Test of Proportions

Discrete cases like $X \sim B i n o m$ , where we have

\frac{occurrences}{total trials}

Two-sided tests

H_{0} : θ = θ_{0}, v s H_{a} : θ \neq θ_{0}

For a two-sided test, we want to define the critical regions as: if

\begin{aligned} X \geq K (\frac{α}{2}) & or & X \leq K^{'} (\frac{α}{2}) \\ evidende to suggest θ > θ_{0} & evidence to suggest θ < θ_{0} \end{aligned}

we reject $H_{0}$

$K (\frac{α}{2})$ is the smallest integer for which $P (X \geq K | H_{0}) \leq \frac{α}{2}$
$K^{'} (\frac{α}{2})$ is the largest integer for which $P (X \leq K^{'} | H_{0}) \leq \frac{α}{2}$

H_{0} : θ = θ_{0} v s H_{a} : θ > θ_{0}

We define the critical region for an $α$ level test as:

We can use the Central Limit Theorem and use the normal approximation to the binomial distribution

E (X) = n θ_{0}, v a r (X) = n θ_{0} (1 - θ_{0})

Z = \frac{X - E (X)}{\sqrt{V a r (X)}} = \frac{X - n θ_{0}}{\sqrt{n θ_{0} (1 - θ_{0})}} \overset{a p p r o x}{\sim} N (0, 1) under H_{0}

we reject $H_{0}$ if $| Z_{o b s} | \geq Z_{1 - \frac{α}{2}}$

When n is moderate (not very large)

Z = \frac{(x \pm \frac{1}{2}) - n θ_{0}}{\sqrt{n θ_{0} (1 - θ_{0})}} \overset{approx}{\sim} N (0, 1)

when

{\begin{cases} x \geq n θ_{0} \to + \frac{1}{2} \\ x < n θ_{0} \to - \frac{1}{2} \end{cases}

for continuity correction