Sampling Distribution of the Sample Proportion (7.4)

Understanding Sampling Distributions and Proportions
📌 A sampling distribution is formed by repeatedly taking samples, calculating a statistic (like $\hat{p}$ or $\bar{x}$) for each, and graphing the results.
📊 A proportion represents the fraction of favorable outcomes relative to the whole, calculated as (Number of Favorable Outcomes) / (Total Number of Outcomes).
📝 For a sample proportion, the symbol is $\hat{p}$; for the population proportion, the symbol is $p$. For an example, a sample size of 10 with 2 green-eyed people yields a proportion $\hat{p} = 2/10 = 0.2$.

Properties of the Sampling Distribution of the Sample Proportion ($\hat{p}$)
🔬 If the sampling distribution of $\hat{p}$ is normal (applies Central Limit Theorem), the mean ($\mu_{\hat{p}}$) equals the population proportion ($p$).
📏 The standard deviation ($\sigma_{\hat{p}}$) is calculated as $\sqrt{\frac{p(1-p)}{n}}$, where $Q = 1-p$ (the proportion of unsuccessful outcomes) and $n$ is the sample size.
📐 The standardization formula (Z-score for proportions) is $Z = \frac{\hat{p} - p}{\sigma_{\hat{p}}}$, allowing the use of the Z-score table for calculating areas.

Central Limit Theorem (CLT) Conditions for Proportions
✅ The Central Limit Theorem applies to the sampling distribution of $\hat{p}$ only if two specific conditions are met:
1. $n \cdot p \geq 10$
2. $n \cdot (1 - p) \geq 10$
🚧 This differs from the sample mean distribution where the CLT generally applies when $n \geq 30$.

Key Points & Insights
➡️ The mean of all sample proportions ($\mu_{\hat{p}}$) converges to the true population proportion ($p$).
➡️ To use the Z-table for proportions, ensure both $np \geq 10$ and $n(1-p) \geq 10$ are satisfied.
➡️ The standard error ($\sigma_{\hat{p}}$) for proportions is $\sqrt{\frac{p(1-p)}{n}}$, which is crucial for calculating Z-scores.

📸 Video summarized with SummaryTube.com on Dec 03, 2025, 14:49 UTC