Power of a hypothesis test mathes probabality mca syllabus for probability and stats
- 1. Power of a hypothesis test
- 2. The power of a hypothesis test is the probability that the test correctly rejects the null hypothesis (H0 ) when the alternative hypothesis (H1 ) is true. In other words, it measures the test’s ability to detect a true effect. Mathematical Definition • The power of a test is given by: • Power=1−β where: • β is the probability of making a Type II error (failing to reject H0when H1is true). • 1−β represents the probability of correctly rejecting H0when H1is true.
- 3. Key Factors Affecting Power 1.Sample Size (n): Larger sample sizes reduce variability, making it easier to detect differences and increasing power. 2.Significance Level (α): A higher significance level (e.g., 0.05 vs. 0.01) increases power but also raises the chance of a Type I error. 3.Effect Size: Larger differences between the null and alternative hypotheses are easier to detect, increasing power. 4.Variability (Standard Deviation, σ): Lower variability in the data leads to higher power. 5.Test Type: One-tailed tests generally have more power than two-tailed tests when the direction of the effect is known.
- 4. Interpretation A high power (e.g., 0.8 or 80%) means the test is likely to detect a real effect. A low power means there's a high chance of missing a real effect (Type II error).
- 5. Problem :- A researcher wants to test if a new tutoring program improves exam scores. Historical data shows scores average 75 (σ = 10). The program is expected to increase scores to 79 (a 4-point gain). Using a sample of 25 students and a 5% significance level, calculate the power of the test. Solution 1.Hypotheses: H0:μ=75H0 :μ=75 (No improvement). H1:μ>75H1 :μ>75 (One-tailed test: scores increase). 2.Significance Level: α=0.05α=0.05. 3.Critical Value: For α=0.05α=0.05 (one-tailed), the critical zz-value is 1.645
- 6. Step :- 4 Find β (Type II Error Probability): β is the probability that we fail to reject H0 when H1 is actually true (i.e., the mean is really 79). We compute the z-score for Xc=78.29 under H1(mean = 79): z=78.29−79 2 =-0.355 From the z-table, the probability of getting a z-score less than -0.355 is 0.3612. Thus, β=0.3612(36.12%), meaning there is a 36.12% chance of failing to detect an improvement.
- 7. Step :- 5 Calculate Power: Using the formula: Power =1−β =1−0.3612 =0.6388 the power of the test is 63.88%. consclusion :- Since the power is around 64%, it's moderately strong, but researchers often prefer power to be at least 80%. If a higher power is needed