Question

The probability that a rental car will be stolen is 0.001. if 25,000 cars are rented from hertz, what is the normal approximation to the probability that fewer than 20 will be stolen?

Answer 1

N=25000, p=0.001, np=25000*0.001=25 >10, so the normal approximation should apply.
fewer than 20 => 0,1,2,3...19
Applying the continuity correction, we have X<19.5.
&mu;=np=25
&sigma;&sup2;=npq=25000*.001*.999=24.975 =>
&sigma;=&radic;(24.975)=4.9975
Z(X<19.5)=(19.5-25)/4.9975=-1.100550
P(X<19.5)=P(Z<-1.10055)=0.1355463 [from normal probability table, left tail]
Hence normal approximation to probability that fewer than 20 cars will be stolen is 0.1355.
[actual value by binomial distribution is 0.133450. approximation is quite close]