Question

CNNBC recently reported that the mean annual cost of auto insurance is 978 dollars. Assume the standard deviation is 299 dollars. You take a simple random sample of 92 auto insurance policies.

Find the probability that a single randomly selected value is less than 988 dollars.
P(X < 988) = ___________

Find the probability that a sample of size n=92n=92 is randomly selected with a mean less than 988 dollars.
P(M < 988) = ____________

Answer 1

a)

Given,

= 978 , = 299

We convert this to standard normal as

P(X < x) = P(Z < ( x - ) / )

So,

P(X < 988) = P(Z < (988 - 978) / 299)

= P(Z < 0.03)

= 0.5120

b)

Using central limit theorem,

P( < x) = P(Z < ( x - ) / ( / sqrt(n) ) )

So,

P( < 988) = P(Z < ( 988 -978) / (299 / sqrt(92) ) )

= P(Z < 0.32)

= 0.6255