Final answer:
The probability that the average loss exceeds $300 is approximately 37.56%, assuming the company's report is accurate. To be 90% certain of maintaining profitability, the insurance company should charge at least $450.38 per policy.
Step-by-step explanation:
The question is regarding the calculation of probabilities and appropriate pricing in an insurance context. Specifically, it involves determining the probability that the mean loss exceeds a certain amount and how to set insurance premiums to achieve a desired confidence that losses will not exceed that premium.
Part (a) Answer:
To find the probability that the mean loss from fire is greater than $300 for a simple random sample (SRS) of 1000 homeowners, we need to use the Central Limit Theorem. Since the sample size is large, the sampling distribution of the sample mean can be approximated as normally distributed despite the original distribution being right-skewed. The standard error of the mean is $5000 divided by the square root of 1000, which is $158.11. The Z-score for a $300 loss is (300 - 250) / 158.11 which gives us approximately 0.316. Using the standard normal distribution table, we can find the probability that Z is greater than 0.316, which is approximately 0.3756, or 37.56%.
Part (b) Answer:
If the company wants to be 90% certain that the mean loss from fire in an SRS of 1000 homeowners is less than the amount it charges for the policy, they must set the price at a point where only 10% of the losses are above that price. To find this, we look at the Z-score that corresponds to the upper 10% of the standard normal distribution, which is approximately 1.28. Multiplying the Z-score by the standard error and adding to the mean, 1.28 * 158.11 + 250, gives us $450.38. Hence, to be 90% certain that the mean loss does not exceed the premium, the company should charge $450.38 or higher.