Q.63 - The Practice of Statistics for AP

Question

More on insurance An insurance company knows that in the entire population of homeowners, the mean annual loss from fire is $\mu =250$ and the standard deviation of the loss is $\sigma =300$. The distribution of losses is strongly right-skewed: many policies have $0$ loss, but a few have large losses. If the company sells $10,000$ policies, can it safely base its rates on the assumption that its average loss will be no greater than $275$? Follow the four-step process

Step-by-Step Solution

Answer

From the given information, the company can safely base its rates on the assumption that is average loss will be no longer greater than$275$  it sells, $10,000$ policies.

Step 1: Given Information

It is given in the question that, the mean annual loss, $\mu =250$

the standard deviation of the loss, $\sigma =300$

follow the four-step process.

Step 2: Explanation

The central limit theorem states that if the sample size of a sampling distribution is $300$ or more, then the sample mean is approximately normal whose mean is $\mu$ and the standard deviation is $\frac{\sigma }{\sqrt{n}}$.

The $z$ value of a distribution  can be found by dividing the difference between the population mean and sample mean by  the standard deviation  that is, $z=\frac{x-\mu }{\frac{s^r}{\sqrt{n}}}.$

Since the sample size of $10,000$policies is at least $30$; so we can apply the central limit theorem.

Find the $z$value by using the formula $z=\frac{x-\mu }{\frac{s^r}{\sqrt{n}}}.$

Step 3: Explanation

Substitute $275$ for $x$,$250 for \mu ,300 for \sigma ,and 10,000 for n$ the above formula and simplify.

$z=\frac{275-250}{\frac{300}{\sqrt{10000}}}$

  $=\frac{25}{\frac{306}{100}}$

  $=8.33$

Thus, the corresponding probability is:

$P(\overline{x}>275)=P(z>8.33)=P(Z<-8.33)=0.0001$

Thus, the company can safely assume that the average loss will be no greater than $275$ because the probability is almost zero.

Accordingly, the company can safely base its rates on the assumption that its average loss will be no greater than $275$ it sells $10,000$ policies.