by DAN CALLOWAY
Published 13 August 2010
WEAVERVILLE, NC – The t-test is a statistical test for two means. It is most often used to compare the means of two experimental conditions and, thus, two applications can be distinguished. The first one applies to experiments with two independent groups; that is, when subjects are assigned randomly to an experimental group and also when subjects are assigned at random to an experimental group and a control group. The null hypothesis is then that the population mean scores are equal for the two conditions. That is to say, there is no difference if we could compare the entire population of scores for the experimental and the control conditions. H0 : μE – μC = 0, where μE and μC represent the population means of the experimental and the control groups, respectively. Here, the alternative hypothesis would be Ha: μE – μC ≠ 0. If the direction of difference were detected ahead of time, a one-tailed version of Ha would be shown as μE – μC > 0. The t-test assumes that the scores in each condition are normally distributed and that the two distributions have equal variance (Levin, 1999).
When more than two means are analyzed, such as the case with more than two independent variables, then the t-test becomes an inappropriate test and gives way to statistical test of more than two means: analysis of variation, also called the ANOVA. The associated statistic of the ANOVA is the F-test. There are two versions of the ANOVA and its associated statistic. The first version of ANOVA is known as the one-way ANOVA, in which there is only one independent variable, but this independent variable may have many different levels as when you are comparing various dosages of a drug or varying amounts of reinforcement (Levin, 1999). In this case, the null hypothesis is H0: μ1 = μ2 = . . . = μk, where μi is the population mean for level i. Here, the alternative hypothesis would state that the population means are not all equal. The ANOVA is similar to an extension of the t-test, but the t-test is only able to test the means of two variables, each containing one level. If, for example, there were eight groups to be compared, and one used a t-test to compare every possible pair of groups, then one would have to conduct 28 different t-tests. If each individual t-test was conducted with
α = 0.05, then you could easily see that 28 t-tests could easily lead to one or more Type I errors because the expected number of errors would be 28 X 0.05 = 1.40. Thus, instead, a single analysis of variation is preferred because it would not be as likely to lead to a Type I error.
The F-test, in ANOVA, is defined as the ratio of two sample variances, that is to say,
F = s12 / s22; hence the term, analysis of variance. In the case of a one-way ANOVA, the variance term in the numerator is referred to as the between-groups variance because it is a measure of how much the k different group means vary from one another. The variance term in the denominator is called the within-groups variance because it is a measure of the average variance of scores within each experimental condition. Thus, the denominator is a measure of sampling error, while the numerator is a measure of sampling error plus any differences between experimental conditions that go beyond sampling error (Levin, 1999).
Reference:
Levin, I. P. (1999). Relating Statistics and Experimental Design: An Introduction (p. 90). Thousand Oaks, CA: Sage Publications, Inc.
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