Hypothesis Testing

Hypothesis testing: using a data observed from a distribution with unknown parameters, we hypothesise that the parameters of this distribution take particular values and test the validity of this hypothesis using statistical methods

Confidence intervals: provide probabilistic level of certainty regarding parameters of a distribution

Example:
1. $X_1, X_2,..., X_n$
2. unknown mean value $\mu$
3. known $\sigma$

normal distribution: $N(\mu, \sigma^2)$
estimate of $\mu$ : $\bar X=\frac{X_1, X_2,..., X_n}{n}$
distribution of $\bar x$ : $N(\mu, \frac{\sigma^2}{n})$

Suppose:
$P(\bar X\leq \mu+2)$
$P(\bar X-\mu\leq 2)$
$P(\frac{\bar X-\mu}{\sigma/\sqrt{n}}\leq \frac{2}{\sigma/\sqrt{n}})$