Student's t Test

Inferences about μ1 − μ2 Based on Independent Samples

The null and alternative hypotheses for a two-sided test (i.e., a nondirectional test) regarding equality of population means μjs (j = 1,2) for the two groups (e.g., treatment [group 1] versus control [group 2]) are:

The directional hypotheses are:

or

[Page 1462]To test either null hypothesis (only one set of null and alternatives hypotheses are used in any one situation), the Student's two independent sample test statistic is the statistic of choice, and is equal to

where

and are the means of the samples of observations from populations one and two, respectively, E( −) refers to the expected value of the difference between the two sample means, a value given by the null hypothesis, which typically is stipulated to be zero (though the null hypothesis can be that the population difference equals any real number, e.g., H0: μ1 − μ2 = 100), and

est

stands for the estimate of the variance of the difference between two independent sample means.

Because the standard deviation of the difference between two independent sample means equals

The test statistic was derived under the restriction that the population variances for the groups are equal. Accordingly, the statistic can be expressed as

where σ20 is the common value of variance, and N1 and N2 are the sample sizes for groups 1 and 2, respectively.

Thus, the t statistic, given the usual null hypothesis of equal population means, is equal to

where

and s21 and s22 are the unbiased estimates of the common population variance from groups 1 and 2, respectively, and are thus pooled into one estimate, s2p.

William Gosset, publishing under the pseudonym of “Student,” derived the probability density function for a t variable, which is

This probability density function is often called the t distribution. An examination of the density function reveals a lot about characteristics of the distribution of t variables. First, the probability of observing any value of t (the statistic), by chance and chance alone, is determined by just one parameter ν (nu). Thus, the t distribution is referred to as a one parameter family of distributions; that is, by knowing ν, which is also referred to as the degrees of freedom, one can determine the probability of having observed a value of t, that is, the value of the test statistic. One should also note that the value of t enters the density function as a squared value; consequently, the distribution of t values must be symmetric; that is, positive and negative values of t with the same squared value have the same probability of occurrence. In addition, because all the constants in the density function are positive and the term involving t (in the square brackets) is raised to a negative value, the largest positive density value is when t = 0. From the previous equation, it can be observed that that the t distribution is a symmetric unimodal distribution, bell shaped in form, resembling a normal distribution.

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