[Basic Stats] 01.Hypothesis Testing

This series will be followed a more advanced series in mathematical statistics which I'm still getting the hang of!

This post is meant to be a refresher on the topic!
If this is your first time with these topics, please checkout more thorough materials.

Hypothesis Testing

Terminology

Hypothesis : A statement on something we are trying to investigate. It's normally a statement that is based on a belief about the population.

Null Hypothesis : The originial statement. The statement about the population we want to test.

Alternative Hypothesis : The opposite of the null hypothesis.

Type I Error : Thee probability of rejecting the null hypothesis although it is true. If we say tht the possibility of rejecting H0 is $a$ then type I error is $1 - a$.

Type II Error : The probability of accepting the H0 despite the fact that it is wrong.

P-Value : The probability that the H0 will be true. It is a number between 0 ~ 1.

Two-side Test v.s. One-side Test

	Two-tailed test	Left-tailed test	Right-tailed test
Sign in $H_{a}$ rejection region	$\neq$	<	>

One Sample Hypothesis Test

Assumptions about the population mean - population variance is KNOWN

If we know the population variance we can utilize the $z-score$ regardless of the size of the population.

$a = 0.05$, $z =\frac{\bar{x} - \mu}{\sigma/ n^{0.5}}$

a) $|z_{0}| \geq z_{a/2}$ Reject $H_{0}$
b) $z_{0} \geq z_{a}$ Reject $H_{0}$
c) $z_{0} \leq -z_{a}$ Reject $H_{0}$

Otherwise we FAIL to reject $H_{0}$

Assumptions about the population mean - the population variance is UNKNOWN

$a = 0.05$, $z =\frac{\bar{x} - \mu}{s/ n^{0.5}}$ ~ $t(n-1)$

$n>30$ The T-distribution becomes similar to a normal distribution as the degree of freedom increases

Assumptions about the population proportion

Remember that the sampling distribution of $\hat{p}$ has a mean of $p$ and a standard deviation of $\sqrt{p(1-P)/n}$

$a = 0.05, z= \frac{\hat{p}-p}{\sqrt{p(1-P)/n}}$