Question

Insurance companies need to maintain a certain amount in reserved funds in order to pay anticipated claims. The average monthly claim amount for the last 60 months for company A was $7,500,000 and the (sample) standard deviation was $1,200,000.

(a) Find a 95% upper confidence bound on the average monthly claim amount.
(b) The regulations on the reserves will be strengthened in the near future. According to the new regulations, insurance companies that do not have sufficient amount in reserve will be subject to a significant penalty. Company A wants to adjust the target reserve amount accordingly, by computing a new upper bound on the average monthly claim amount. Should the company recalculate an upper confidence bound with a higher or a lower level of confidence? Briefly explain why. Then compute a 99.95% upper confidence bound on the average monthly claim amount.

Answer 1

a ) Upper bound of CI is 7803483.87

b) The new upper bound is 8039767.74

Explanation:

From sample data:

sample size n = 60

sample mean x = 7500000

Sample standard deviation s = 1200000

a) Confidence Interval 95 % then significance level α = 5 % o α = 0.05

α/2 = 0.025 z(c) from z-table is z(c) = 1.96

CI 95 % = ( x ± z(c) * s/√n

CI 95 % = ( 7500000 ± 1.96 * 1200000/√60

CI 95 % = ( 7500000 ± 303483.87 )

CI 95 % = ( 7196516.13 ; 7803483.87)

Then upper bound of CI is 7803483.87

b) The company has to decrease the significance level equivalent to widen the confidence interval.

If CI now is 99.95 % significance level is α = 0.0005 and

α/2 = 0.00025 z(c) for that α/2 is from z-table z(c) ≈ 3.486

CI 99.95 % = ( x ± z(c)*s/√n

CI 99.95 % = 7500000 ± 3.486*1200000/ √60

CI 99.95 % = (7500000 ±539767.74)

CI 99.95 % = ( 6960232.26 ; 8039767.74)

The new upper bound is 8039767.74