Solved Problem 1: An Insurance Company Is Entering A New Are

Solved Prob 1an Insurance Company Is Entering In A New Area Which Is

An insurance company is entering a new area that is demographically similar to other regions where it insures. Historical data suggests an average claim amount of $20,450. However, the company suspects that claims in this new area may be higher. To test this, they randomly select 50 claims and find a sample mean claim amount of $21,050, with the population standard deviation known to be $4,500. Using a significance level of α = 0.05, they want to determine if the claims are indeed higher, indicating a valid concern.

The hypotheses are formulated as follows: the null hypothesis (H0) states that the mean claim amount has not increased, i.e., μ = 20,450 (or μ ≤ 20,450), and the alternative hypothesis (HA) posits that the claims have increased, i.e., μ > 20,450.

Since the population standard deviation is known, the test is performed using a z-test. The test statistic is calculated as:

z = (sample mean - population mean) / (population standard deviation / √n) = (21,050 - 20,450) / (4,500 / √50) ≈ 600 / (4,500 / 7.07) ≈ 600 / 636.44 ≈ 0.941

For a one-sided test at α = 0.05, the critical value (cutoff point) is approximately 1.645. Since the calculated z-value of 0.941 is less than 1.645, we do not reject the null hypothesis. Alternatively, calculating the p-value:

p-value = 1 - NORM.S.DIST(0.941) ≈ 1 - 0.8264 ≈ 0.1736

As the p-value exceeds 0.05, the evidence is insufficient to support the claim that the mean claim amount is higher in this new area.

Paper For Above instruction

The process of hypothesis testing in insurance claims analysis is fundamental to understanding whether observed data substantiate claims of increased risks or costs. In this scenario, the insurance company’s concern is whether the average claim amount in the new area exceeds the historical average of $20,450. Establishing a formal statistical test involves creating null and alternative hypotheses and choosing the appropriate test statistic based on the known parameters.

The null hypothesis (H0): μ ≤ $20,450, suggesting no increase. The alternative hypothesis (HA): μ > $20,450, indicating a potential rise in claims. Given the population standard deviation is known, a z-test is appropriate. The sample size of 50 claims provides sufficient power to this test, allowing for the calculation of the z-statistic with the sample mean, population mean, known standard deviation, and sample size. The calculation yields a z-value of approximately 0.941.

Comparing this z-value to the critical value of 1.645 (for a one-sided test at 0.05 significance level), it is evident that the data do not provide sufficient evidence to reject H0. The p-value (~0.1736) corroborates this conclusion. Therefore, based on this sample, the insurance company’s suspicion of higher claims cannot be statistically confirmed. Even though the sample mean is higher than the historical average, the difference is not statistically significant at the 5% level.

In practical terms, this analysis suggests that the claims in the new area are not conclusively higher than in previous regions, providing the company with a measure of reassurance. Nevertheless, continued monitoring and data collection would be advisable to confirm this initial finding over time. This process exemplifies how hypothesis testing aids in decision-making under uncertainty within actuarial and insurance contexts, facilitating evidence-based strategies and risk assessment.

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