Two Means (Paired Samples)

Two Means
(Paired Samples)

Type of Plot:
Histogram

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

Parameter: $\mu_{diff}$
Point Estimate: $\bar{X}_{diff}$

Hypothesis Test

Hypotheses:
$ H_0: \mu_{diff} = \mu_0 = 0 $
1) $ H_a: \mu_{diff} < 0 $
2) $ H_a: \mu_{diff} \ne 0 $
3) $ H_a: \mu_{diff} > 0 $

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

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

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

Conditions for Distributional Approximation (To $ T$):

Independent observations among pairs
Nearly normal population of differences
OR
large number of pairs ( $ n \ge 30 $ )

Confidence Interval

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

Conditions for Distributional Approximation (To $ T$):

Independent observations among pairs
Nearly normal population of differences
OR
large number of pairs ( $ n \ge 30 $ )