Question

Power of a statistical test is the probability of rejecting the null hypothesis when the alternate hypothesis is true.

Which of the following statements is/are CORRECT?

• A. Power is computed as \(1-\beta\), where \(\beta\) is the probability of Type II error
• B. Power is interpreted as the probability of correctly rejecting a false null hypothesis
• C. Power is computed as \(1-\alpha\), where \(\alpha\) is the probability of Type I error
• D. Power is interpreted as the probability of falsely rejecting a correct null hypothesis

Hint:

Use the definitions of Type I error (\(\alpha\)) and Type II error (\(\beta\)) and match Power to \(1-\beta\), the probability of correctly rejecting a false null hypothesis.

Answer 1

The Correct Option is A, B

Step 1: State the definitions involved in hypothesis testing.
In a statistical test of \(H_0\) versus \(H_1\), a Type I error, probability \(\alpha\), is rejecting \(H_0\) when \(H_0\) is actually true, a false alarm. A Type II error, probability \(\beta\), is failing to reject \(H_0\) when \(H_1\), the alternative, is actually true, a missed detection. The question itself defines Power as the probability of rejecting the null hypothesis when the alternate hypothesis is true.

Step 2: Relate Power to \(\beta\).
By definition, \(\beta = P(\text{fail to reject } H_0 \mid H_1 \text{ true})\), and Power \(= P(\text{reject } H_0 \mid H_1 \text{ true})\). Since reject and fail to reject are complementary events given that \(H_1\) is true, \[ \text{Power} + \beta = 1 \implies \text{Power} = 1 - \beta \] This exactly matches option (A), which is therefore CORRECT.

Step 3: Check option (B) against the question own definition.
Option (B) says Power is the probability of correctly rejecting a false null hypothesis. Correctly rejecting a null hypothesis necessarily means the null hypothesis is false, so rejecting it is the right decision, which is the same event as rejecting \(H_0\) when \(H_1\) is true. This is a direct restatement of the question own definition of Power, so (B) is CORRECT.

Step 4: Check option (C).
\(1 - \alpha\) is the probability of correctly failing to reject \(H_0\) when \(H_0\) is true, sometimes called the confidence level or the test specificity, not the Power of the test. Power is defined using \(\beta\), behaviour under \(H_1\), not \(\alpha\), behaviour under \(H_0\). So (C) is INCORRECT.

Step 5: Check option (D).
The probability of falsely rejecting a correct null hypothesis is, by definition, \(\alpha\), the Type I error rate, not the Power of the test. So (D) is INCORRECT.

Step 6: Conclude.
The correct statements are (A) and (B).

\[ \boxed{\text{Options (A) and (B) are correct}} \]