Question

In the following problem, check that it is appropriate to use the normal approximation to the binomial. Then use the normal distribution to estimate the requested probabilities.

Do you try to pad an insurance claim to cover your deductible? About 43% of all U.S. adults will try to pad their insurance claims! Suppose that you are the director of an insurance adjustment office. Your office has just received 140 insurance claims to be processed in the next few days. Find the following probabilities. (Round your answers to four decimal places.)

(b) fewer than 45 of the claims have been padded

(d) more than 80 of the claims have not been padded

Answer 1

Answer:

Given,

sample n = 140

p = 0.43

q = 1 - p

= 1 - 0.43

= 0.57

Mean = np = 140*0.43 = 60.2 [have padded]

Standard deviation = sqrt(npq) = sqrt(140*0.43*0.57) = 5.86

b)

P(X < 45) = P(X < 44.5)

= P(z < (44.5 - 60.2)/5.86)

= P(z < -2.68)

= 0.0036811 [since from z table]

= 0.0037

d)

Mean = np = 140*0.57 = 79.8 [haven't padded]

Standard deviation = sqrt(npq) = sqrt(140*0.43*0.57) = 5.86

P(X > 80) = P(X > 80.5)

= P(z > (80.5 - 79.8)/5.86)

= P(z > 0.12)

= 0.4522416 [since from z table]

= 0.4522