Chapter 2:

• $\mu = \frac{sum\ of\ population}{N}$
• $\bar X = \frac{sum\ of\ samples}{n}$
• $Sampling\ error = \bar X - \mu$
• $\mu_{\bar X} = \frac{\sum \bar X}{n}$
• $\sigma_{\bar X} = \frac{\sigma}{\sqrt n}$
• Z distribution

if $n \lt 30$ AND distribution is normal: $z = \frac{\bar X - \mu}{\sigma / \sqrt n}$ if $n \ge 30$ OR distribution is NOT normal: $z = \frac{\bar X - \mu}{s / \sqrt n}$

$P(z)$ is found from the Z - distribution table $P(z_1 \gt \bar X_1) = 0.5 - P(z_1)$ $P(z_2 \lt \bar X_2) = 0.5 - P(z_2)$ $P(\bar X_2 \lt z \lt \bar X_1) = P(z_1) + P(z_2)$

Remember: $\mu =\mu_{\bar X}$ if all possible samples can be selected from a population.

Chapter 3:

Confidence intervals

| - $\sigma$ is known

• population is not normal | $\bar X \pm z \frac{s}{\sqrt n}$ | $for\ 95\%, z = 1.96$ $for\ 99\%, z = 2.58$ | | - - $\sigma$ is unknown - $n \lt 30$
• population is approximately normal | $\bar X \pm t_{\alpha, n-1} \frac{s}{\sqrt n}$ | In $t$ distribution table $\alpha = 95\% \ or\ 99\%$ (column) $n -1$ (row) |
• $n = (\frac{z \times s}{E})^2$

Chapter 4:

• $\bar X = \frac{sum\ of\ samples}{n}$
• $s^2 = \frac{\sum [X_i - \bar X ]^2}{n-1}$
• $\mu = \bar X \pm t_{n-1,\alpha} \frac{s}{\sqrt n}$

NOTE: $\frac {s}{\sqrt n} = \sqrt {\frac{s^2}{n}}$

Welsh procedure

$\bar Y_i(w) = \begin{cases} \frac{\displaystyle\sum_{s=-(i-1)}^{(i-1)} \bar Y_{i+s}}{2i-1} &\text{if } i = 1,...,w \\ \frac{\displaystyle\sum_{s=-w}^{w} \bar Y_{i+s}}{2w+1} &\text{if } i = w+1, ... m-w \end{cases}$

Use first rule when $i \le w$ and the second rule when $i \gt w$