Question

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evaluate the formula ( z = \frac { hat { p } - p } { sqrt { \frac { p q } { n } } } ) when ( hat { p } = \frac { 150 } { 550 } ), ( n = 550 ), ( p = 0.25 ), and ( q = 1 - p ).
( z = ) (round to two decimal places as needed.)

Explanation:

Step1: Calculate $\hat{p}$

$\hat{p}=\frac{150}{550}\approx0.27$

Step2: Calculate $q$

Since $q = 1 - p$ and $p = 0.25$, then $q=1 - 0.25=0.75$

Step3: Calculate the denominator

First, calculate $pq=0.25\times0.75 = 0.1875$. Then $\sqrt{\frac{pq}{n}}=\sqrt{\frac{0.1875}{550}}\approx\sqrt{0.0003409}\approx0.01846$

Step4: Calculate the numerator

The numerator is $\hat{p}-p=0.27 - 0.25=0.02$

Step5: Calculate $z$

$z=\frac{\hat{p}-p}{\sqrt{\frac{pq}{n}}}=\frac{0.02}{0.01846}\approx1.08$

Answer:

$1.08$