Question

The mean annual cost of automobile insurance is $939 (CNBC, February 23, 2006). Assume that the standard deviation is σ = $245. What is the probability that a sample of 50 automobile insurance policies will have a mean within $25 of the population mean? Please show all steps.

Answer 1

Let X be the random variable denoting the mean annual

cost of automobile insurance.

X ~ N(939, 245) i.e. (X - 939)/245 ~ N(0,1)

Let M be the sample mean of annual automobile insurance

cost of 50 automobile insurance policies.

E(M) = 939, s.d.(M) = 245 / = 34.6482.

The probability that the sample mean will be within $25 of

the population mean = P(939 - 25 < M < 939 + 25)

= P(914 < M < 964) = P[(914 - 939)/34.6482 <

(M - 939)/34.6482 < (964 - 939)/34.6482]

= P[-0.7215 < (M - 939)/34.6482 < 0.7215]

= (0.7215) - (-0.7215) = 0.7647 - 0.2353 = 0.5294. (Ans).