Claims filed under auto insurance policies follow a normal distribution with mean 19,400 and standard deviation 5,000.

Calculate the probability that the average of 25 randomly selected claims exceeds 20,000.

• 0.01
• 0.15
• 0.27
• 0.33
• 0.45

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.

Solution: C

Let [math]X_1,\ldots,X_n[/math] denote the 25 collision claims, and let

[[math]] \overline{X} = \frac{1}{25}(X_1 \cdots + X_n). [[/math]]

We are given that each [math]X_i \, (i=1,\ldots,25)[/math] follows a normal distribution with mean 19,400 and standard deviation 5,000. As a result [math]\overline{X}[/math] also follows a normal distribution with mean 19,400 and standard deviation 25^-1/2 5000 = 1000. We conclude that

[[math]] \begin{align*} \operatorname{P}[\overline{X} \gt 20000] &= \operatorname{P}[\frac{\overline{X}-19400}{1000} \gt \frac{20000-19400}{1000}] \\ &= \operatorname{P}[\frac{\overline{X}-19400}{1000} \gt 0.6] \\ &= 1 - \Phi(0.6) = 1-0.7257 \\ & = 0.2743. \end{align*} [[/math]]

Copyright 2023. The Society of Actuaries, Schaumburg, Illinois. Reproduced with permission.