The power curve provides the probability of
A. incorrectly accepting the null hypothesis
B. correctly rejecting the alternative hypothesis
C. correctly rejecting the null hypothesis
D. correctly accepting the null hypothesis
The Correct Answer and Explanation is:
Correct Answer: C. correctly rejecting the null hypothesis
Explanation:
In statistical hypothesis testing, the power curve is a graphical representation of the power of a test, which is defined as the probability of correctly rejecting the null hypothesis (H₀) when it is actually false. In other words, it shows how effective a test is at detecting an effect when there truly is one.
Let’s break down the key concepts to understand this better:
1. Hypothesis Testing Basics:
In any hypothesis test, we make two main hypotheses:
- Null hypothesis (H₀): Assumes no effect or no difference.
- Alternative hypothesis (H₁ or Ha): Assumes there is an effect or a difference.
There are four possible outcomes in hypothesis testing:
- Type I error (α): Incorrectly rejecting a true null hypothesis.
- Type II error (β): Failing to reject a false null hypothesis.
- Correct retention of H₀: When H₀ is true and we do not reject it.
- Correct rejection of H₀ (Power): When H₀ is false and we correctly reject it.
2. What Is Power?
Power is mathematically defined as:
$$
\text{Power} = 1 – \beta
$$
Where β (beta) is the probability of a Type II error (failing to reject a false H₀).
So, power is the probability that the test will detect a true effect when one exists — that is, correctly rejecting H₀ when it’s false.
3. Power Curve Purpose:
A power curve visually shows how the power of a test increases with:
- Larger effect sizes
- Larger sample sizes
- Lower variability
- Higher significance levels (α)
This helps researchers plan studies by determining the sample size needed to achieve a desired power level (commonly 0.8 or 80%).
Summary:
The power curve gives the probability of correctly rejecting the null hypothesis, making Option C the correct answer. This concept is essential in designing effective experiments and ensuring that real differences are not overlooked due to insufficient statistical power.