One-Tailed Test

Alternative Hypotheses: Directional versus Nondirectional

In hypothesis testing, null hypotheses (H0) are tested against statistical alternative hypotheses (Ha). Alternative hypotheses can be set up as nondirectional or directional. A nondirectional Ha states that the parameter differs from the value in the null hypothesis with no indication of the direction of the difference. For example, given H0 stating that the population mean for reading achievement is 100 (H0: μ = 100), the nondirectional Ha states that the population mean is different from 100 (Ha: μ≠100). Thus, the nondirectional Ha does not specify if the population mean is greater or less than 100. On the other hand, a directional Ha not only states that the parameter deviates from the value in the null hypothesis, but also specifies the direction of the deviation. For the aforementioned H0: μ = 100, a directional Ha can be that the population mean is either greater than 100 (Ha: μ > 100) or less than 100 (Ha: μ < 100).

The type of hypothesis testing in which H0 is tested against a nondirectional Ha is called a two-tailed test, whereas the one in which H0 is tested against a directional Ha is called a one-tailed test. The procedures for conducting one- or two-tailed tests are fundamentally similar. The difference between one- and two-tailed tests lies in the location of the region of rejection in sampling distributions.

Each hypothesis testing, whether one-tailed or two-tailed, starts with setting up the null and alternative hypotheses before collecting data. Thus, researchers need to decide whether they will conduct a one-tailed or a two-tailed test before data collection. The second step in hypothesis testing that takes place before data collection is deciding the level of significance, alpha value. The level of significance is the probability of rejecting H0 when it is actually true (i.e., the probability of Type I error). As with other probability values in hypothesis testing, alpha is obtained from sampling distributions. For a statistic of interest (e.g.,

or SD), a sampling distribution is a distribution of an infinite number of sample statistics where the samples are the same size random samples as the actual sample from a population where H0 is true. In a sampling distribution, the proportion of the area covered by any range of sample statistics represents the probability of that range of sample statistics obtained for a random sample from a population where H0 is true. Accordingly, the alpha value represents the proportion of the area in a sampling distribution that is equal to the probability of Type I error. For example, for a sample of size 50, if the probability of Type I error is 5%, in the sampling distribution, which consists of means for every random sample of size 50, the area covered by the range of values that correspond to Type I error is 5% of the whole area. The area represented by alpha is also referred to as the region of rejection.

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