Module 20: Distribution of Sample Means

Key Topic: How Sample Size Affects Variability of Sample Means

• Objective: Understand the impact of sample size on the variability of sample means.

Example: Birth Weight Sample

• Population mean birth weight (μ) = 3,500 g
• Standard deviation (σ) = 500 g
• Initial observation:
  • Sample mean of 9 babies = 3,400 g
  • Conclusion: Does not suggest the town's mean birth weight is less than 3,500 g.

Investigating Sample Size Impact

• Research Question: Does increasing sample size change the conclusion about the mean birth weight?
• Simulation Details:
  • Sample sizes tested: n = 9, 25, 100
  • 1,000 samples recorded for each size

Simulation Results

• Sample Mean Observations:
  • Sample size n = 9: Mean centered at 3,500 g, wider spread.
  • Sample size n = 25: Mean centered at 3,500 g, spread closer to the mean.
  • Sample size n = 100: Mean centered at 3,500 g, narrowest spread.

Analysis of the Results

• Center:
  • Unaffected by sample size; remains approximately the population mean (3,500 g).
• Spread:
  • Smaller with larger samples.
  • Standard deviation decreases as sample size increases.
• Shape:
  • Sampling distributions appear approximately normal.

Conclusion on Sample Size Effect

• Impact on Conclusions:
  • Larger sample sizes (e.g., n = 100) lead to less variability in sample means.
  • Sample mean of 3,400 g is further from the mean when n = 100, suggesting evidence of a lower population mean.

Theoretical Probability Model

• Model Description:
  • Population with mean μ and standard deviation σ.
  • Sampling distribution of sample means:
    • Mean = μ
    • Standard deviation = σ / √n
• Implications:
  • Theoretical model supports observed trends in sample means regarding center and spread.

This module provides insights into how increasing the sample size reduces variability in sample means, thus affecting the strength of conclusions drawn regarding a population mean.