TARGET DECK: probStat/chapter12

Hypothesis Testing

Description:

• 

Using Confidence interval:

• To test claims about a population mean $\mu$, we construct a confidence interval (a, b).
  • Claim: $\mu ={\mu }_0$ Do not reject test-claim if ${\mu }_0\in (a,b)$, reject otherwise.
  • Claim: $\mu <{\mu }_0$. Do not reject test-claim if $b<{\mu }_0$, reject otherwise.
  • Claim: $\mu >{\mu }_0$. Do not reject test-claim if ${\mu }_0<a$ , reject otherwise.

Null Hypothesis and Alternative Hypothesis

One-tailed vs two-tailed:

1. $H_0:\mu \ge {\mu }_0,H_a:\mu <{\mu }_0$, lower one-tailed
2. $H_0:\mu \le {\mu }_0,H_a:\mu >{\mu }_0$, upper one-tailed
3. $H_1:\mu ={\mu }_0,H_a:\mu \ne {\mu }_0$, two-tailed

Test statistic:

• Used to compare against another critical value to evaluate hypothesis
• All test statistic value must lie outside of Confidence interval $1-\alpha$, to be rejected

For p-value:

• Use Normal distribution, $ts=\frac{\bar{x}-{\mu }_0}{{\sigma }_x}$
• for two-tailed, $p$-value is divided by 2

For t-value:

• Use t distribution, $ts=\frac{\bar{x}-{\mu }_0}{s/\sqrt{n}}$ at $n-1$ def
• then convert to $p$-value (area under graph) at $n-1$ df

For Proportion:

• Use Population proportion
• $ts=\frac{\bar{p}-p_0}{\sqrt{\frac{p_0.(1-p_0)}{n}}}$

For Difference random variable of 2 variables:

• For known $\sigma$
  • Because ${\bar{x}}_1-{\bar{x}}_2\sim N(D_0,\sqrt{\frac{{\sigma }_1^2}{n_1}+\frac{{\sigma }_2^2}{n_2}})$
  • $ts=z=\frac{{\bar{x}}_1-{\bar{x}}_2-D_0}{\sqrt{\frac{{\sigma }_1^2}{n_1}+\frac{{\sigma }_2^2}{n_2}}}$
    • $D_0={\mu }_1-{\mu }_2$
• For unknown $\sigma$
  • Then use t-distribution with degree of freedom
  • $df=\frac{(\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}{)}^2}{\frac{1}{n_1-1}(\frac{s_1^2}{n_1}{)}^2+\frac{1}{n_2-1}(\frac{s_2^2}{n_2}{)}^2}$
    • Round down for more conservative interval estimation

Type 1 Error vs Type 2 Error

Level of significance:

• Same as probability of making a Type 1 Error
• When the null hypothesis is true as an equality
• If the cost of making a Type I error is high, small values of α are preferred.
  • Like medication, must have small LoS
• If the cost of making a Type I error is not too high, larger values of α are typically used.
  • it is ok to make mistake