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 130 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

here as np=130*0.45 =58.5 and np(1-p)=71.5 both are greater then 10 ; therefore we can use normal approximation

here mean of distribution=μ=np=		58.5
and standard deviation σ=sqrt(np(1-p))=		5.6723

for normal distribution z score =(X-μ)/σx
therefore from normal approximation of binomial distribution and continuity correction:

a)

half or more of the claims have been padded :

probability =

P(X>64.5)

=

P(Z>1.06)=

1-P(Z<1.06)=

1-0.8554=

0.1446

b)

fewer than 45 of the claims have been padded

probability =

P(X<44.5)

=

P(Z<-2.47)=

0.0068

c)

from 40 to 64 of the claims have been padded :

probability =

P(39.5<X<64.5)

=

P(-3.35<Z<1.06)=

0.8554-0.0004=

0.8550

d)

more than 80 of the claims have not been padded =at most 49 have been padded:

probability =

P(X<49.5)

=

P(Z<-1.59)=

0.0559