Student Center | Instructor Center | Information Center | Home

Statistics for the Behavioral Sciences, 4/e

Michael Thorne, Mississippi State University -- Mississippi State
Martin Giesen, Mississippi State University -- Mississippi State

Significance of the Difference Between Two Sample Means

Chapter Overview

The chapter begins with the derivation of the sampling distribution of the differences for independent samples. First, pairs of random samples are taken from a population, and means for some characteristic are computed. Next, the difference between the pairs of means is determined. A frequency distribution of the differences is constructed, and the distribution is used to plot a frequency polygon. The distribution is called the sampling distribution of the mean differences.

The properties of the sampling distribution of the mean differences are (1) its mean is equal to 0; (2) the larger the size of the samples drawn from the population, the more closely the sampling distribution approximates the normal curve; (3) the larger the size of the samples, the smaller the standard deviation of the sampling distribution. The standard deviation is called the standard error of the difference between means and is symbolized by

<a onClick="window.open('/olcweb/cgi/pluginpop.cgi?it=gif:: ::/sites/dl/free/0072832517/55319/chap10.gif','popWin', 'width=NaN,height=NaN,resizable,scrollbars');" href="#"><img valign="absmiddle" height="16" width="16" border="0" src="/olcweb/styles/shared/linkicons/image.gif"> (1.0K)</a> Both raw-score and defining formulas for estimating the standard error of the differences are given. The estimated standard error is symbolized by

As in Chapter 9, hypothesis testing involves computing how far from the mean of the sampling distribution our observed difference in sample means lies in estimated standard error units. We first compute a t score and then compare the computed t score with values known to cut off the deviant 5% (or 1%) of the distribution of t for samples of a given size converted to degrees of freedom. The t score or t ratio is the ratio of the difference between a pair of sample means to the estimated standard error of the differences. The degrees of freedom for the t test for independent samples is N₁ + N₂ – 2, and the critical values again are found in Table B.

A two-tailed test looks at both ends of the distribution of the test statistic, t in this chapter. Only one end of the distribution is considered in the one-tailed test. The end considered in the one-tailed test is the one predicted by the experimenter before conducting the experiment. The one-tailed test is a more powerful test if the outcome of the experiment is in the predicted direction. If you can reasonably predict the outcome’s direction before data collection, use of the one-tailed test is warranted because it is more powerful. To determine the one-tailed probabilities, halve the probability values in Table B.

Assumptions required by the two-sample t test for independent samples are that the populations from which the samples are drawn are normal, that the population variances are homogeneous (equal), and that the samples are independent. The first two assumptions apparently can be violated with little effect upon the conclusions made with the test. This property of a statistical test to give valid conclusions even when its assumptions are violated is called robustness. If you fear violation of the assumptions, it is recommended that you use fairly large samples of equal size. The power of the test can be increased by using dependent samples. Dependent samples can be formed by using matched pairs of unrelated individuals in which subject pairs, matched as closely as possible on relevant characteristics, are selected. Then, one member of each pair is assigned to one treatment group, and the other member is assigned to the other group. Another procedure for obtaining dependent samples is to use each subject as his or her own control; that is, the same individuals are given each experimental treatment. This type of experiment is called a repeated measures design or within-subjects comparison.

The power of the test using dependent samples is increased through a decrease in the standard deviation of the sampling distribution of mean differences. The standard deviation is called the standard error of the mean differences and is symbolized by

<a onClick="window.open('/olcweb/cgi/pluginpop.cgi?it=gif:: ::/sites/dl/free/0072832517/55319/10c.GIF','popWin', 'width=NaN,height=NaN,resizable,scrollbars');" href="#"><img valign="absmiddle" height="16" width="16" border="0" src="/olcweb/styles/shared/linkicons/image.gif"> (1.0K)</a> is estimated by

<a onClick="window.open('/olcweb/cgi/pluginpop.cgi?it=gif:: ::/sites/dl/free/0072832517/55319/10d.GIF','popWin', 'width=NaN,height=NaN,resizable,scrollbars');" href="#"><img valign="absmiddle" height="16" width="16" border="0" src="/olcweb/styles/shared/linkicons/image.gif"> (0.0K)</a> , which is called the estimated standard error of the mean differences.

The direct difference method is used to compute a t ratio for dependent or related samples. In the final equation discussed, t is computed by dividing the mean of the differences by the standard error of the mean differences. The computed value of t is then compared with the critical values of t from Table B with df = N – 1, where N is the number of pairs of scores.

2003 McGraw-Hill Higher Education
Any use is subject to the Terms of Use and Privacy Notice.
McGraw-Hill Higher Education is one of the many fine businesses of The McGraw-Hill Companies.