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 37% 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 138 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

To use normal approximation following conditions must be satisfied:

np ≥ 5   and n(1 − p) ≥ 5

In this case, n = 138 and p = 0.37

n*p = 138*0.37 = 51.06 ≥ 5

n*(1-p) = 138*0.63 = 86.94 ≥ 5

Since both the condition are satisfied we can use the normal approximation to the binomial for this problem.

Please note for all the below answers calculator has been used. If standard normal table is used there may be slight variation in answer as z values need to be rounded off to two decimal places. Result using calculator are more accurate.

Answer a)

Answer b)

Answer c)

Answer d)

37% of all U.S. adults try to pad their insurance claim

So, percentage of U.S. adults who do not try to pad their insurance claim = 100-37 = 63%