Chapter 7.3, Problem 63E

To determine

Whether the company can safely base its rates on the assumption that its average loss will be no greater than $275 or not if it sells 10,000 policies.

Answer to Problem 63E

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.

Explanation of Solution

Given information:

The mean annual loss, $\mu =\$250$

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

Formula used:

The central limit theorem states that if the sample size of a sampling distribution is 30 or more, then the sample mean is approximately normal whose mean is $\mu$and standard deviation is $\frac{\sigma }{\sqrt{n}}$ .

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

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 $z=\frac{x-\mu }{\frac{\sigma }{\sqrt{n}}}$ .

Substitute 275 for x, 250 for $\mu$ , 300 for $\sigma$and 10000 for $n$in the above formula and simplify.

  $\begin{array}{c}z=\frac{275-250}{\frac{300}{\sqrt{10000}}}\\ =\frac{25}{\frac{3\cancel{00}}{1\cancel{00}}}\\ =8.33\end{array}$

Thus, the corresponding probability is:

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

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

Hence, 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.