Question

What is the first big change that American drivers made due to higher gas prices? According to an Access America survey, 30% said that it was cutting recreational driving. However, 28% said that it was consolidating or reducing errands. If these figures are true for all American drivers, and if 20 such drivers are randomly sampled and asked what is the first big change they made due to higher gas prices?

a. What is the probability that exactly 8 said that it was consolidating or reducing errands?
b. What is the probability that none of them said that it was cutting recreational driving?
c. What is the probability that more than 7 said that it was cutting recreational driving?

Answer 1

a)
Here, n = 20, p = 0.28, (1 - p) = 0.72 and x = 8
As per binomial distribution formula P(X = x) = nCx * p^x * (1 - p)^(n - x)

We need to calculate P(X = 8)
P(X = 8) = 20C8 * 0.28^8 * 0.72^12
P(X = 8) = 0.0924
0

b)

Here, n = 20, p = 0.3, (1 - p) = 0.7 and x = 0
As per binomial distribution formula P(X = x) = nCx * p^x * (1 - p)^(n - x)

We need to calculate P(X = 0)
P(X = 0) = 20C0 * 0.3^0 * 0.7^20
P(X = 0) = 0.0008

c)
Here, n = 20, p = 0.3, (1 - p) = 0.7 and x = 7
As per binomial distribution formula P(X = x) = nCx * p^x * (1 - p)^(n - x)

We need to calculate P(X <= 7).
P(X <= 7) = (20C0 * 0.3^0 * 0.7^20) + (20C1 * 0.3^1 * 0.7^19) + (20C2 * 0.3^2 * 0.7^18) + (20C3 * 0.3^3 * 0.7^17) + (20C4 * 0.3^4 * 0.7^16) + (20C5 * 0.3^5 * 0.7^15) + (20C6 * 0.3^6 * 0.7^14) + (20C7 * 0.3^7 * 0.7^13)
P(X <= 7) = 0.0008 + 0.0068 + 0.0278 + 0.0716 + 0.1304 + 0.1789 + 0.1916 + 0.1643
P(X <= 7) = 0.7722

P(x> 7) =1 - P(x< =7)
= 1 - 0.7722
= 0.2278