Alternative Hypothesis

The Alternative and Null Hypotheses: Similarities and Differences

Both the alternative hypothesis (symbolized as H1 or as Ha) and the null hypothesis (symbolized as H0) are statements as to the possible state of affairs in the population(s) of interest. These statements are similar in two other respects: In any given study, both the null and alternative hypotheses must deal with the same statistical concept. Thus, if the null hypothesis deals with the difference between two population means (μ1 and μ2), then the alternative hypothesis must also deal with the difference between μ1 and μ2. Moreover, in the usual applied situation, neither H1 nor H0 can be proven true on the basis of the study's data.

Although the alternative hypothesis and the null hypothesis are alike in certain ways, they differ in three important ways. First, H1 and H0 are “opposites” in the sense that they say different things about a study's population(s). Second, the hypothesis testing procedure is focused more on the null hypothesis than on the alternative hypothesis. The null hypothesis is always stated first, and it is H0 that will or will not be rejected after the sample data are analyzed. Finally, it is the alternative hypothesis (and not H0) that causes a statistical test to be either one-tailed or two-tailed.

Directional and Nondirectional Alternative Hypotheses

The directionality of the alternative hypothesis determines whether a statistical test is conducted in a one-tailed or a two-tailed manner. The alternative hypothesis is said to be directional if it stipulates that the population parameter is positioned on one particular side of the number specified in H0. For example, the alternative hypothesis would be directional if it said that a population mean is greater than 20 while the null hypothesis said that 20 is the value of the population mean. Stated symbolically, this situation could be summarized as follows:

Of course, the alternative hypothesis in this example would also be directional if it were set up to say H1: μ < 20. Regardless of which way the directional H1 points, such alternative hypotheses lead to one-tailed tests. This is because the critical region is positioned entirely in one tail of the test statistic's sampling distribution.

The alternative hypothesis is said to be nondirectional if it stipulates that the population parameter is positioned on either side of the number specified in H0. For example, the alternative hypothesis would be nondirectional if it says that a population correlation, ρ, has either a positive or negative value while the null hypothesis says that ρ is equal to zero. Stated symbolically, this situation could be summarized as

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