Sampling Distribution Models for Proportions Study Notes

Key Variables

• $P$ (Population Proportion): The true proportion of a trait within a population.
• $p$ (Sample Proportion): The observed proportion from a specific sample.
• Assumptions: Sample size ($n$) must be sufficiently large to justify using a normal model.

Example 1: College Retention

• Scenario: National retention rate is $P = 0.73$. A college has $p = 505/597 \approx 0.846$.
• Statistical significance: Calculated using the Z-score formula:
  $Z = \frac{p - P}{\sqrt{\frac{P(1-P)}{n}}}$
• Here, $n = 597$, $P = 0.73$, and $p = 0.846$.

Example 2: Polling for Budget Support

• Scenario: True support is $P = 0.55$ with a sample of $n = 450$.
• Normal Approximation Model:
  • Mean: $\mu_p = P = 0.55$
  • Standard Deviation: $\sigma_p = \sqrt{\frac{P(1-P)}{n}} = \sqrt{\frac{0.55(0.45)}{450}}$
• Objective: Find the probability that the sample proportion ($p$) is less than $0.50$ by calculating the Z-score for $0.50$ and using the cumulative normal distribution.