Module 20: Distribution of Sample Means

Overview

• Comparison of sampling distributions for sample means and sample proportions.
• Variables: Categorical vs. Quantitative.
  • Examples: Categorical (Gender), Quantitative (Age).
• Parameters:
  • P: Population proportion
  • μ: Population mean
  • σ: Population standard deviation
• Statistics:
  • P̂: Sample proportion
  • X̄: Sample mean

Sampling Distribution

• Proportions:
  • Center: P
  • Spread: ( \sqrt{ \frac{P(1-P)}{n} } )
  • Shape: Normal if ( nP \geq 10 ) and ( n(1-P) \geq 10 )
• Means:
  • Center: μ
  • Spread: ( \frac{σ}{\sqrt{n}} )
  • Shape: Discussed below

Key Concepts

• The mean of the sampling distribution is the population parameter.
• Variability of the sampling distribution is affected by the sample size.
• Normality of shape depends on specific conditions.

Conditions for Normal Sampling Distribution

• For proportions: Conditions based on sample size and population proportion.
• For means: Conditions based on sample size and population distribution.

Investigating Sample Means

• Simulation with a skewed population distribution (e.g., commute times).
• Small Sample Size (n=5):
  • Skewed distribution of sample means.
  • Sample mean approximately equals population mean (25 minutes in example).
  • Sample standard deviation approximately matches theoretical prediction.
• Large Sample Size (n=30):
  • Distribution of sample means appears normal.
  • Decreased standard deviation of sample means.

Central Limit Theorem (CLT)

• Key Result: For large samples, the sampling distribution of sample means is approximately normal.
• Importance:
  • Allows for inference procedures (hypothesis tests, confidence intervals).
  • Ensures applicability of normal probability models for sample means.
• Sample Size Guidelines:
  • General Rule: Sample size (n) > 30 for normal distribution of sample means.
  • Larger samples needed if population distribution is more skewed.

Summary

• Conditions for normal shape of sampling distributions:
  • Proportions: Based on nP and n(1-P).
  • Means: Normal if n > 30 or if population distribution is normal.
• Larger sample sizes for skewed population distributions ensure normal sampling distribution.