Bootstrap Confidence Intervals and P-Value Interpretation

Bootstrap Confidence Intervals

Understanding Confidence Intervals Through Percentiles

• Definition: A confidence interval (CI) is constructed using percentiles of simulated values to estimate a population parameter. It defines a range within which the true population parameter is likely to fall.

• Structure: Comprises a lower bound (LB) and an upper bound (UB), which are simulated values that encapsulate a specified middle percentage of the most frequently simulated values.

• Example: $90\%$ Confidence Interval:

  • The middle $90\%$ area contains the most frequently simulated values.

  • The remaining $10\%$ of simulations are equally divided into two tails: a lower tail and an upper tail.

  • Each tail represents $5\%$ of the simulated values.

  • Percentile Rank of Lower Bound: The lower bound is the $5^{\text{th}}$ percentile of the simulations. This means it is greater than $5\%$ of all simulated values.

  • Percentile Rank of Upper Bound: The upper bound is the $95^{\text{th}}$ percentile of the simulations. This is because it is greater than the $5\%$ in the lower tail plus the $90\%$ in the middle, totaling $95\%$ (5\% + 90\% = 95\%$).

• General Principle: The percentile rank indicates the proportion of values that are less than a given number.

Bootstrap Simulation Procedure

• Objective: To generate a distribution of simulated proportions (p\hat{\text{s}}$) that reflect the variability expected from repeated sampling from the original sample's characteristics.</p></li><li><p><strong>Steps (Example: Vision Impairment)</strong>:</p><ol><li><p><strong>Represent the Sample</strong>: If$75$out of$300$$ residents have vision impairment, create a conceptual