An insurance company estimates that it should make an annual profit of $150 on each homeowner's policy written,with a standard deviation of $6700.If it writes 12,469 of these policies,what is the probability that the company will be profitable? Assume that policies are independent of each other and that the company's total profit follows a Normal model.

A) 0.977
B) 0.994
C) 0.989
D) 0.011
E) 0.006

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adrian rodriguez · Accepted Answer

Final Answer : B, Explanation :   Given that the insurance company estimates an annual profit of $150 per homeowner's policy with a standard deviation of $6700, and it writes 12,469 policies, we can calculate the probability of profitability using the Central Limit Theorem. The mean total profit (  μ t o t a l \mu_{total} μ t o t a l ​  ) is the number of policies (12,469) multiplied by the profit per policy ($150), and the standard deviation of the total profit (  σ t o t a l \sigma_{total} σ t o t a l ​  ) is the standard deviation per policy ($6700) multiplied by the square root of the number of policies (  12469 \sqrt{12469} 12469 ​  ). The question asks for the probability that the company will be profitable, which means we are looking for P(X > 0).Using the Central Limit Theorem, we find that the distribution of the total profit is approximately normal. To find the probability of being profitable, we calculate the Z-score for a profit of $0 and then use the standard normal distribution to find the corresponding probability.  Z = X − μ t o t a l σ t o t a l Z = \frac{X - \mu_{total}}{\sigma_{total}} Z = σ t o t a l ​ X − μ t o t a l ​ ​  Given that the mean and standard deviation are based on the total number of policies, we can calculate the Z-score for $0 profit and then find the corresponding probability from the standard normal distribution. The calculation will show that the probability of the company being profitable (i.e., having a total profit greater than $0) is very high, which corresponds to choice B) 0.994. This high probability is due to the large number of policies written, which, according to the Central Limit Theorem, results in the total profit distribution being closely approximated by a normal distribution centered around a positive mean profit.

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