Question

CNNBC recently reported that the mean annual cost of auto insurance is 967 dollars. Assume the standard deviation is 286 dollars. You take a simple random sample of 59 auto insurance policies.

Find the probability that a single randomly selected value is more than 975 dollars. P(X > 975) =

Find the probability that a sample of size n = 59 is randomly selected with a mean that is more than 975 dollars. P(M > 975) =

Answer 1

a)

Here, μ = 967, σ = 286 and x = 975. We need to compute P(X >= 975). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z = (975 - 967)/286 = 0.03

Therefore,
P(X >= 975) = P(z <= (975 - 967)/286)
= P(z >= 0.03)
= 1 - 0.512 = 0.4880

b)

Here, μ = 967, σ = 37.234 and x = 975. We need to compute P(X >= 975). The corresponding z-value is calculated using Central Limit Theorem

z = (x - μ)/σ
z = (975 - 967)/37.234 = 0.21

Therefore,
P(X >= 975) = P(z <= (975 - 967)/37.234)
= P(z >= 0.21)
= 1 - 0.5832 = 0.4168