In the following problem, check that it is appropriate to use the normal approxi

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 44% 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 128 insurance claims to be processed in the next few days. Find the following probabilities. 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

Explanation / Answer

Here n = 128 and p = 0.44

and np = 0.44*128 =56.32 and n(1-p) = 128*0.56 = 71.68, since np and n(1-p) are both greater than 10, we can use the normal approximation

Mean = np = 56.32

Standard deviation = Sqrt(128*0.44*0.56) = 5.616

a) P(X>64)

= P(z>(64+0.5-56.32)/5.616) =

= 0.0726

b) P(x<45)

= P(z<(45-0.5-56.32)/5.616)

= 0.0177

c) P(40<x<64)

= P((40+0.5-56.32)/5.616<z<(64-0.5-56.32)/5.616)

= 0.8970

d) For this mean = 128*0.56 = 71.68

P(X>80)

P(z>80+0.5-71.68)/5.616)

= 0.0581

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