Question

A nationwide survey of college seniors by a university revealed that almost 70​% disapprove daily pot​ smoking, according to a report in a magazine. If 14 seniors are selected at random and asked their​ opinion, find the probability that the number who disapprove of smoking pot daily is​ (a) anywhere from 7 to 9​, ​(b) at most 5 and​ (c) not less than 8.

Answer 1

Answer:

a) P ( 7 ≤ X ≤ 9) = 0.384

b) P(X ≤ 5) = 0.0083

c) P( X ≥ 8) = 0.904

Step-by-step explanation:

Proportion of college seniors that disapprove daily pot smoking, p = 0.7

Proportion of college seniors that do not disapprove daily pot smoking,

q = 1 - p = 1 - 0.7

q = 0.3

14 seniors are selected i.e. sample size, n = 14

This is a binomial distribution question:

$p(X = r) = nCr p^{r} q^{n-r}$

a) probability that the number who disapprove of smoking pot daily is anywhere from 7 to 9​

P ( 7 ≤ X ≤ 9) = P(X=7) + P(X=8) + P(X=9)

$P(X=7) = 14C7 p^{7} q^{14-7} \\P(X=7) = 14C7 * 0.7^{7} 0.3^{7}\\P(X=7) = 3432 * 0.7^{7} 0.3^{7}\\P(X=7) = 0.062$

$P(X=8) = 14C8 p^{8} q^{14-8} \\P(X=8) = 14C8 * 0.7^{8} 0.3^{6}\\P(X=8) = 3003 * 0.7^{8} 0.3^{6}\\P(X=8) = 0.126$

$P(X=9) = 14C9 p^{9} q^{14-9} \\P(X=9) = 14C9 * 0.7^{9} 0.3^{5}\\P(X=9) = 2002 * 0.7^{9} 0.3^{5}\\P(X=9) = 0.196$

P ( 7 ≤ X ≤ 9) = P(X=7) + P(X=8) + P(X=9)

P ( 7 ≤ X ≤ 9) = 0.062 + 0.126 + 0.196

P ( 7 ≤ X ≤ 9) = 0.384

b) Probability that the number who disapprove of smoking pot daily is​ at most 5

P(X ≤ 5) = P(X=0) + P(X=1) + P(X=2) + P(X=3) + P(X=4) + P(X=5)

$P(X=0) = 14C0 p^{0} q^{14-0} \\P(X=0) = 14C0 * 0.7^{0} 0.3^{14}\\P(X=0) = 1 * 0.7^{0} 0.3^{14}\\P(X=0) = 0.0000000478$

$P(X=1) = 14C1 p^{1} q^{14-1} \\P(X=1) = 14C1 * 0.7^{1} 0.3^{13}\\P(X=1) = 14 * 0.7^{1} 0.3^{13}\\P(X=1) = 0.00000156$

$P(X=2) = 14C2 p^{2} q^{14-2} \\P(X=2) = 14C2 * 0.7^{2} 0.3^{12}\\P(X=2) = 91 * 0.7^{2} 0.3^{12}\\P(X=2) = 0.0000237$

$P(X=3) = 14C3 p^{3} q^{14-3} \\P(X=3) = 14C3 * 0.7^{3} 0.3^{11}\\P(X=3) = 364 * 0.7^{3} 0.3^{11}\\P(X=3) = 0.000221$

$P(X=4) = 14C4 p^{4} q^{14-4} \\P(X=4) = 14C4 * 0.7^{4} 0.3^{10}\\P(X=4) = 1001 * 0.7^{4} 0.3^{10}\\P(X=4) = 0.00142$

$P(X=5) = 14C5 p^{5} q^{14-5} \\P(X=5) = 14C5 * 0.7^{5} 0.3^{9}\\P(X=5) = 2002 * 0.7^{5} 0.3^{9}\\P(X=5) = 0.00662$

P(X ≤ 5) = 0.0000000478 + 0.00000156 + 0.0000237 + 0.000221 + 0.00142 + 0.00662

P(X ≤ 5) = 0.0083

c) Probability that the number who disapprove of smoking pot daily is​ not less than 8.

P( X ≥ 8) = P(X=8) + P(X=9) + P(X=10) + P(X=11) + P(X=12) + P(X=13) + P(X=14)

$P(X\geq 8) = (14C8 * 0.7^{8} 0.3^{6}) + (14C9 * 0.7^{9} 0.3^{5}) + (14C10 * 0.7^{10} 0.3^{4}) + (14C11 * 0.7^{11} 0.3^{3}) + (14C12 * 0.7^{12} 0.3^{2}) + ( 14C13 * 0.7^{13} 0.3^{1}) + (14C14 * 0.7^{14} 0.3^{0})$

P( X ≥ 8) = 0.126 + 0.194 + 0.229 + 0.194 + 0.113 + 0.041 + 0.00678

P( X ≥ 8) = 0.904