Question not on textbook: an insurance company has 25,000 automobile policy holders. if the yearly claim of a policy holder is a random variable with mean 320 and standard deviation 540, use clt to approximate the probability that the total yearly claim exceeds 8.3 million

There is 0.00022 probability that the yearly income of the insurance claims will exceed 8.3 millions.

Step-by-step explanation:

Given data:

Sample Size = n = 25,000

Mean of yearly claim = Mu = 320

Standard Deviation of yearly claim = SD = 540

What we have to find out?

Probability of total yearly claims exceeding 8.3 million?

As the total no. of policy holders is 25,000 therefore the targeted value of claim will be calculated as follows:

Targeted value of policy claim = x = Total Yearly claim / No. of Policy holders

x = (8.3 * 1.000.000) / 25,000

x = 332

Central Limit Theorem (CLT):

From Central theorem we know that: z(x) = (x - Mu)/(SD/√n) ; Equation 1

By putting values in equation 1 we have:

z(332) = (332-320)/(540 / √25,000)

z(332) = 12 / (540/158.114)

z(332) = 3.51

By using the normal distribution tables we get:

P(x <= 332) = Phi(z) = Phi(3.51)

P(x <= 332) = 0.99978

By using the probability unity method we will get:

P(x > 332) = 1 - P(x <= 332)

P(x > 332) = 1 - 0.99978

P( x > 332) = 0.00022

Thus we get the probability of total yearly claim exceeding 8.3 million equals to 0.00022.