Читать книгу Applied Univariate, Bivariate, and Multivariate Statistics - Daniel J. Denis - Страница 60

Just as the t‐test for one sample is a generalization of the z‐test for one sample, for which we use s² in place of σ², the t‐test for two independent samples is a generalization of the z‐test for two independent samples. Recall the z‐test for two independent samples:

where and denote the expectations of the sample means and respectively (which are equal to μ₁ and μ₂).

When we do not know the population variances and , we shall, as before, obtain estimates of them in the form of and . When we do so, because we are using these estimates instead of the actual variances, our new ratio is no longer distributed as z. Just as in the one‐sample case, it is now distributed as t:

(2.6)

on degrees of freedom v = n₁ − 1 + n₂ − 1 = n₁ + n₂ − 2.

The formulization of t in (2.6) assumes that n₁ = n₂. If sample sizes are unequal, then pooling variances is recommended. To pool, we weight the sample variances by their respective sample sizes and obtain the following estimated standard error of the difference in means:

which can also be written as

Notice that the pooled estimate of the variance is nothing more than an averaged weighted sum, each variance being weighted by its respective sample size. This idea of weighting variances as to arrive at a pooled value is not unique to t‐tests. Such a concept forms the very fabric of how MS error is computed in the analysis of variance as we shall see further in Chapter 3 when we discuss the ANOVA procedure in some depth.