One Proportion

Parameter: $P$
Point Estimate: $\hat{P}$

Hypothesis Test

Confidence Interval

Hypotheses:
$ H_0: P = p_0 $
1) $ H_a: P < p_0 $
2) $ H_a: P \ne p_0 $
3) $ H_a: P > p_0 $

Type of Plot:
Bar Graph

Test Statistic Random Variable (Assuming $H_0$ is true):
$ Z =\dfrac{ \hat{P} - p_0}{\sqrt{\dfrac{p_0(1 - p_0)}{n} }} \sim N(0, 1) $

Observed Test Statistic:
$ z_{obs} = \dfrac{\hat{p}_{obs} - p_0}{\sqrt{\dfrac{p_0(1 - p_0)}{n} }} $

$ \mathbf{\textit{P}}$-value:
1) $ \mathbb{P}(\hat{P} \le \hat{p}_{obs}) = \mathbb{P}(Z \le z_{obs}) $
2) $ 2 \cdot \mathbb{P}(\hat{P} \ge \hat{p}_{obs}) = \mathbb{P}(\big| Z \big| \ge \big| z_{obs} \big|) $
3) $ \mathbb{P}(\hat{P} \ge \hat{p}_{obs}) = \mathbb{P}(Z \ge z_{obs}) $

Conditions for Distributional Approximation (To $ Z$) (Assuming $ H_0 $ is true):

Independent Observations
Number of expected successes and expected failures is at least 10 $ \left[ n p_0 \ge 10 \text{ and } n (1 - p_0) \ge 10 \right] $

Formula for CI:
$ \hat{p}_{obs} \pm z^* \sqrt{ \dfrac{\hat{p}_{obs}(1 - \hat{p}_{obs})}{n} }$

Conditions for Distributional Approximation (To $ Z$):

Independent Observations
Number of observed successes and observed failures is at least 10 $ \left[ n \hat{p}_{obs} \ge 10 \text{ and } n (1 - \hat{p}_{obs}) \ge 10 \right] $

Example Problem

qplot(x = response, data = dataframe, geom = "bar")