Page 183 - Intermediate Statistics for Dummies

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Part III: Comparing Many Means 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 that its
values represent counts or measurements).
The y-values for each population have a normal distribution (however,
their means may be different; that is 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.
Although you have seen t-tests before in your intro stats class, it may be good
to review the main ideas. The t-test tests the hypotheses Ho: µ 1 = µ 2 versus
Ha: µ 1 is ≤, ≥, or ≠µ 2 , where the situation dictates which of these hypotheses
you use. (Just a note that with ANOVA, you extend this idea to k different
means from k different 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
populations, using two independent samples. To form the test statistic (the
t-statistic), 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 1 + n 2 – 2 degrees of freedom and find the p-value.
If the p-value is less than the prespecified 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.