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Part III: Analyzing Variance with ANOVA
Comparing Two Means with a t-Test

The two-sample t-test is designed to test to see whether two population means
are different. The conditions for the two-sample t-test are the following:

✓ The two populations are independent. In other words, their outcomes
don’t affect each other.
✓ The response variable (y) is a quantitative variable, meaning its values
have numerical meaning and represent quantities of some kind.
✓ The y-values for each population have a normal distribution. However,
their means may be different; that’s what the t-test determines.
✓ The variances of the two normal distributions are equal.

For large sample sizes when you know the variances, you use a Z-test for the
two population means. However, a t-test allows you to test two population
means when the variances are unknown or the sample sizes are small. This
occurs quite often in situations where an experiment is performed and the
number of subjects is limited. (See your Stats I text or Statistics For Dummies
(Wiley) for info on the Z-test.)
Although you’ve seen t-tests before in your Stats I class, it may be good to
review the main ideas. The t-test tests the hypotheses Ho: μ = μ versus Ha:
1 2
μ is ≤, ≥, or ≠ μ , where the situation dictates which of these hypotheses you
1 2
use. (Note: With ANOVA, you extend this idea to k different means from k dif-
ferent populations, and the only version of Ha of interest is ≠.)
To conduct the two-sample t-test, you collect two data sets from the two popu-
lations, using two independent samples. To form the test statistic (the t-statis-
tic), you subtract the two sample means and divide by the standard error (a
combination of the two standard deviations from the two samples and their
sample sizes). You compare the t-statistic to the t-distribution with n + n – 2
1 2
degrees of freedom and find the p-value. If the p-value is less than the prede-
termined α level, say 0.05, you have enough evidence to say the population
means are different. (For information on hypothesis tests, see Chapter 3.)

For example, suppose you’re at a watermelon seed-spitting contest where
contestants each put watermelon seeds in their mouths and spit them as
far as they can. Results are measured in inches and are treated with the
reverence of the shot-put results at the Olympics. You want to compare the
watermelon seed-spitting distances of female and male adults. Your data set
includes ten people from each group.

You can see the results of the t-test in Figure 9-1. The mean spitting distance
for females was 47.8 inches; the mean for males was 56.5 inches; and the differ-
ence (females – males) is –8.71 inches, meaning the females in the sample spit
seeds at shorter distances, on average, than the males. The t-statistic for the

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