Question

In one study it was found that  84%  of all homes have a functional smoke detector. Suppose this proportion is valid for all homes. Find the probability that in a random sample  of  150  homes, between  70 %  and  90%  will have a functional smoke detector.

Answer 1

$P(0.70<p<0.90) \\ P(X<p) = P(Z< \frac{p-p_0}{sd}) \\ p_0 = 0.84 \\ sd = \sqrt{\frac{p_0(1-p_0)}{n}} \\ n=150 \\ sd = \sqrt{\frac{0.84(1-0.84)}{150}} \\ sd = 0.0299 \\ P(0.70<p<0.90) = P(p< 0.90) -P(p<0.70) \\ = P(Z< \frac{0.90 -0.84}{0.0299}) -P(Z< \frac{0.70-0.84}{0.0299}) \\ = P(Z< 2.0067) -P(Z< -4.6823) \\ = 0.9776 -0.0000014 \\ = 0.9776$