1. **State the problem:** We want to understand the relationship between sample size, power level, and effect size in a multiple linear regression with 4 predictors. 2. **Formula and concepts:** Power analysis for multiple regression uses the effect size $f^2$, significance level $\alpha$, number of predictors $k$, and sample size $N$ to determine the power level. The effect size $f^2$ is defined as $$f^2 = \frac{R^2}{1-R^2}$$ where $R^2$ is the proportion of variance explained. 3. **Given data:** - Medium effect size: $f^2 = 0.15$ - Number of predictors: $k = 4$ - Significance level: $\alpha = 0.05$ - Minimum sample size for power 0.85: $N = 95$ - Capped sample size: $N = 200$ - Power at $N=200$ is 0.95 4. **Interpretation:** - At $N=95$, power is 0.85, meaning an 85% chance to detect the medium effect size. - Increasing $N$ to 200 raises power to 0.95, increasing the likelihood of detecting the effect. 5. **Conclusion:** - The sample size cap of 200 participants ensures a high power level (0.95) for detecting a medium effect size with 4 predictors at $\alpha=0.05$. - This balance allows sufficient time for data analysis while maintaining statistical rigor. This analysis confirms the researchers' choice of sample size cap to optimize power and project timeline.