Day 18

Math 216: Statistical Thinking

Bastola

Sampling Distributions using Proportions

  • Objective: Understand how the sample proportion \(\hat{p}\) estimates the population proportion \(p\).
  • Context: Whether estimating voter preferences, customer behaviors, or biological occurrences, knowing how to infer population characteristics from sample data is crucial.

Motivation and Real-World Examples

  • Voter Preference: Estimate the proportion of voters in favor of a new bill.
  • Customer Behavior: Understand the percentage of customers using store credit cards.
  • Conservation Efforts: Track the fraction of endangered species born in captivity.

Sampling distribution for \(\hat{p}\)

Properties of Sampling Distribution of \(\hat{p}\)

  1. Mean of \(\hat{p}\)
    • The mean equals the population proportion \(p\).
    • \(\hat{p}\) is an unbiased estimator of \(p\). \[ E(\hat{p}) = \mu_{\hat{p}} = p \]
  2. Standard Deviation of \(\hat{p}\)
    • The variability of \(\hat{p}\) decreases as sample size \(n\) increases. \[ \sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}} \]

Central Limit Theorem for \(\hat{p}\)

  • Approximately Normal: For large samples, \(\hat{p}\)’s distribution is nearly normal.
  • Rule of Thumb:
    • Consider large if \(np \geq 15\) and \(n(1-p) \geq 15\). \[ n \cdot \hat{p} \geq 15 \quad \text{and} \quad n(1-\hat{p}) \geq 15 \]

Normal Approx. for Binomial

  • Binomial Properties: For \(X \sim \text{Binomial}(n, p)\):
    • Mean: \(\mu = np\)
    • Variance: \(\sigma^2 = np(1-p)\)
    • Normal approximation: Valid if \(np \geq 15\) and \(n(1-p) \geq 15\).
  • Sampling Distribution of Sample Proportion:
    For proportions \(\hat{p} = X/n\), the sampling distribution is approximately normal:
    \[ \hat{p} \approx N\left(\mu_{\hat{p}} = p,\ \sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}}\right) \]