For Sample Distribution, we need to know…

1. The sample variance $S^2 = \frac{1}{n-1}\sum\limits^n_{i=1}(X_i-\bar X)^2$
2. If the population is normal ($X\sim N(\mu,\sigma^2)$), then $\bar X\sim N(\mu,\frac{\sigma^2}{n})$ If not normal, but $n>30$, we can get $\bar X\sim N(\mu,\frac{\sigma^2}{n})$ by CLT**.**
3. For 2 samples, $\bar X_A$ and $\bar X_B$, $\bar X_A - \bar X_B\sim N(\mu_A - \mu_B,\frac{\sigma^2_A}{n_A} + \frac{\sigma^2_B}{n_B})$. Note: $\mu$ subtraction, $\frac{\sigma^2}{n}$ addition.

General Question:

Given $\mu$ and $\sigma$ of a population, given $n$ of a sample, ask to find the $P(X>a)$.

• Solution

A. $\chi^2$ distribution

Task (main idea)

Compare the value of computed $\chi^2$ and chi-squared distribution table to confirm “will the event happen”.

If $\chi^2_{0.975} < \chi^2 < \chi^2_{0.025}$, we can convince the event will happen.

$X=Z^2_1+Z^2_2+\cdots+Z^2_n$ follows a chi-squared distribution with $n$ DOF, denoted by $X\sim\chi^2_n$.

If $X_1\sim\chi^2_{n_1}$ and $X_2\sim\chi^2_{n_2}$,

then $X_1 + X_2\sim\chi^2_{n_1+n_2}$.

$\chi^2\sim(n-1)\frac{S^2}{\sigma^2}$ has a chi-squared distribution with DOF $v=n-1$. Note: DOF is $n-1$ rather than $n$.

• Chi-Square Distribution

B. $t$ distribution

The use of the 𝑡-distribution and the sample size consideration is not related to the Central Limit Theorem.

Given $Z\sim N(0,1)$ and $V\sim\chi^2_v$ ($v = n-1$ is DOF), if $Z$ and $V$ are independent, then $t=\frac{Z}{\sqrt{V/v}}$.

If $X_1,X_2,\cdots,X_n$ is normal, then$X_1,X_2,\cdots,X_n$$T=\frac{\bar X-\mu}{S / \sqrt n}$.

• t-distribution table

C. $F$ distribution

Symmetric Questions

For questions like find $P(-\frac{1.5}{\sigma} \le Z\lt \frac{1.5}{\sigma})\lt0.99$, we have this figure, where the specific $\sigma$ value(position) is what we want.

However, we CAN’T get this by z-score table, because the shape of z-score table is like: