Which of these statements is true about hypothesis testing?[Fill in the blank]

Question

BlackTom AI · Accepted Answer

Type I error is P(reject H0 | H0 is true) = α, and Type II error is P(fail to reject H0 | H0 is false) = β. These definitions are the central concepts for the choices given.

1) Relevant knowledge points and concepts
- Definition of Type I: rejecting a true null hypothesis (probability α, the significance level).
- Definition of Type II: failing to reject a false null hypothesis (probability β).
- Relationship between α and β: for a fixed sample size, decreasing α tends to increase β (a trade-off), unless sample size is increased.
- Interpretation of “reject” vs “fail to reject”: rejecting H0 provides evidence in favor of the alternative; failing to reject H0 indicates insufficient evidence for the alternative.

2) Step-by-step reasoning
- Translate each option into the formal definitions:
  a) “Failing to reject H0 when H0 is false” = P(fail to reject H0 | H0 false) = β → Type II. This directly matches the definition.
  b) “Lowering the significance level increases probability of Type I error.” Lowering α reduces P(reject H0 | H0 true), so it decreases Type I error, not increases. So (b) is false.
  c) “Rejecting H0 means we have insufficient evidence for the alternative.” Rejecting H0 means we have sufficient evidence against H0 and therefore support for the alternative; the statement reverses the correct logic. So (c) is false.
  d) “Decreasing Type I error also decreases Type II error.” For a fixed sample size, decreasing α usually increases β (opposite). Only by increasing sample size can you reduce both α and β. So (d) is false in general.

3) Intermediate/formula steps shown
- α = P(reject H0 | H0 true)
- β = P(fail to reject H0 | H0 false)
- Option (a) maps exactly to the expression for β.

4) Why the correct answer is correct
Option (a) restates the definition of a Type II error exactly: failing to reject a false null hypothesis is the definition of Type II.

5) Why other options are incorrect (brief)
- (b) is incorrect: lowering α decreases Type I error.
- (c) is incorrect: rejecting H0 implies sufficient evidence for the alternative, not insufficient.
- (d) is incorrect: decreasing Type I error typically increases Type II error unless sample size changes.

Conclusion: The correct statement is a. Failing to reject the null hypothesis when the null is false in reality results in a Type II error, which matches the provided answer.

Which of these statements is true about hypothesis testing?[Fill in the blank]单项选择题

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