One- and Two-Tailed Tests

The one-tailed or two-tailed test is a part of a much more elaborate procedure called hypothesis testing or tests of statistical significance. Prior to doing a hypothesis test, a researcher will have formulated a problem, identified the variables, formulated the hypotheses, and collected the data.

In hypothesis testing, there are five basic steps:

• Stating the null hypothesis
• Stating the alternative hypothesis
• Computing the test statistic
• Formulating the decision rule and making a decision
• Drawing a conclusion

At the very basic level, the researcher would be comparing either a sample mean against a population mean or the difference between two sample means. Hypothesis tests are not restricted to tests on means. Other comparisons and evaluations could involve variances, proportions, and correlations, to name a few. Furthermore, these tests are not restricted to comparing only two means.

The statement of the alternative hypothesis is a statement expressed in statistical terms concerning an investigator's research interest. For example, the investigator may have a question concerning the effects of alcohol consumption on perceptual judgment. If the investigator feels from personal experience or observation that alcohol would have a negative effect on perceptual ability, the alternative hypothesis would reflect that. For research problems in which the data are considered parametric, the null and alternative hypotheses would be expressed by the use of population parameters. The simplest case would be between two means, μ1 and μ2.

In the example, alcohol is given to an experimental group of participants, μ1 and a placebo, an inactive simulation of alcohol, is given to a control group, μ2. Each participant is measured on the number of correct judgments made on a perceptual task. The alternative hypothesis would state that the group receiving alcohol (the experimental group) would perform worse than the group receiving the placebo (the control group). Statistically, the alternative hypothesis would be written as H1: μ1 −μ2 < 0 or H1: μ1 < μ2. The alternative hypothesis is not directly testable. Note that it just says that one group is worse than the other. It does not say by how much. To test this hypothesis directly would require a large, perhaps infinitely large, number of tests in which each difference was to be tested. It is much easier to create a hypothesis that is the opposite of the alternative hypothesis and, with empirical evidence, demonstrate that it cannot be tenable. If this opposite hypothesis is not tenable, then by inference, the alternative hypothesis must be true. This opposite hypothesis is called the null hypothesis. Some part of the null hypothesis points directly to a testable value, such as zero. So for this example, the null hypothesis, written as H0, is H0: μ1 −μ2 ≥ 0. Note that the null hypothesis contains the equal sign.

The null hypothesis is used to help direct the hypothesis test. The investigator would assume the null hypothesis to be true and then, through empirical data, demonstrate that it cannot be tenable. As a result

of this, the alternative hypothesis is shown to be tenable. However, it is the alternative hypothesis that dictates whether the test against the null hypothesis should be one-sided or two-sided. Whenever the investigator's research question indicates that one treatment group is better, more improved, more impaired, weaker, faster, or something else along these lines, than another group, the alternative hypothesis would be considered as one-tailed, or one-sided. When no direction is given by the investigator as to which group will be better or worse than the other group, the alternative hypothesis would be two-tailed. Consider the example given. The investigator had hypothesized that alcohol consumption would lead to impaired performance compared to the performance of those who did not consume alcohol. The alternative hypothesis was H1: μ1 < μ2. This is a one-tailed test. If the investigator had hypothesized that those that consumed alcohol would demonstrate improved performance over those with no alcohol, the alternative hypothesis would have been written as H1: μ1 > μ2. This still would have been a one-tailed test. However, if the investigator had stated uncertainty as to whether or not alcohol consumption would lead to a positive or negative change in performance in comparison to those that did not consume alcohol, the test would be two-tailed. It is two-tailed because the uncertainty would make it possible that either μ1 < μ2 or μ1 > μ2 could be true if the null hypothesis were shown not to be tenable. For this situation, the null hypothesis would have been stated as H0: μ1 −μ2 = 0 or H0: μ1 = μ2. Such a test is also called a nondirectional test.

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