Formulas for Final Exam

• Definitions:
  $\sigma$ - population standard deviation
  $s$ - sample standard deviatioxn
  $\pi$ - population proportion
  $p$ - sample proportion
  $\mu$ - population mean
  $\bar{x}$ - sample mean
  $n$ - sample size or number of trials

• The binomial probability distribution is as follows. $${P(x~given~n~trials) = {}_n C_x \pi^x ( 1 - \pi)^{n-x}}$$. $${{}_n C_x = \frac{n!}{(n-x)! x!}}$$.

• Normalizing a variable: $${z = \frac{x - \mu}{\sigma}}$$.

• Standard deviation of the sampling distribution for the sample mean: $${\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}}$$.

• Standard deviation of the sampling distribution for the sample proportion when the population proportion ($\pi$) is known: $${\sigma_{p} = \sqrt{\frac{\pi ( 1 - \pi)}{n}}}$$.

• Standard deviation of the sampling distribution for the sample proportion when the population proportion ($\pi$) is not known: $${\sigma_{p} = \sqrt{\frac{p ( 1 - p)}{n}}}$$.

• The sampling distribution for the sample proportion is normally distributed if $pn \geq 10$ and $(1-p)n \geq 10$.

• $${P(A \vert B) = \frac{P(A~and~B)}{P(B)}}$$.

• A confidence interval is always of the form: point estimate $\pm$ (critical value) $\times$ (standard deviation of sampling distribution).

• Expected value of a discrete distribution: $\Sigma x p(x)$.

• $${\bar{x} = \frac{\Sigma x}{n}}$$.

• $${s = \frac{\Sigma (\bar{x} - x)^2}{n - 1}}$$.