Bootstrapping & Statistical Inference Overview

• Bootstrapping is a resampling technique for estimating the distribution of a statistic using a single sample.

• It helps estimate population parameters and construct confidence intervals when only one sample is available.

Key Concepts in Sampling Distributions

• Population Mean: Average of the entire population.

• Sample Mean: Average from a sample, an estimate of the population mean.

• Sampling Distribution: Theoretical distribution of sample means from all possible samples of a fixed size, centered at the population mean and approximating normality with large samples.

• Standard Error: Reflects variability of sample means, decreasing with larger samples.

Bootstrapping Process

1. Generate Bootstrap Sample: Draw observations with replacement until a sample equal to the original size is formed.

   • Randomly draw an observation from the original sample (which was drawn from the population)

   • Record the observation's value

   • Return the observation to the original sample

   • Repeat the above the same number of times as there are observations in the original sample

2. Point Estimation: Calculate the statistic (mean, median, etc.) for each bootstrap sample.

3. Construct Bootstrap Distribution: Repeat the previous steps (e.g., 10,000 iterations) to create a distribution of statistics for analysis.

Bootstrap distribution for a point estimate: a list of point estimates calculated from bootstrap samples drawn with replacement from a single sample (that was drawn from the population)

Producing Confidence Intervals

• Percentile Method: Sort bootstrap estimates and identify percentiles (2.5th and 97.5th) to form a 95% confidence interval, e.g., $119.28 to $203.63 for a mean of $155.80.

• Confidence Level Impact: Higher confidence levels lead to wider intervals, enhancing certainty but reducing precision.

Limitations of Bootstrapping

• Relies on the original sample being representative; biased samples yield biased estimates.

• Cannot replace true sampling distribution for actual variability assessments.

Applications in R

• Use R functions like rep_sample_n for bootstrap resampling and mean calculations.

• Visualize bootstrap distributions against theoretical distributions for clarity.

Summary of Bootstrapping Benefits

• Enables estimation of statistics and uncertainty from a single sample.

• Supports confidence interval computation, even without full population data.

Conclusion

• Bootstrapping is crucial in data science for assessing uncertainty in sample estimates, preparing individuals for advanced analysis in real-world scenarios.

Code Function Table