Answer to Question #196704 in Statistics and Probability for jack

Question

A car dealership has both Electric and Internal Combustion Engine (ICE) vehicles available for sale. 30% of customers looking to purchase a new car are interested in Electric vehicles, while the rest want ICE vehicles. On a particular day, the dealership had 17 people coming in to purchase a new car. What is the probability that the number of people who are interested in an Electric vehicle is more than two standard deviations away from the mean value?

Answer 1

Let X be the number of people.

$X\sim B(n,p) \\ X\sim B(17,0.30)\\$

Mean $E(X)=np=17\times 0.3=5.1$

Standard deviation-

$s=\sqrt{np(1-p)}=\sqrt{17\times 0.3\times (1-0.3)}=1.8894$

By using Normal approximation of Binomal distribution.

$X\sim N(\mu=5.1,\sigma=1.8894)$

Probability that the number of people who are interested in an Electric vehicle is more than two standard deviations away from the mean value is-

$=P(X<\mu-2\sigma)+P(X>\mu+2\sigma) \\[9pt] =P(Z<-2)+P(z>2) \\[9pt] =0.0228+0.0228\\[9pt]=0.0456$