One Mean

Parameter: $\mu$
Point Estimate: $\bar{X}$

Hypothesis Test

Hypotheses:
$ H_0: \mu = \mu_0 $
1) $ H_a: \mu < \mu_0 $
2) $ H_a: \mu \ne \mu_0 $
3) $ H_a: \mu > \mu_0 $

Test Statistic Random Variable (Assuming $H_0$ is true):
$ T =\dfrac{ \bar{X} - \mu_0}{ S / \sqrt{n} } \sim t (df = n - 1) $

Observed Test Statistic:
$ t_{obs} = \dfrac{\bar{x}_{obs} - \mu_0}{s_{obs} \, / \, \sqrt{n}} $

$ \mathbf{\textit{P}}$-value:
1) $ \mathbb{P}(\bar{X} \le \bar{x}_{obs}) = \mathbb{P}(T \le t_{obs}) $
2) $ 2 \cdot \mathbb{P}(\bar{X} \ge \bar{x}_{obs}) = \mathbb{P}\left(\big| T \big| \ge \big| t_{obs} \big| \right) $
3) $ \mathbb{P}(\bar{X} \ge \bar{x}_{obs}) = \mathbb{P}(T \ge t_{obs}) $

Confidence Interval

Formula for CI:
$ \bar{x}_{obs} \pm t^*_{df} \cdot \dfrac{s_{obs}}{\sqrt{n} } $

Conditions for Distributional Approximation (To $ T$):

Independent Observations
Nearly normal population
OR
large sample size ( $ n \ge 30 $ )

Example Problem

Conditions for Distributional Approximation (To $ T$):

Independent Observations
Nearly normal population
OR
large sample size ( $ n \ge 30 $ )

Type of Plot:
Histogram

qplot(x = response, data = dataframe, geom = "histogram", color = I("white") )