CourseKata Chapter 6: Part II

Sampling

Mansour Abdoli, PhD

Today: From Samples to Distributions

Session Goals

By the end of today, you can:

• Simulate a sampling distribution of the mean
• Connect LLN to sampling distributions
• Identify unusual (unexpected) events
• See why normal distributions appear naturally

Important Idea

Statistics Are Random Variables

• A statistic summarizes a sample.
• Categorical → proportion
• Quantitative: mean, SD, median, …

Random samples → Variable statistics

• Distribution of Statistics:
  • What is its mean?
  • What is its variance?
  • What is its distribution shape?

Law of Large Numbers (LLN)

• Estimating the Distribution of a Statistic:
  • Use a large number of observed statistic
  • The mean converges faster than the distribution

Sampling Distribution of the Mean

Steps:

1. Draw random sample
2. Compute mean
3. Repeat many times
4. Plot distribution of sample means

Sampling Distribution for Size \(n\)

• What do you notice?
• What happens if a larger \(n\) is used?

Key Patterns

As sample size increases:

• Mean of sample means ≈ population mean (LLN)
• Variance decreases \[Var(\bar{Y}) = \frac{\sigma^2}{n}\]
• Distribution becomes smoother
  • More possible sample means.
• Shape becomes more symmetric (bell-shaped)
  • Central Limit Theorem

Central Limit Theorem (CLT)

• As \(n\) increases,
  • \(\bar Y\) distribution approaches a Normal shape
  • with a mean equal to the population mean \(\mu\)
  • and a smaller variance \(\frac{\sigma^2}{n}\)
• In other words: for large \(n\)’s \[\bar Y \hat \sim N(\mu, \sigma/\sqrt{n})\]

What is a Large \(n\)?

• Any \(n\) works for \(Y\sim N(\mu, \sigma)\)
• \(n\ge30\) is good for approximately symmetric \(Y\)-distributions.
• For heavily skewed distribution of \(Y\), use simulation to check.

Checling \(n\) by Simulation

Normal Density & Empirical Rule

• Normal Distribution & Empirical Rule:
  • 68% of values fall in \((\mu-\sigma, \mu+\sigma)\)
  • 95% of values fall in \((\mu-2\sigma, \mu+2\sigma)\)
  • 99.7% of values fall in \((\mu-3\sigma, \mu+3\sigma)\)

Unusual Events

Expected events (e.g. sample mean = population mean) could be unusual:

Unexpected Unusual Events (Tails)

Events away from expected events

• Farther away from the center: Lower tail probability

Testing Claims (Hypothesis Testing)

• Claim: The die used is fair (\(H_0\))
• Roll a dies 40 times and get \(\bar Y = 4.3\)
• Is it too far? Is it too unlikely?