( $15$ points) $\newline$ For Company A there is a $65\%$ chance that no claim is made during the coming year. If one or more claims are made, the total claim amount is normally distributed with mean $10,000$ and standard deviation $1,500$ . $\newline$ For Company B there is a $75\%$ chance that no claim is made during the coming year, If one or more claims are made, the total claim amount is normally distributed with mean $9,000$ and standard deviation $2,000$ . $\newline$ Assuming that the total claim amounts of the two companies are independent, what is the probability that, in the coming year, Company B's total claim amount will exceed Company A's total claim amount? $\newline$ Solution

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Interpret confidence intervals for population means

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Q. (

15

points)

\newline

For Company A there is a

65\%

chance that no claim is made during the coming year. If one or more claims are made, the total claim amount is normally distributed with mean

10,000

and standard deviation

1,500

\newline

For Company B there is a

75\%

chance that no claim is made during the coming year, If one or more claims are made, the total claim amount is normally distributed with mean

9,000

and standard deviation

2,000

\newline

Assuming that the total claim amounts of the two companies are independent, what is the probability that, in the coming year, Company B's total claim amount will exceed Company A's total claim amount?

\newline

Solution

Understand the Problem: Understand the problem and determine what is being asked. $\newline$ We need to find the probability that Company B's total claim amount will be greater than Company A's total claim amount. Given that the claim amounts are normally distributed and the probabilities of no claim are $65\%$ for Company A and $75\%$ for Company B, we need to consider the cases where claims are made by both companies.
Calculate Probability of Claims: Calculate the probability of at least one claim being made for both companies. $\newline$ For Company A, the probability of at least one claim being made is $1 - 0.65 = 0.35$ . $\newline$ For Company B, the probability of at least one claim being made is $1 - 0.75 = 0.25$ .
Set Up Problem: Set up the problem to find the probability that Company B's claim amount exceeds Company A's claim amount. $\newline$ We need to find $P(B > A | A \text{ makes a claim and } B \text{ makes a claim}) \times P(A \text{ makes a claim}) \times P(B \text{ makes a claim})$ .
Calculate Probability of Exceeding: Calculate the probability that Company B's claim amount exceeds Company A's claim amount given that both companies make a claim. $\newline$ Since the claim amounts are independent and normally distributed, the difference of the claim amounts will also be normally distributed. The mean of the difference will be the difference of the means, and the variance of the difference will be the sum of the variances (since they are independent). $\newline$ Mean difference = Mean of B - Mean of A = $9,000$ - $10,000$ = $-1,000$ . $\newline$ Variance of the difference = Variance of B + Variance of A = $(2,000)^2$ + $(1,500)^2$ = $4,000,000$ + $2,250,000$ = $6,250,000$ . $\newline$ Standard deviation of the difference = $\sqrt{\text{Variance}}$ = $\sqrt{6,250,000}$ = $10,000$ $0$ .
Find Z-Score: Find the z-score for the mean difference. $Z = \frac{X - \text{Mean}}{\text{Standard Deviation}} = \frac{0 - (-1,000)}{2,500} = \frac{1,000}{2,500} = 0.4$ .
Use Z-Score to Find Probability: Use the z-score to find the probability that B's claim amount exceeds A's claim amount. We look up the z-score of $0.4$ in the standard normal distribution table or use a calculator to find the probability. The probability corresponding to $z = 0.4$ is approximately $0.6554$ . This is the probability that B's claim amount is greater than A's claim amount given that both companies make a claim.
Combine Probabilities: Combine the probabilities to find the overall probability that Company B's claim amount will exceed Company A's claim amount. $\newline$ Overall probability = $P(B > A \,|\, A \text{ makes a claim and } B \text{ makes a claim}) \times P(A \text{ makes a claim}) \times P(B \text{ makes a claim}) = 0.6554 \times 0.35 \times 0.25$ .
Calculate Final Answer: Calculate the final answer. $\newline$ Overall probability = $0.6554 \times 0.35 \times 0.25 = 0.0573$ or $5.73\%$ .