10. Explain how the power of a hypothesis test is influenced by each of the following. Assume that all other factors are held constant.
a. Increasing the alpha level from .01 to .05.
b. Changing from a one-tailed test to a two-tailed test.
a. Increasing the alpha level from .01 to .05:
To understand how increasing the alpha level from .01 to .05 influences the power of a hypothesis test, we need to first understand what the alpha level represents. The alpha level, also known as the level of significance, is a predetermined threshold that is used to determine whether to reject or fail to reject the null hypothesis.
In hypothesis testing, we start with a null hypothesis (H0) that assumes no effect or relationship, and an alternative hypothesis (H1) that postulates the presence of an effect or relationship. The goal of the hypothesis test is to gather evidence from the sample data to either reject the null hypothesis in favor of the alternative hypothesis or fail to reject the null hypothesis.
The alpha level represents the probability of making a Type I error, which is rejecting the null hypothesis when it is actually true. A lower alpha level, such as .01, reduces the probability of making a Type I error but increases the chances of committing a Type II error, which is failing to reject the null hypothesis when it is actually false.
By increasing the alpha level from .01 to .05, we are increasing the probability of making a Type I error. This means that we are becoming more lenient in rejecting the null hypothesis. As a result, the power of the hypothesis test increases.
Power, in this context, refers to the ability of the test to correctly reject the null hypothesis when it is false. Increasing the alpha level to .05 widens the acceptance region for the test statistic, making it easier to reject the null hypothesis. Thus, more alternative hypotheses that are actually true will be correctly detected, resulting in an increase in power.
b. Changing from a one-tailed test to a two-tailed test:
A one-tailed test is used when we have a specific direction or inequality in mind for the alternative hypothesis, whereas a two-tailed test is used when we are interested in detecting any kind of difference or relationship, regardless of its direction.
When we change from a one-tailed test to a two-tailed test, it has implications for the distribution of our test statistic and the critical region(s) we use to determine if the null hypothesis should be rejected.
In a one-tailed test, the critical region is located entirely on one side of the distribution, while in a two-tailed test, the critical region is split into two equal areas, on both sides of the distribution.
By changing from a one-tailed test to a two-tailed test, we are reallocating a portion of the alpha level from one tail to the other tail. This means that the area of each tail is reduced, resulting in a narrower critical region for each tail.
As a consequence, the power of the hypothesis test is affected. Since the critical region becomes narrower, it becomes harder to reject the null hypothesis based on the observed test statistic. This means that the test becomes less sensitive to detecting the alternative hypothesis, leading to a decrease in power.
In summary, increasing the alpha level from .01 to .05 increases the power of a hypothesis test, as it becomes more lenient in rejecting the null hypothesis. On the other hand, changing from a one-tailed test to a two-tailed test decreases the power of the hypothesis test, as it narrows the critical region and makes it harder to reject the null hypothesis.
a. Increasing the alpha level from .01 to .05:
The power of a hypothesis test is influenced by the significance level, also known as the alpha level. By increasing the alpha level from .01 to .05, the power of the test generally increases. This is because a higher alpha level allows for a larger critical region, which means there is a greater chance of rejecting the null hypothesis when it is actually false. In other words, the test becomes more likely to detect a true alternative hypothesis. However, it is important to note that increasing the alpha level also increases the probability of making a Type I error (rejecting the null hypothesis when it is actually true). So, while increasing the alpha level can increase the power of the test, it is accompanied by a higher risk of making a Type I error.
b. Changing from a one-tailed test to a two-tailed test:
The power of a hypothesis test is also influenced by whether the test is one-tailed or two-tailed. Changing from a one-tailed test to a two-tailed test can potentially decrease the power of the test. This is because in a one-tailed test, all the critical region is concentrated on one side of the distribution, allowing for a higher chance of detecting a true alternative hypothesis. On the other hand, in a two-tailed test, the critical region is divided between both tails of the distribution, reducing the area available for each tail. As a result, the power of the test decreases as there is less sensitivity to detecting significant differences in both directions of the distribution. However, it is important to choose the appropriate test (one-tailed or two-tailed) based on the research question and hypothesis being tested.