Computes statistics for mean and variance inferences using samples from two normal populations.

The routine TWOMV computes the statistics for making inferences about the means and variances of two normal populations, using independent samples in X and Y. For inferences concerning parameters of a single normal population, see routine UVSTA. For two samples that are paired, see routine ATWOB (see Chapter 3, “Correlation”), since the pairs can be considered to be blocks.

Let μX and

be the mean and variance, respectively, of the first population, and μY and

be the corresponding quantities of the second population. The routine TWOMV is used for testing μX = μY and

or for setting confidence intervals for μX ‑ μY and

The basic quantities in STAT(1) through STAT(4) are

where nx and ny are the respective sample sizes (in STAT(5) and STAT(6)).

The test for the equality of means of two normal populations depends on whether or not the variances of the two populations can be considered equal. If the variances are equal, the test is the two-sample t test, which is equivalent to an analysis of variance test see (Chapter 4, “Analysis of Variance”). In this case, the statistics returned in STAT(7) through STAT(12) are appropriate for testing μX = μY. The pooled variance (in STAT(7)) is

The t statistic (in STAT(8)) is

For testing μX = μY + c, for some constant c, the confidence interval for μ X ‑ μY can be used. (If the confidence interval includes c, the null hypothesis would not be rejected at the significance level 1 ‑ CONPRM/100.)

If the population variances are not equal, the ordinary t statistic does not have a t distribution; and several approximate tests for the equality of means have been proposed. (See, for example, Anderson and Bancroft 1952, and Kendall and Stuart 1979.) The name Fisher-Behrens is associated with this problem, and one of the earliest tests devised for this situation is the Fisher-Behrens test, based on Fisher’s concept of fiducial probability. Another test is called Satterthwaite’s procedure. The routine TWOMV computes the statistics for this approximation, which was suggested by H.F. Smith and modified by F.E. Satterthwaite (Anderson and Bancroft 1952, page 83). The test statistic is

where

Under the null hypothesis of equal population means, this quantity has an approximate t distribution with degrees of freedom f (in STAT(17)), given by

The F statistic for testing the equality of variances is given by

is the larger of

is the smaller. If the variances are equal, this quantity has an F distribution with nx ‑ 1 and ny ‑ 1 degrees of freedom.

It is generally not recommended that the results of the F test be used to decide whether to use the regular t test or the modified tʹ on a single set of data. The more conservative approach is to use the modified tʹ (Satterthwaite’s procedure) if there is doubt about the equality of the variances.

This example is taken from Conover and Iman (1983, page 294). It involves scores on arithmetic tests of two grade school classes. The question is whether a group taught by an experimental method has a higher mean score. The data are shown below.

It is assumed that the variances of the two populations are equal so the statistics of interest are in STAT(8) and STAT(9). It is seen from the output below that there is strong reason to believe that the two means are different (t-value of ‑4.804). Since the lower 97.5% confidence limit does not include zero, the null hypothesis that μx ≤ μy would be rejected at the 0.05 significance level. (The closeness of the values of the sample variances provides some qualitative substantiation of the assumption of equal variances.)

For a second example, the same data set is used to illustrate the use of the IDO parameter to bring in the data one observation at a time. Since there are more “Y” values than “X” values, NROWX is set to zero on the later calls to TWOMV.