Z test

• uses $\sigma$, if we don't know that parameter then we need to use the T test
• $Z=\frac{\overline{X}-\mu }{\sigma /\sqrt{n}}$ has the standard normal distribution
  • converts our $\overline{X}$ to $N\sim (0,1)$

For proportions

$Z=\frac{{\hat{p}}_H-{\hat{p}}_N}{S{E}_0({\hat{p}}_H-{\hat{p}}_N)}=\frac{\hat{p}-p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$

T test

• used when we dont know $\sigma$
• test statistic $t=\frac{\overline{X}-\mu }{s/\sqrt{n}}$ has the t distribution with $n-1$ degrees of freedom
• the observed value of the test(t) statistic tells us how many standard errors of the value of $\overline{X}$ is from the hypothesized value of $\mu$
  • $\text{Test Statistic}=\frac{\text{Estimator}-\text{Hypothesized Value}}{SE\text{(Estimator)}}$
  • $t=\frac{\overline{X}-{\mu }_0}{SE(\overline{X})}$
    • SE = standard error
  • $t=\frac{\overline{X}-{\mu }_0}{s/\sqrt{n}}$
• when comparing the difference in two means
  • $t=\frac{{\overline{X}}_1-{\overline{X}}_2}{SE({\overline{X}}_1-{\overline{X}}_2)}$
    • $\mu$'s cancel each other

Chi-Squared Test

F test

F distribution