About 42 of all US adults will try to pad the insurance clai

About 42% of all US adults will try to pad the insurance claim to be processed in the next few days. Suppose that you are director of an insurance adjustment office, your office has just recorded 140 insurance claims to be processed in the next few days, Find the following probabilities.a, that half or more of the claim are padded. b, fewer than 45 are padded are padded. c, from 40 to 64 are padded and d, more than 80 are padded

Solution

Let X be the random variable that US adults will try to pad the insurance claim to be processed in the next few days.

X follows binomial distribution with n = 140 and p=0.42

Here n is very large.

np = 140*0.42 = 58.8

n*(1-p) = 140*(1-0.42) = 81.2

both are > 5.

So we can do it by normal distribution.

X has normal distribution wirh mean = n*p = 140 * 0.42 = 58.8

variance = n*p*(1-p) = 140 * 0.42*(1-0.42) = 34.104

sd = sqrt(variance) = sqrt(34.104) = 5.8399

Find the following probabilities.

a, that half or more of the claim are padded.

half of 140 claims are padded.

half of 140 = 140 / 2 = 70

that is we have to find P(X = 70).

by making continuiyy correction ,

P(70-0.5 < X < 70+0.5) = P(69.5 < X < 70.5)

Now convert X into z-score.

z = (x - mean) /sd

z-score for x=69.5,

z = (69.5 - 58.8) / 5.8399 = 1.8322

z-score for x = 70.5

z = (70.5 - 58.8) / 5.8399 = 2.0035

that is P(1.8322 < Z < 2.0035) = P(Z <=2.0035) - P(Z < = 1.8322)

These probabilities we can find by using EXCEL.

syntax :

=NORMSDIST(z)

P(1.8322 < Z < 2.0035) = 0.9774 - 0.9665 = 0.0109

b, fewer than 45 are padded are padded

P(X < 45)

convert x = 45 into z-score.

z = (45 - 58.8) / 5.8399 = -2.3631

P(Z < -2.3631) = 0.0091

c, from 40 to 64 are padded

P(40 < X < 64)

convert x= 40 and x=64 into z-score.

z = (40 - 58.8) / 5.8399 = -3.2193

z = (64-58.8) / 5.8399 = 0.8904

P(-3.2193 < Z < 0.8904) = P(Z < = 0.8904) - P( Z < = -3.2193)

= 0.8134 - 0.0006 = 0.8128

d, more than 80 are padded

P(X > 80) =

convert x=80 into z-score.

z = (80 - 58.8)/5.8399 = 3.6302

P(Z > 3.6302) = 1 - P(Z <=3.6302) = 1 - 0.9999 = 0.0001