STAT101 - 3.3 (1 slide per page)

Page 1: Overview of Bootstrap Confidence Intervals

• Section 3.3: Constructing Bootstrap Confidence Intervals

• Course: Statistics 101, University of Canterbury

Page 2: Learning Objectives

• Unit B: Understanding Inference

  • Confidence Intervals

    • Sampling Distributions

    • Understanding and Interpreting Confidence Intervals

    • Constructing Bootstrap Confidence Intervals

    • Bootstrap Confidence Intervals using Percentiles

  • Hypothesis Testing

    • Introducing Hypothesis Tests

    • Measuring Evidence with P-values

    • Determining Statistical Significance

    • A Closer Look at Testing

• Key Skills:

  • Selecting a bootstrap sample and computing a bootstrap statistic

  • Using technology to create a bootstrap distribution

  • Estimating the standard error from bootstrap distributions

  • Constructing a 95% confidence interval based on the sample statistic and standard error from a bootstrap distribution

Page 3: Key Terms and Formulas

• Standard Error (SE):

  • Standard deviation of the sampling distribution

• Margin of Error (MoE):

  • For a 95% CI: MoE = 2 × SE

• Confidence Interval:

  • formula: statistic ± MoE

  • Calculate statistic for each sample

Page 4: Bootstrap Introduction

• Concern: Only one sample available, leading to uncertainty on how sample statistics vary.

• Bootstrapping as a method to assess this variability.

Page 5: Concept of Bootstrapping

• Metaphor: "Pull yourself up by your bootstraps" illustrates achieving the seemingly impossible without outside help—applies to statistical inference.

Page 6: Case Study - Proportion of Blue M&Ms

• Learning goal: Where might the "actual" proportion (p) of blue M&Ms be found?

Page 7: Importance of Random Sampling

• Assumptions needed: Sample must be representative of the hypothetical population to generalize results.

• Random Sampling: Essential for accurate statistical inference.

Page 8: Hypothetical Population Example

• Visualizes the concept of sampling from a hypothetical population composed of the original sample.

Page 9: Simulating Sampling Distribution

• How repeated samples from a hypothetical population are simulated.

• Practicality: Sample drawn with replacement from the existing sample elements.

Page 10: Example of Sampling with Replacement

• Scenario: Random sample of 9 dogs used to demonstrate bootstrapping.

Page 11: Bootstrap Sample Visuals

• Illustrates the generation of a bootstrap sample from an original sample.

Page 12: Definitions of Bootstrap Components

• Bootstrap Sample: Random sample with replacement of the same size as the original sample.

• Bootstrap Statistic: Statistic computed from a bootstrap sample.

• Bootstrap Distribution: Distribution of numerous bootstrap statistics.

Page 13: Possible Bootstrap Samples

• Example provided for testing if generated samples can be valid bootstrap samples from the original values.

Page 14: Bootstrap Statistics Overview

• Samples vs. Statistics: Original sample, bootstrap samples, bootstrap statistics visualized.

Page 15: Bootstrap Distribution Analysis

• Dotplot representation of bootstrap samples and comparative statistics (mean and standard error).

Page 16: Centering of Distributions

• Sampling distribution centered around population parameter, whereas bootstrap distribution centers on sample statistic.

• Bootstrapping used to assess precision of original estimates—not to improve point estimates.

Page 17: Variability Assessment

• Variability in bootstrap statistics reflects variability in sample statistics.

• Standard error can be estimated from bootstrap distribution's standard deviation.

Page 18: Statistical Inference Formula

• Formula for confidence intervals: statistic ± 2 × SE, where SE is derived from bootstrap distribution.

Page 19: Formulas Recap

• Standard Error: Standard deviation of the bootstrap distribution.

• Margin of Error: MoE = 2 × SE, for 95% CI.

• Construct confidence interval: statistic ± MoE.

Page 20: Bootstrapping Applications

• Bootstrapping can assess uncertainty in various sample statistics:

  • Proportion (p)

  • Difference in means (µ1 - µ2)

  • Difference in proportions (p1 - p2)

Page 21: Example - Mustang Prices

• Scenario: Randomly sampled n = 25 Mustangs to estimate average price.

Page 22: Estimating Average Prices

• Calculated statistics from sampled Mustangs:

  • Sample Size (n) = 25

  • Mean price estimate (x̄) = $15.98K

  • Standard deviation (s) = $11.11K

  • Precision assessment follows.

Page 23: Steps in Mustang Bootstrapping

• Procedure:

  1. Means calculated from bootstrap samples.

  2. Repetitive calculations yield distribution data.

Page 24: Mustang Distribution Plot

• Dotplot for bootstrap distribution showing mean and standard error.

Page 25: Global Warming Belief Survey

• Survey on 2,251 individuals was conducted regarding beliefs on global warming.

• Result showed 1,328 saying "yes", leading to confidence interval analysis.

Page 26: Global Warming Bootstrap Distribution

• Bootstrap analysis showing mean proportions and standard error from 1,000 samples.

Page 27: Political Party Beliefs

• Comparison of beliefs in global warming by political party

  • 79% of Democrats vs. 38% of Republicans.

  • Confidence interval needs determination for the difference in proportions.

Page 28: Bootstrapping Difference in Proportions

• Distribution representation for the difference in belief proportions between Democrats and Republicans.

Page 29: Generating a Bootstrap Distribution

• Steps:

  1. Generate bootstrap samples by systematic sampling with replacement.

  2. Compute bootstrap statistics.

  3. Assemble bootstrap statistics into a distribution.

• Resulting distributions can estimate confidence intervals if broadly symmetrical.