Answer to Question #244353 in Statistics and Probability for Flyyboyy

Question

In a particular country on the African continent, 35% of the population is estimated to have at
at least one smart phone. If a small sample of 40 people is selected from the population
for a statistical investigation, Use the Binomial distribution to estimate the probability that,
the number of people in the sample that have at least one smart phone is;
a) at most 15;
b) more than 12 but fewer than 18;
c) exactly equal to the mean of the distribution

Answer 1

$p=0.35 \\ n=40 \\ q=p-1 = 0.65 \\ P(X=x) = C^n_x p^x q^{n-x} \\ a) \; P(X≤15) = \sum^{15}_{x=0} C^{40}_x (0.35)^x (0.65)^{40-x} \\ P(X≤15) = 0.6946 \\ b) \; P(12<X<18) = \sum^{17}_{13} C^{40}_x (0.35)^x (0.65)^{40-x} \\ = \sum^{17}_{x=0} C^{40}_x (0.35)^x (0.65)^{40-x} - \sum^{12}_{x=0} C^{40}_x (0.35)^x (0.65)^{40-x} \\ = 0.8761-0.3143 \\ = 0.5618 \\ c) \; Mean = np = 40 \times 0.35 = 14 \\ P(X=14) = C^{40}_{14} (0.35)^{14} (0.65)^{26} \\ P(X=14) = 0.1313$