Activity 18

MATH 216: Statistical Thinking

Sampling Distribution of Sample Proportions

Time Allocation: 15 minutes total (5 min reading, 10 min individual work)

Part 1: Conceptual Understanding (3 minutes)

Instructions: Answer the following questions:

  1. What is the Central Limit Theorem for proportions, and what conditions must be satisfied for normal approximation?
  1. How does the standard error formula \(\sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}}\) relate to sample size and precision?
  1. Why is the sampling distribution of \(\hat{p}\) approximately normal for large samples, regardless of the population distribution?

Part 2: Standard Error Calculations (4 minutes)

Calculate mean and standard error of \(\hat{p}\):

  1. Political polling: \(p = 0.52\), \(n = 1000\) \(\mu_{\hat{p}}\) = , \(\sigma_{\hat{p}}\) =

  2. Market research: \(p = 0.30\), \(n = 100\) \(\mu_{\hat{p}}\) = , \(\sigma_{\hat{p}}\) =

  3. Quality control: \(p = 0.08\), \(n = 200\) \(\mu_{\hat{p}}\) = , \(\sigma_{\hat{p}}\) =

  4. Medical trial: \(p = 0.45\), \(n = 400\) \(\mu_{\hat{p}}\) = , \(\sigma_{\hat{p}}\) =

Show your work for two calculations:

Part 3: Probability Applications (3 minutes)

Calculate probabilities using normal approximation:

  1. Medical trial: \(p = 0.45\), \(n = 400\) \(P(\hat{p} < 0.40)\) =

  2. Political polling: \(p = 0.52\), \(n = 1000\) \(P(\hat{p} > 0.55)\) =

  3. Quality control: \(p = 0.08\), \(n = 200\) \(P(0.05 < \hat{p} < 0.11)\) =

Check CLT conditions: For each scenario, verify if \(np \geq 15\) and \(n(1-p) \geq 15\)

Critical Thinking: Why do we need both success-failure conditions (\(np \geq 15\) and \(n(1-p) \geq 15\)) for the CLT to apply to proportions?