Bootstrapping & Wrap-up - Lecture Notes

Lecture Overview

• Topic: Bootstrapping & wrap-up

• Course Evaluation: SEI Surveys currently active.

• Final Project and Exam: Comprehensive details provided during the session.

Population Distributions Review

• Presented an extensive overview of the population distribution relating to the price per night for Airbnb listings in Vancouver.

• Discussed the sample distribution derived from an analysis of 40 Airbnb listings, highlighting variations in pricing.

Sampling Distribution of Sample Means

• Analyzed the sampling distribution of sample means across 20,000 samples, each consisting of size 40. Key inquiry points included:

  1. Impact of sample size on the peak and range of the resultant histogram.

  2. Difficulties encountered in establishing an accurate distribution in real-world data analysis.

  3. Potential remedies for challenges associated with sampling.

• General Insight: Smaller sample sizes contribute to a broader spread in data and increased uncertainty.

Bootstrapping Basics

• Definition: Bootstrapping is a statistical technique that employs a single sample to approximate the sampling distribution of a statistic.

• With a sufficiently large sample size, the sample is reflective of the wider population, facilitating robust analysis. Notably, the Standard Error of the Mean (SE) diminishes as the sample size expands.

Generating Bootstrap Samples

• Steps to produce a single bootstrap sample:

  1. Randomly select an observation from the original sample.

  2. Document the observation's value.

  3. Return the observation to the sample pool.

  4. Repeat this process until the desired size matches that of the original sample.

Differences Between Sampling and Bootstrap Distribution

• The bootstrap mean is centered around the initial sample mean, which represents the unknown population mean.

• Key Concepts:

  • Traditional sampling is executed from the population, whereas bootstrapping is conducted using the original sample.

  • Bootstrapping employs sampling with replacement, allowing for repeated use of observations from the original sample.

Using Bootstrap for Estimation

• Steps to derive a plausible range for means using bootstrapping:

  1. Draw a bootstrap sample.

  2. Calculate the mean or other relevant estimates from that sample.

  3. Repeat the previous steps to form a bootstrap sampling distribution.

  4. Compute confidence intervals, such as a 95% confidence level.

• Discussion surrounding the limitations of bootstrap distributions and their applicability in creating 95% confidence intervals.

Bootstrapping Example: Airbnb Data

• Demonstrated the bootstrapping process utilizing actual prices from Airbnb listings:

  1. Conducted an example that represented the means of sample prices.

  2. Reported the calculated mean as $155.80.

  3. Compared histograms of bootstrap samples to the original sample distribution visually.

  4. Presented approximations of bootstrap distributions through repeated sampling to illustrate variability.

Constructing Confidence Intervals

• Procedures for forming 95% percentile bootstrap confidence intervals:

  1. Organize the bootstrap distribution in ascending order.

  2. Determine the 2.5% and 97.5% percentiles to identify lower and upper bounds.

• The confidence interval was established between $119.28 and $203.63, representing a plausible range for the true population mean.

Summary of Key Points

• Bootstrapping as a method allows for estimation drawn from a single sample, approximating sampling distributions effectively.

• Highlighted the significance of confidence intervals in conveying accuracy and uncertainty in estimates.

• Acknowledged that true population parameters often remain unknown, positioning bootstrapping as a valuable method for practical estimation.

Closing Remarks

• Concluded the lecture by emphasizing that foundational knowledge serves as a gateway to more complex statistical techniques and analysis methods.

• Encouraged further exploration of additional resources for continued learning in areas of statistics and inference.