision process either accepts or rejects the statement.

More precisely, the hypothesis testing procedure can be broken down into three steps:

  1. 1.

    Formulation of a (hypothetical) statement.

    The hypothetical statement formed is called the null hypothesis, or H0. For example, in testing whether a coin is “fair”, it is tossed 100 times and the number of heads are counted. H0 could be { The number of heads = 50}. Accompanying the null hypothesis H0 is the alternative hypothesis Ha. The statement in Ha is the compliment of the statement in H0 (the universe is the sample space). For example, Ha would be { The number of heads 50}.

  2. 2.

    Testing of the statement.

    This is usually the most mathematical part of the procedure. To test H0, first assume H0 is true. Then apply an appropriate test statistic using values obtained from the study. There are many test statistics, depending on H0, Ha, and the nature of the study.

    Based on the distributional forms of these test statistics, four major types of tests are of interest. A t test is based on a test statistic that has a t-distribution. An f test (http://planetmath.org/fdistribution) and a χ2 test (http://planetmath.org/ChiSquaredRandomVariable) are so named for the same reason. A z-test is one that is based on a test statistic having a normal or Gaussian distribution.

    Before calculating the test statistic, a value of the significance level of the test needs to be specified. The significance level, known as α, is the probability of rejecting H0 (or accepting Ha) when in fact, H0 is true: α=P(HaH0).

  3. 3.

    Deciding whether to accept or reject the statement.

    Once the value of the test statistic is obtained, it is used to find a corresponding probability of obtaining such a statisticMathworldMathworldPlanetmath given that H0 is true. This probability is called the p-value. This p-value is then compared α, the significance level of the test. If p-value <α, then the usual next step is to reject the null hypothesis H0 (and Ha accepted). Otherwise, H0 will be accepted.

When a statement (whether it is null hypothesis or the alternative hypothesis) is accepted, it merely says that, statistically, there is not enough evidence to reject the statement. Acceptance of a hypothetical statement does not prove that the underlying statement is true.

The concept of statistical hypothesis testing can be found in any standard introductory statistics textbooks, as well as numerous internet websites (for example, http://www.google.com/search?hl=en&lr=&q=hypothesis+testingclick to find the result of a Google search). The purpose of this entry is to give a very brief description of hypothesis testing and to serve as a link reference for other entries.

Title hypothesis testing
Canonical name HypothesisTesting
Date of creation 2013-03-22 14:42:12
Last modified on 2013-03-22 14:42:12
Owner CWoo (3771)
Last modified by CWoo (3771)
Numerical id 5
Author CWoo (3771)
Entry type Definition
Classification msc 62A01
Related topic ChiSquaredStatistic
Defines null hypothesis
Defines alternative hypothesis
Defines significance level
Defines t test
Defines f test
Defines chi-squared test
Defines chi-square test
Defines χ2 test
Defines p-value