THE t TEST

Suppose we have a research hypothesis that says "homicide investigators who take a short course on the causes of HIV will be less fearful of the disease than investigators who have not taken the course," and test it by conducting an experiment in which a random sample of investigators is assigned to take the course and another random sample is designated as the control group. Let's suppose that at the end of the experiment the experimental group gets a mean of 16.61 on a fear of HIV scale and the control group gets a mean of 29.67 (where the higher the score, the greater the fear of HIV). These means support our research hypothesis. But can we be certain that our research hypothesis is correct? If you've been reading the topics on statistics in order from the beginning, you already know that the answer is "no" because of the null hypothesis, which says that there is no true difference between the means; that is, the difference was created merely by the chance errors created by random sampling. (These errors are known as sampling errors.) Put another way, unrepresentative groups may have been assigned to the two conditions quite at random.

The t test is often used to test the null hypothesis regarding the observed difference between two means? For the example we are considering, a series of computations (which are beyond the scope of this book) would be performed to obtain a value of t (which, in this case, is 5.38) and a value of degrees of freedom (which, in this case, is df = 179). These values are not of any special interest to us except that they are used to get the probability (p) that the null hypothesis is true. In this particular case, p is less than .05. Thus, in a research report, you may read a statement such as this:

The difference between the means is statistically significant (t = 5.38, df= 179, P < .05V

As you know from Topic 38, the term statistically significant indicates that the null hypothesis has been rejected. You should recall that when the probability that the null hypothesis is true is .05 or less (such as .01 or .001), we reject the null hypothesis. (When something is unlikely to be true because it has a low probability of being true, we reject it.)

Having rejected the null hypothesis, we are in a position to assert that our research hypothesis probably is true (assuming no procedural bias was allowed to affect the results, such as testing the control group immediately after a major news story on a famous person with AIDS, while testing the experimental group at an earlier time).

What leads a t test to give us a low probability? Three things:

1. Sample size. The larger the sample, the less likely that an observed difference is due to sampling errors. (You should recall from the sections on sampling that larger samples provide more precise information.) Thus, when the sample is large, we are more likely to reject the null hypothesis than when the sample is small.

2. The size of the difference between means. The larger the difference, the less likely that the difference is due to sampling errors. Thus, when the difference between the means is large, we are more likely to reject the null hypothesis than when the difference is small.

3. The amount of variation in the population. You should recall from Topic 22 that when a population is very heterogeneous (has much variability) there is more potential for sampling error. Thus, when there is little variation (as indicated by the standard deviations of the sample), we are more likely to reject the null hypothesis than when there is much variation.

A special type of t test is also applied to correlation coefficients. Suppose we drew a random sample of 50 students and correlated their hand size with their GPAs and got an r of .19. The null hypothesis says that the true correlation in the population is O.OO-that we got .19 merely as the result of sampling errors. For this example, the t test indicates that p > .05. Since the probability that the null hypothesis is true is greater than 5 in 100, we do not reject the null hypothesis; we have a statistically insignificant correlation coefficient. (In other words, for n = 50, an r of .19 is not significantly different from an r of 0.00.) When reporting the results of the t test for the significance of a correlation coefficient, it is conventional not to mention the value of t. Rather researchers usually indicate only whether or not the correlation is significant at a given probability level.

¹ You probably recall that we prefer random sampling because it precludes any bias in the assignment of subjects to the groups and because we can test for the effect of random errors with significance tests; we cannot test for the effects of bias.

² To test the null hypothesis between two medians, the median test is used; it is a specialized form of the chi square test, whose results you already know how to interpret.

³ Sometimes researchers leave out the abbreviation df and present the result as t(179) = 5.38, p < .05.

EXERCISE ON TOPIC

1. What does the null hypothesis say about the difference between two sample means?

2. Is the value of t usually of any special interest to consumers of research?

3. Suppose you read that for the difference between two means, t = 2.000, df= 20,p > .05. Using conventional standards, should you conclude that the null hypothesis should be rejected?

4. Suppose you read that for the difference between two means, t = 2.859, df= 40,p < .01. Using conventional standards, should you conclude that the null hypothesis should be rejected? .

5. Based on the information in question 4, should you conclude that the difference between the means is statistically significant?

16. When we use a large sample, are we more or less likely to reject the null hypothesis than when we use a small sample?

7. When the size of the difference between means is large, are we more or less likely to reject the null hypothesis than when the size of the difference is small?

8. If we read that for a sample of 92 subjects, r = .41, P < .001, should we reject the null hypothesis?

9. Is the value of r in question 8 statistically significant?

10. Of the three things that lead to a low probability, which one is most directly under the control of a researcher?