AP Stats Hypothesis Testing 2 — Questions and Answers
Question 1: A researcher conducts a two-sided t-test and obtains a p-value of 0.03. Which conclusion is most appropriate at the α = 0.05 level?
- Fail to reject H₀; there is insufficient evidence.
- Reject H₀; the results are statistically significant. (Correct answer)
- Accept H₀; the null hypothesis is true.
- Reject Hₐ; the alternative hypothesis is false.
Correct answer: Reject H₀; the results are statistically significant.
Since 0.03 < 0.05, we reject H₀ and conclude the results are statistically significant at the 5% level.
Question 2: In a hypothesis test, a Type II error occurs when you:
- Reject a true null hypothesis.
- Fail to reject a false null hypothesis. (Correct answer)
- Set α too small.
- Use a two-sided test instead of one-sided.
Correct answer: Fail to reject a false null hypothesis.
A Type II error (β) is failing to reject H₀ when H₀ is actually false — missing a real effect.
Question 3: The power of a hypothesis test is best described as:
- The probability of a Type I error.
- 1 minus the probability of a Type II error. (Correct answer)
- The probability of failing to reject a true H₀.
- α divided by the sample size.
Correct answer: 1 minus the probability of a Type II error.
Power = 1 − β, the probability of correctly rejecting a false null hypothesis.
Question 4: Which of the following actions will INCREASE the power of a hypothesis test, all else equal?
- Decreasing the sample size.
- Decreasing the significance level α.
- Increasing the sample size. (Correct answer)
- Using a two-sided test instead of one-sided.
Correct answer: Increasing the sample size.
Larger samples reduce variability, making it easier to detect a true difference and thus increasing power.
Question 5: A one-proportion z-test requires which condition regarding expected counts?
- np ≥ 5 and n(1−p) ≥ 5 using the null proportion. (Correct answer)
- The sample mean must be at least 30.
- The population must be normally distributed.
- The sample size must exceed 100.
Correct answer: np ≥ 5 and n(1−p) ≥ 5 using the null proportion.
For a z-test on proportions, both np₀ and n(1−p₀) must be at least 10 (some curricula use 5) to ensure the sampling distribution is approximately normal.
Question 6: A paired t-test is preferred over a two-sample t-test when:
- The two groups have equal variances.
- Observations in the two groups are naturally matched or related. (Correct answer)
- The sample sizes are very large.
- The data are categorical.
Correct answer: Observations in the two groups are naturally matched or related.
Paired t-tests account for within-pair correlation by analyzing differences, reducing extraneous variability.
Question 7: If you reduce α from 0.05 to 0.01 without changing anything else, what happens to the probability of a Type II error?
- It decreases.
- It stays the same.
- It increases. (Correct answer)
- It equals α.
Correct answer: It increases.
Lowering α makes it harder to reject H₀, so the probability of failing to reject a false H₀ (Type II error) increases.
AP Stats Hypothesis Testing 2
A researcher conducts a two-sided t-test and obtains a p-value of 0.03.
Which conclusion is most appropriate at the α = 0.05 level?