Answer
(a) $-z_\frac{α}{2}\lt z_0\lt z_\frac{α}{2}$: null hypothesis is not rejected.
(b) $P$-value $\gtα$: null hypothesis is not rejected.
Work Step by Step
$np_0(1-p_0)=500\times0.9(1-0.9)=45\gt10$
$p̂ =\frac{x}{n}=\frac{440}{500}=0.88$
$z_0=\frac{p̂ -p_0}{\sqrt {\frac{p_0(1-p_0)}{n}}}=\frac{0.88-0.9}{\sqrt {\frac{0.9(1-0.9)}{500}}}=-1.49$
(a) $z_\frac{α}{2}=z_{0.025}=1.96$
Also, $-z_\frac{α}{2}=-1.96$
Since $-z_\frac{α}{2}\lt z_0\lt z_\frac{α}{2}$, we do not reject the null hypothesis.
(b) $P$-value $=2P(z\gt|z_0|)=2P(z\gt1.49)=2(1-0.9319)=0.1362$
Since $P$-value $\gtα$, we do not reject the null hypothesis.
