An Insurance Company Has A Random Number Of Claims In A Give

An Insurance Company Has A Random Number Of Claims In a Given Year Wit

An insurance company has a random number of claims in a given year with an average of 1000 claims. The claim size is also random, with an average of $1200. The company wants to determine the expected total payout to meet the claims in a year, considering both the average number of claims and the average claim size.

Additionally, a separate statistical estimation problem involves constructing a 90% confidence interval for a population mean. A sample of size 100 is taken, yielding a sample mean of 1300. Given that the population standard deviation is 25, the task is to calculate the margin of error for this confidence interval.

Paper For Above instruction

Introduction

Insurance companies operate by managing uncertain risks, primarily stemming from the claims they need to pay out in a given period. Quantifying expected payouts and understanding variability through confidence intervals are vital aspects of risk management, financial planning, and decision-making in the insurance industry. This paper discusses the calculation of the expected total claims payout based on claim frequency and severity, as well as how to determine the margin of error for a population mean estimate using sample data.

Expected Total Claims Payout

The total amount an insurance company expects to pay in claims within a year depends on the frequency of claims and their individual sizes. The problem provides an expected claim count (the mean number of claims per year) as 1000 and an average claim size of $1200. Assuming independence between these variables and in the absence of information about the distribution, the expected total payout can be derived using basic principles of expectation in probability theory.

The total claims payout \( T \) can be modeled as the sum of individual claim sizes for all claims, i.e.,

T = N * X

where \( N \) is the number of claims, a random variable with an expected value \( E[N] = 1000 \), and \( X \) is the claim size, with a mean \( E[X] = \$1200 \). Given the independence typically assumed in such models, the expected total payout is calculated as:

E[T] = E[N  X] = E[N]  E[X] = 1000 * 1200 = \$1,200,000

Therefore, the insurance company should expect to pay approximately \$1,200,000 in claims per year, on average. This estimation helps in setting appropriate reserve levels, premium calculations, and risk assessment.

Constructing a 90% Confidence Interval & Calculating Margin of Error

Separately, the problem involves statistical inference where a sample of size 100 yields a sample mean of 1300, and the population standard deviation is known to be 25. The goal is to determine the margin of error (MOE) for a 90% confidence interval around the population mean.

Since the population standard deviation is known, the confidence interval for the population mean is constructed using the normal distribution. The formula for the margin of error in this context is:

MOE = Z_{α/2} * (σ / √n)

where \( Z_{α/2} \) is the critical value corresponding to the desired confidence level, \( σ \) is the known population standard deviation, and \( n \) is the sample size.

For a 90% confidence level, the critical z-value is approximately 1.645, as derived from standard normal distribution tables.

Plugging in the numbers:

MOE = 1.645  (25 / √100) = 1.645  (25 / 10) = 1.645 * 2.5 = 4.1125

Thus, the margin of error for the 90% confidence interval is approximately \$4.11. This means the estimate of the population mean (1300) will be within ±\$4.11 with 90% confidence. This small margin of error indicates a high precision of the sample mean estimate given the sample size and known population variability.

Conclusion

In summary, the insurance company can expect to pay around \$1.2 million annually to settle claims, based on average claim frequencies and sizes. This figure underpins financial planning, reserve setting, and risk management strategies. From a statistical perspective, when analyzing sample data, understanding how to calculate the margin of error for confidence intervals helps quantify the uncertainty around the estimated mean. In this case, a margin of approximately \$4.11 allows for confident inference about the population mean based on the sample data provided.

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