Sampling distribution. 抽样分布 (Sampling distribution) Sampling distribution is distribution of sample statistics like $\bar{x}$, $\hat{p}$. (Illustrations showing drawing samples and calculating statistics). $\mu_x$ population mean. 所有人身高加起来除以总人数 (sum of everyone's height divided by total number of people). $\bar{x}$ sample mean. 样本里的人身高加起来除以样本人数 (sum of heights in sample divided by sample size). $\mu_{\bar{x}}$ mean of sampling distribution. population 的分布 (distribution of population), e.g., 全校学生身高分布 (height distribution of all students in school) – individual information. sample 的分布 (distribution of sample), e.g., 某班学生的身高分布 (height distribution of students in a class). Sample statistics 的分布 (distribution of sample statistics), e.g., 每个班级算一个平均身高 $\bar{x}$, 所有 $\bar{x}$ 的分布 (each class calculates an average height $\bar{x}$, the distribution of all $\bar{x}$). 当满足了条件时* (When conditions are met*) $\bar{x} \sim N(\mu_x, \frac{\sigma}{\sqrt{n}})$ $\hat{p} \sim N(p, \sqrt{\frac{p(1-p)}{n}})$ →记 (Remember) 使 $\bar{x} \sim N(\mu_x, \frac{\sigma}{\sqrt{n}})$ 的条件 (Conditions for $\bar{x} \sim N(\mu_x, \frac{\sigma}{\sqrt{n}})$): 1) SRS. Sample were selected randomly. $\Rightarrow$ unbiased $\Rightarrow \mu_{\bar{x}} = E(\bar{x}) = \mu_x$. 2) 10% condition: $n < 0.1N \Rightarrow$ independent sampling $\Rightarrow \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$. 3) Large number condition/Normal condition: $n \geq 30$ OR population is normal $\Rightarrow \bar{x} \sim N$. 使 $\hat{p} \sim N(p, \sqrt{\frac{p(1-p)}{n}})$ 的条件 (Conditions for $\hat{p} \sim N(p, \sqrt{\frac{p(1-p)}{n}})$): 1) SRS. $\Rightarrow$ unbiased $\Rightarrow \mu_{\hat{p}} = p$. 2) 10% condition $\Rightarrow$ independent $\Rightarrow \sigma_{\hat{p}} = \sqrt{\frac{p(1-p)}{n}}$. 3) Large count: $np \geq 10$ and $n(1-p) \geq 10 \Rightarrow \hat{p} \sim N$. $\hat{p} = \frac{X}{n} \leftarrow$ 属于标类别的货 (category count) $n \leftarrow$ sample size. $X \sim Binomial(n,p)$. if unbiased $E(X) = np$, $E(\hat{p}) = E(\frac{X}{n}) = \frac{np}{n} = p$. if independent $Std(X) = \sqrt{np(1-p)}$. $Std(\hat{p}) = Std(\frac{X}{n}) = \frac{\sqrt{np(1-p)}}{n} = \sqrt{\frac{p(1-p)}{n}}$. if $np \geq 10$, $n(1-p) \geq 10$. $X \sim Binomial \Rightarrow X \sim N$. $\Rightarrow \hat{p} = \frac{X}{n} \sim N$. $\bar{x} = \frac{\sum X_i}{n}$. $E(\bar{x}) = E(\frac{\sum X_i}{n}) = \frac{\sum E(X_i)}{n}$. if unbiased $E(\bar{x}) = \frac{n \mu_x}{n} = \mu_x$. $Std(\bar{x}) = Std(\frac{\sum X_i}{n})$. if independent $Std(\bar{x}) = \frac{1}{n} \sqrt{n} Std(X) = \frac{\sigma_x}{\sqrt{n}}$. 云记 (Remember) if $n \geq 30$ Central Limit Theorem. $\Rightarrow \sum X_i \sim N$. $\bar{x} = \frac{\sum X_i}{n} \sim N$. OR if $X \sim N$, $\sum X_i \sim N$. $\bar{x} = \frac{\sum X_i}{n} \sim N$.