Lecture Notes: Chapter 12A - Airbnb and Sampling Distribution for Averages

Introduction

• Focus on Airbnb, specifically housing prices in New York City under $500.
• The average Airbnb listing in NYC is $130 per night (as of 2019).

Key Concepts

• Transition from Proportions to Averages:
  • Previous focus on proportions (p-hat, central limit theorem for confidence intervals).
  • Now applying similar concepts to means (x-bar).

Sampling Distribution for Averages

• Sampling Distribution of Sample Mean (X̄):
  • As sample size (n) increases:
    • The distribution of the sample mean becomes closer to normal.
    • Variability of the sample means (standard deviation) decreases.
  • For large samples, the sample mean distribution's mean equals the population mean.
• Central Limit Theorem (CLT):
  • Applies to means as well as proportions.
  • Allows normal approximations for probabilities of sample means.

In-Class Activity Details

• Focus on Airbnb Prices in NYC:
  • Limiting analysis to listings under $500.
  • Simulating samples (n = 25) to determine expected mean and variability.

Simulations

• Activity:
  • Use a tool to simulate different sample sizes from Airbnb data.
  • Examine the effect of increasing sample sizes on the distribution of the sample mean.

Observations from Simulated Data

1. Sample Size = 2:
   • Distribution is skewed, not normal.
   • Mean = 131, Standard Deviation ≈ 58.
2. Sample Size = 10:
   • Distribution begins to normalize.
   • Mean = 131, Standard Deviation ≈ 26.8.
3. Sample Size = 50:
   • Distribution becomes more normal.
   • Mean = 130, Standard Deviation ≈ 12.3.

Mathematical Formulas (CLT for Averages)

• Mean of Sample Means (μX̄):
  • Equals population mean (μ).
• Standard Deviation of Sample Means (σX̄):
  • Formula: σ / √n (where σ is the population standard deviation).
• Comparison of Simulated and Calculated Values:
  • Simulated standard deviations closely match calculated values.

Case Study: LA vs. NYC Airbnb Prices

• Examine whether LA Airbnb listings are more expensive than NYC based on a sample mean of $152.
• Z-Score Calculation for Sample Mean:
  • Z = (X̄ - μ) / (σ/√n)
• Probability Calculation:
  • Calculate the probability of observing a sample mean of $152 or higher using normal distribution.
  • Decision based on calculated p-value in relation to a chosen alpha level.

Conclusion

• Recap of sampling distribution concepts applied to averages.
• Importance of sample size in determining the shape and variability of the sample mean distribution.
• Use of central limit theorem for making probabilistic decisions about sample means.