Solved Prob 1: An Insurance Company Is Entering A New Area

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

An insurance company is entering a new area, which is demographically similar to other areas where it operates. Historically, the average claims in these areas are $20,450 with a known standard deviation of $4,500. The company is concerned that claims in this new area might be higher. A random sample of 50 claims in the new area yields a sample mean of $21,050. Using a significance level of α = 0.05, we are asked to test whether the claims are significantly higher in the new area, thereby validating the company's concern.

First, formulate the hypotheses: the null hypothesis (H₀) is that the population mean claims remain at $20,450, i.e., H₀: μ = 20,450. The alternative hypothesis (H₁) is that the claims are higher, H₁: μ > 20,450. Since the standard deviation of claims is known and the population is assumed normal or the sample size is sufficiently large, we deploy a z-test for the mean.

Calculate the test statistic: z = (x̄ - μ₀) / (σ / √n) = (21,050 - 20,450) / (4,500 / √50). Plugging in values: z = 600 / (4,500 / 7.07) ≈ 600 / 636.68 ≈ 0.941. The critical value for a one-tailed test at α = 0.05 is 1.645 since we are testing for higher claims.

Decision rule: if z > 1.645, reject H₀. Since 0.941

Alternatively, determine the p-value: p = 1 - Φ(z) = 1 - Φ(0.941). Using standard normal distribution tables or software, Φ(0.941) ≈ 0.8264, thus p ≈ 0.1736, which exceeds 0.05. Therefore, insufficient evidence exists to conclude that claims are higher in the new area.

Conclusion: Based on the sample data, we cannot reject the null hypothesis at the 5% significance level. The evidence does not support the concern that claims are significantly higher in this new area.

Paper For Above instruction

The process of hypothesis testing forms a cornerstone of inferential statistics, allowing organizations to make data-driven decisions regarding observations in a population based on sample data. The scenario presented involves testing whether a new area has higher insurance claims than the historical average, which necessitates employing a statistical hypothesis test to evaluate the claim effectively.

In this context, the null hypothesis (H₀: μ = 20,450) states that the population mean claims in the new area remain unchanged from previous areas. Conversely, the alternative hypothesis (H₁: μ > 20,450) posits that claims are higher, which would justify concern and potential adjustments in insurance policies or premiums. Using a significance level of α = 0.05, analysts determine the critical value for a one-sided z-test, which is approximately 1.645.

The calculation of the test statistic involves the sample mean, the hypothesized mean, the known population standard deviation, and the sample size. Specifically, the z-score quantifies how many standard errors the sample mean is away from the hypothesized mean. For this case, z ≈ 0.941, which falls short of the critical value, indicating a lack of statistical significance.

The p-value approach further supports this conclusion. The p-value of approximately 0.174 exceeds the significance threshold of 0.05, emphasizing that the observed sample mean is not sufficiently unlikely under the null hypothesis to warrant rejection. This result implies that, based on the data, there is no compelling statistical evidence to suggest claims are significantly higher in the new area.

Understanding these statistical tools enables organizations to avoid hasty conclusions based solely on sample fluctuations. In this case, the analysis indicates that the company's concern about increased claims is not statistically justified at the typical significance level. Consequently, the company may decide to continue monitoring claims or proceed with their current policies without immediate concern for elevated claim amounts.

In real-world applications, such hypothesis testing allows insurers to assess whether observed differences in claims are due to random variation or represent genuine changes in the underlying population. Proper interpretation ensures resources are allocated effectively, and risk management strategies are based on solid statistical evidence.

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