20. 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.
Increasing alpha from .01 to .05 increases the power of the test. The higher the level, the more powerful the test.
If the test is one-tailed, the test will be more powerful because one is less likely to miss a statistically significant test if the predicted direction is accurate.
I hope this helps.
a. Ah, the alpha level! It's like changing the rules of hide-and-seek. So, when you increase the alpha level from .01 to .05, you're basically saying, "Okay, I'll give you a bigger head start before I start seeking." This means that the null hypothesis has a higher chance of surviving the test, which lowers the power of the hypothesis test. So, the power decreases when you make it easier for the null hypothesis to hide.
b. Switching from a one-tailed test to a two-tailed test is like going from playing table tennis to playing doubles tennis. Suddenly, you've got twice as many opponents to face! In terms of hypothesis testing, it means you're considering both sides of the distribution for evidence against the null hypothesis. So, more evidence is needed to reject the null hypothesis and support the alternative hypothesis, reducing the power of the test. It's like trying to win a game with double the competition – not an easy task, my friend!
a. Increasing the alpha level from .01 to .05:
The power of a hypothesis test is influenced by the significance level or alpha level. Increasing the alpha level from .01 to .05 means that the test will be more permissive in accepting the alternative hypothesis. This implies that the criteria to reject the null hypothesis becomes less stringent. Consequently, the probability of committing a Type II error (failing to reject the null hypothesis when it is actually false) decreases and the power of the test increases. In other words, a higher alpha level increases the chances of correctly rejecting the null hypothesis when it is indeed false.
b. Changing from a one-tailed test to a two-tailed test:
Changing from a one-tailed test to a two-tailed test affects the power of the hypothesis test. In a one-tailed test, the alternative hypothesis is specified in one direction only (e.g., the mean is greater than a given value). However, in a two-tailed test, the alternative hypothesis is specified in both directions (e.g., the mean is not equal to a given value).
By conducting a two-tailed test, we are essentially testing for the possibility of a significant difference in both directions, which means the significance level is split equally between the two tails. This split in critical region leads to a decrease in the power of the test compared to the one-tailed test because the test has to detect a significant deviation from the null hypothesis in both directions. Consequently, the power of the hypothesis test is generally higher in a one-tailed test compared to a two-tailed test, assuming all other factors are held constant.
a. Increasing the alpha level from .01 to .05: The power of a hypothesis test is the probability of correctly rejecting the null hypothesis when it is indeed false. When we increase the alpha level from .01 to .05, it means that we are allowing for a larger probability of making a Type I error (rejecting the null hypothesis when it is actually true). This means that we are being more lenient in our decision-making process and are more likely to reject the null hypothesis, even when it might be true. Consequently, the power of the test increases because the probability of correctly rejecting the null hypothesis, when it is false, also increases.
To calculate the power of the hypothesis test, we need the effect size, sample size, and the alpha level.
b. Changing from a one-tailed test to a two-tailed test: In a one-tailed test, the alternative hypothesis is directional, meaning it specifies whether the population parameter is greater than or less than the value specified in the null hypothesis. On the other hand, in a two-tailed test, the alternative hypothesis is non-directional, meaning it tests whether the population parameter is not equal to the value specified in the null hypothesis.
When we change from a one-tailed test to a two-tailed test, we are considering both directions of deviation from the null hypothesis. This means that the test becomes more stringent, as it considers both possibilities of the alternative hypothesis. Consequently, the power of the test decreases because the null hypothesis has to deviate from the expected value in two opposite directions, making it more difficult to detect a significant difference.
To calculate the power of the hypothesis test, we still need the effect size, sample size, and the alpha level. However, in a two-tailed test, the critical value is divided between the two tails of the distribution, leading to a smaller critical region in each tail and hence a lower power compared to a one-tailed test.