Goodness of Fit

Type of Plot:
Bar Graph

Parameters: $P_1, P_2, \cdots, P_k$
Point Estimates: $\hat{P}_1, \hat{P}_2, \ldots, \hat{P}_k$

Confidence Interval
does not apply

Hypothesis Test

Hypotheses:
$ H_0: P_1 = p_{1, 0}, P_2 = p_{2, 0}, \ldots, P_k = p_{k, 0} $
$ H_a: $ At least one $ P_i \ne p_{i, 0} $ for $ i \in { 1, \ldots, k }$

Test Statistic Random Variable (Assuming $H_0$ is true):
$ X^2 = \sum_{\text{all cells}} \dfrac{\text{(observed - expected)}^2}{\text{expected}} \sim \chi^2 (df = k - 1) $

Observed Test Statistic:
$ {x_{obs}}^2 = \sum_i \dfrac{(n_{obs, i} - n_{exp, i})^2}{n_{exp, i}} $

$ \mathbf{\textit{P}}$-value:
$ \mathbb{P}(X^2 \ge {x_{obs}}^2) $

Conditions for Distributional Approximation (To $ \chi^2$) (Assuming $ H_0 $ is true):

Independent Observations
All expected cell counts $ \ge 5 $
Degrees of freedom $ \ge 2 $

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