Day 31

Math 216: Statistical Thinking

Bastola

Review: Mean vs. Proportion

  • Previously studied: Confidence intervals (C.I.) and hypothesis tests for population means.
  • Non-parametric tests were focused on population medians.

Transition to Proportions

  • Now focusing on population proportion (e.g., “What percentage of voters favor candidate A?”)
  • Applies similar principles as the Central Limit Theorem for means, using properties of the sampling distribution of \(\hat{p}\).

Properties of \(\hat{p}\)

  1. Mean: The expected value (mean) of \(\hat{p}\) is equal to the true population proportion \(p\). \[ E(\hat{p}) = p \]
  2. Standard Deviation: For large samples, approximated by \[ \sigma_{\hat{p}} \approx \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \]
  3. Distribution: Becomes approximately normal with large sample sizes.

Large-Sample Hypothesis Testing

  • Test Statistic: Normal (z) statistic for \(H_0: p = p_0\): \[ z_c = \frac{(\hat{p} - p_0)}{\sqrt{\frac{p_0(1-p_0)}{n}}} \]
  • Decision Rule: Reject \(H_0\) if p-value \(< \alpha\), or if \(z_c\) falls into the rejection region.

Hypothesis Test Types

  • One-Tailed Test:
    • \(H_a: p < p_0\) or \(H_a: p > p_0\)
    • Rejection region: \(z_c < -z_{\alpha}\) or \(z_c > z_{\alpha}\)
  • Two-Tailed Test:
    • \(H_a: p \neq p_0\)
    • Rejection region: \(z_c < -z_{\alpha/2}\) or \(z_c > z_{\alpha/2}\)

Conditions for Testing

  1. Random sample from the population.
  2. Large enough sample size, typically \(n\hat{p} \geq 15\) and \(n\hat{q} \geq 15\).

Confidence Interval for \(\hat{p}\)

  • Large-sample confidence interval: \[ \hat{p} \pm z_{\alpha/2} \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \]
  • Valid when \(n\) is large for normal approximation.

Sample Size Determination

  • To specify sampling error (SE) and confidence level: \[ n = \frac{(z_{\alpha/2})^2 \hat{p}(1-\hat{p})}{SE^2} \]
  • Conservative estimate uses \(p \approx 0.5\) when \(\hat{p}\) is unknown.