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: Build a 2 by 2 decision table of the four possible test outcomes.
There are two possible truths, $H_0$ true or $H_1$ true, and two possible decisions, reject $H_0$ or fail to reject $H_0$, giving four cells: (i) $H_0$ true and we fail to reject it, correct decision, probability $1-\alpha$; (ii) $H_0$ true but we reject it, a Type I error, probability $\alpha$; (iii) $H_1$ true but we fail to reject $H_0$, a Type II error, a miss, probability $\beta$; (iv) $H_1$ true and we reject $H_0$, a correct detection, probability $1-\beta$.

Step 2: Locate Power in this table.
The question defines Power as the probability of rejecting the null hypothesis when the alternate hypothesis is true, which is exactly cell (iv) above, with probability $1-\beta$. Reading directly off the table, this confirms option (A), Power equals $1-\beta$, is CORRECT, and also confirms option (B), since cell (iv) is precisely correctly rejecting a false null hypothesis, the null is false because $H_1$ is true, and rejecting it is therefore the correct call. So (B) is also CORRECT.

Step 3: Locate the quantities in options (C) and (D) in the same table.
Option (C), $1-\alpha$, corresponds to cell (i), correctly failing to reject a true $H_0$, a completely different row of the table, describing behaviour when $H_0$ is true, not when $H_1$ is true, and unrelated to Power. Option (D), falsely rejecting a correct null hypothesis, is cell (ii), which is just $\alpha$ itself, the Type I error rate, again describing behaviour under $H_0$ true, not under $H_1$ true.

Step 4: Conclude from the table.
Power belongs strictly to the $H_1$ true row of the decision table, so only statements that correctly describe that row, options (A) and (B), can be correct, options (C) and (D) both describe the $H_0$ true row and are therefore wrong descriptions of Power.

\[ \boxed{\text{(A) and (B) correctly describe Power}} \]