Which of the following statements about Type 1 and Type 2 error is TRUE?

Question

BlackTom AI · Accepted Answer

Type I error (α) is the probability of rejecting the null hypothesis when it is true; Type II error (β) is the probability of failing to reject the null hypothesis when the alternative is true. Statistical power = 1 − β (the probability of correctly rejecting a false null).

Step-by-step reasoning and demonstration of the trade-off:
1. Definitions: α = P(reject H0 | H0 true). β = P(fail to reject H0 | H1 true). Power = 1 − β.
2. All else equal (fixed sample size, fixed effect size, fixed variance), changing the decision threshold to reduce α makes the rejection region smaller. That directly reduces the chance of rejecting under H1 as well, so β increases.
3. Numerical illustration: assume under H1 the test statistic is normally distributed with mean 2 and SD 1 (a standardized noncentral parameter of 2). For a one-sided test:
   - If α = 0.05, critical z = 1.645. Power = P(Z > 1.645 − 2) = P(Z > −0.355) ≈ 0.639 → β ≈ 0.361.
   - If α = 0.01, critical z = 2.33. Power = P(Z > 2.33 − 2) = P(Z > 0.33) ≈ 0.370 → β ≈ 0.630.
   This shows lowering α (from 0.05 to 0.01) increases β (from 0.361 to 0.630) when sample size and effect size are unchanged.
4. Why the correct answer is correct: the statement “All else equal, a lower probability of Type 1 error usually means a higher probability of Type 2 error” exactly captures this inverse relationship between α and β for fixed design parameters.
5. Why other options are incorrect:
   - “Null hypothesis significance testing is built around minimising Type 2 error”: incorrect — NHST traditionally controls α (Type I), not directly minimizing β.
   - “Type 1 error is when you get a false negative; Type 2 error is when you get a false positive”: definitions are reversed (Type I = false positive, Type II = false negative).
   - “If the true effect size is small, the probability of a Type 1 error is lower”: α is set by the test threshold and does not decrease just because the true effect is small; small effects increase β (reduce power).
   - “Well-controlled and well-designed studies will allow you to avoid both types of error”: you can reduce both (by increasing sample size, reducing variance), but cannot eliminate both simultaneously for a fixed sample without trade-offs.

Conclusion: The correct choice is “All else equal, a lower probability of Type 1 error usually means a higher probability of Type 2 error,” which matches the provided answer.

Which of the following statements about Type 1 and Type 2 error is TRUE?单项选择题

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