Answer
$z_0\gt z_α$: reject the null hypothesis.
There is enough evidence that the proportion of Americans who consider themselves to be liberal is higher than 0.21.
Work Step by Step
$H_0:~p=0.21$ versus $H_1:~p\gt0.21$
$np_0(1-p_0)=199\times0.21(1-0.21)=33.0141\gt10$
$p̂ =\frac{x}{n}=\frac{53}{199}=0.2663$
$z_0=\frac{p̂ -p_0}{\sqrt {\frac{p_0(1-p_0)}{n}}}=\frac{0.2663-0.21}{\sqrt {\frac{0.21(1-0.21)}{199}}}=1.95$
Using the classical method:
$z_α=z_{0.1}$
If the area of the standard normal curve to the right of $z_{0.1}$ is 0.1, then the area of the standard normal curve to the left of $z_{0.1}$ is $1−0.1=0.9$
According to Table V, the z-score which gives the closest value to 0.9 is 1.28.
Since $z_0\gt z_α$, we reject the null hypothesis.
