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--- b:head_first_statistics:estimating_populations_and_samples [2024/11/11 08:11] – [Recap] hkimscil
+++ b:head_first_statistics:estimating_populations_and_samples [2024/11/11 08:23] (current) – [Recap] hkimscil
@@ Line 623: / Line 623: @@
 </code>
 ====== Recap ======
-Distribution of **Sample** <fc #ff0000>**P**</fc>roportion<fc #ff0000>**s**</fc>, <fc #ff0000>$Ps$</fc>
+Distribution of **Sample** <fc #ff0000>**P**</fc>roportion<fc #ff0000>**s**</fc>, <fc #ff0000>$Ps$</fc>,
-Hence,
+when sampling n entities (repeatedly) from a population whose proportion is p.
 \begin{eqnarray*}
-Ps & \sim & N(np, pq) \\
+Ps & \sim & N(p,  \frac{pq}{n}) \\
+\text{hence, } \\
+\text{standard deviation of} \\
+\text{sample proportions} & = & \sqrt{\frac{pq}{n}}
 \end{eqnarray*}
 Distribution of **Sample** <fc #ff0000>Means, $\overline{X}$</fc>
+when sampling a sample whose size is n from a population whose mean is $\mu$ and variance is $\sigma^2$.
+\begin{eqnarray*}
+\overline{X} & \sim & N(\mu,  \frac{\sigma^2}{n}) \\
+\text{hence, } \\
+\text{standard deviation of} \\
+\text{sample means} & = &  \sqrt{\frac{\sigma^2}{n}} \\
+& = &  \frac{\sigma}{\sqrt{n}}
+\end{eqnarray*}