Module 20: Distribution of Sample Means

Overview

• Focus: Quantitative variables
• Objective: Understand variation of sample means in random samples from a population with a known mean.
• Context: Builds on prior knowledge from sample proportions.

Key Concepts

• Sample Distribution of Sample Means
  • Study variations in sample means from random samples.
  • Develop a probability model based on these distributions.
  • Apply model to test claims or estimate population means.

Example: Birth Weights

• World Health Organization (WHO): Monitors variables like birth weight to assess population health.
• Case Study: Town babies in 2005 vs. current random sample.
  • Population Parameter ((\mu)): Mean birth weight of 3,500 grams in 2005.
  • Sample Statistic ((\bar{x})): Current sample mean of 3,400 grams from 9 babies.

Investigative Question

• Does a sample mean of 3,400 grams suggest a decline in the population mean from 3,500 grams?
• Key Inquiry: How much do sample means vary?

Investigative Process

1. Initial Population: Mean birth weight of 3,500 grams.
2. Random Sampling: Take samples of 9 babies.
3. Sample Means: Analyze if a mean of 3,400 grams is likely or unlikely.

Logical Inference

• If likely: Sample could come from a population with a mean of 3,500 grams.
• If unlikely: Sample indicates population mean is less than 3,500 grams.

Conclusion

• Sampling Distribution Investigation: Needed to determine the variability and likelihood of sample means.

Visual Aids

• Diagram illustrating sample collection and mean comparison.

Important Terms

• Parameter: A numerical characteristic of a population ((\mu = 3,500) grams).
• Statistic: A numerical characteristic of a sample ((\bar{x} = 3,400) grams).
• Sampling Distribution: Distribution of a statistic (e.g., sample mean) over many samples.