AP Statistics 3.8 Potential Errors When Performing Tests- Exam Style Questions - MCQs - New Syllabus
▶️ Answer/Explanation
A Type II Error occurs when the null hypothesis is false but we fail to reject it. The probability of a Type II Error (\(\beta\)) becomes smaller when it is easier for a test to detect a true difference from the null hypothesis value.
If the true parameter value moves farther away from the value specified in the null hypothesis, the effect becomes larger and easier to detect. This increases the power of the test and therefore decreases the probability of a Type II Error.
In contrast, decreasing sample size, decreasing \(\alpha\), or increasing standard error generally make it harder to detect a true effect and tend to increase \(\beta\).
✅ Answer: (D)
▶️ Answer/Explanation
A Type II error occurs when we fail to reject the null hypothesis even though the null hypothesis is false. In this problem:
\(H_0:\mu=64\)
\(H_a:\mu>64\)
A Type II error would happen when the true population mean is actually greater than 64, but the test fails to reject \(H_0\). Choice (B) describes exactly this situation. By contrast, choice (C) describes a Type I error because it involves rejecting a true null hypothesis.
✅ Answer: (B)
▶️ Answer/Explanation
The null hypothesis is: \(H_0:\mu_{\text{boys}}-\mu_{\text{girls}}=0\)
This means there is no difference in the average reading speeds of senior boys and senior girls.
A Type I error would therefore occur if the reading specialist concludes that there is a difference (rejects \(H_0\)) when, in reality, there is no difference between the population means.
In context:
Decision: Conclude there is a difference in average reading speed.
Truth: There is actually no difference in average reading speed.
This description matches choice (D).
✅ Answer: (D)