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 45% 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 126 insurance claims to be processed in the next few days. Find the following probabilities. (Round your answers to four decimal places.)

(a) half or more of the claims have been padded


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


(c) from 40 to 64 of the claims have been padded


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

Answer 1

Let X be the number of claims that are padded

The probability of a claim being padded = 0.45

X follows binomial distribution with parameters n = 126 and p = 0.45

We approximate binomial distribution with normal distribution when

Therefore we approximate X with normal distribution

Mean of X = np = 56.7

Standard deviation of X =

a) Half of the claims = 63

  We have used continuity correction

  where Z follows a standard normal distribution

From Z table

Therefore the probability that half or more of the claims have been padded is 0.1469

b)

  We have used continuity correction

  where Z follows a standard normal distribution

From Z table

Therefore the probability that fewer than 45 of the claims have been padded is 0.0139

c)

  We have used continuity correction

  where Z follows a standard normal distribution

From Z table

Therefore the probability that from 40 to 64 of the claims have been padded is 0.9207-0.0009 = 0.9198

d)

P(more than 80 of the claims have not been padded) = P(less than 46 of the claims have been padded)

  We have used continuity correction

  where Z follows a standard normal distribution

From Z table

Therefore the probability that more than 80 of the claims have not been padded is 0.0217