Comparing the power of one-tailed versus two-tailed hypothesis tests.

📚 Quick Study Guide

• 📐 A hypothesis test evaluates evidence for or against a statement about a population.
• 🧭 The null hypothesis ($H_0$) is a statement of no effect or no difference, while the alternative hypothesis ($H_1$) proposes an effect or difference.
• ➡️ A one-tailed test is used when the alternative hypothesis specifies a direction of the effect (e.g., greater than or less than).
• ⬅️ A two-tailed test is used when the alternative hypothesis does not specify a direction (e.g., not equal to).
• 🎯 The p-value is the probability of observing a test statistic as extreme as, or more extreme than, the one computed, assuming the null hypothesis is true.
• 📊 The significance level ($\alpha$) is the probability of rejecting the null hypothesis when it is true (Type I error). Common values are 0.05 and 0.01.
• 🧮 When using a one-tailed test, all of the significance level is placed in one tail of the distribution. For a two-tailed test, the significance level is split equally between both tails.
• 💡 For example:
  • One-tailed (right): $H_0: \mu = 0$, $H_1: \mu > 0$
  • One-tailed (left): $H_0: \mu = 0$, $H_1: \mu < 0$
  • Two-tailed: $H_0: \mu = 0$, $H_1: \mu \neq 0$

Practice Quiz

1. Which of the following is true about a one-tailed test?
   1. It tests for differences in both directions.
   2. It tests for a difference in only one direction.
   3. It always has a higher p-value than a two-tailed test.
   4. It is less powerful than a two-tailed test.
2. When should you use a two-tailed test?
   1. When you expect the effect to be positive.
   2. When you expect the effect to be negative.
   3. When you are unsure of the direction of the effect.
   4. When you have a small sample size.
3. What happens to the significance level ($\alpha$) in a two-tailed test?
   1. It is concentrated in one tail.
   2. It is divided equally between both tails.
   3. It is ignored.
   4. It is doubled.
4. Which hypothesis is tested in a hypothesis test?
   1. Alternative Hypothesis
   2. Null Hypothesis
   3. Both Null and Alternative Hypotheses
   4. Neither Null nor Alternative Hypotheses
5. In a one-tailed test (right-tailed), the critical region is located:
   1. On both sides of the distribution.
   2. On the left side of the distribution.
   3. On the right side of the distribution.
   4. In the center of the distribution.
6. If a p-value is 0.03 and the significance level ($\alpha$) is 0.05, what is the conclusion?
   1. Fail to reject the null hypothesis.
   2. Reject the null hypothesis.
   3. Accept the null hypothesis.
   4. The test is inconclusive.
7. Which test will have more power to detect an effect if the direction of the effect is known?
   1. Two-tailed test
   2. One-tailed test
   3. Both tests have equal power
   4. Power is not related to the type of test