Assume that for a family of four the annual dental care expe

Assume that for a family of four, the annual dental care expenses follow the normal distribution with a mean expense equal to $3,000.00 and a standard deviation of $500. An insurer has decided to offer a dental insurance premium reduction if the policyholder’s dental care expenses do not exceed a specified amount.

What dollar amount should be established if the insurer wants families with the lowest 25 percent of annual dental care expenses to be eligible for the premium reduction?

I know p(x<x0)=0.25 then 0.5-0.25 gives me 0.25 area under the curve. The z value that corresponds with 0.25 is z=-0.68 since its the lowest 25% then its easy to find x0. The answer is approximately 3163? I don\'t get that

Solution

Let X be the total expenditure of dental and it has N(3000,500) distribution

suppose K doller reqired be established if the insurer wants families with the lowest 25 percent of annual dental care expenses to be eligible for the premium reduction

that is P(X<K)=0.25

after calculation we get K = $2662.72. Hence this doller is required