Question

In a certain village 28% of all cars are made by Ford.

(ii) 14 cars are chosen randomly in this village. Find the probability that fewer than 4 of these cars are made by Ford.

(iii) A random sample of 50 cars in the village is taken. Estimate, using a normal approximation, the probability that more than 18 cars are made by Ford.

Answer 1

(ii) $P(X<4)=P(X=0)+P(X=1)+P(X=2)+P(X=3)=$

$=C_{14}^00.28^0(1-0.28)^{14-0}+C_{14}^10.28^1(1-0.28)^{14-1}+C_{14}^20.28^2(1-0.28)^{14-2}+$

$+C_{14}^30.28^3(1-0.28)^{14-3}=0.4187.$

(iii) $\mu=np=50*0.28=14.$

$\sigma=\sqrt{np(1-p)}=\sqrt{50*0.28*0.72}=3.17.$

$P(X>18)=P(Z>\frac{18-14}{3.17})=P(Z>1.26)=0.1038.$