The profits of life insurance companies A and B are normally distributed with the same mean. The variance of company B's profit is 2.25 times the variance of company A's profit. The 14^th percentile of company A’s profit is the same as the [math]p^{\textrm{th}}[/math] percentile of company B’s profit. Calculate [math]p[/math].

• 5.3
• 9.3
• 21.0
• 23.6
• 31.6

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

Solution: D

From the normal table, the 14^th percentile is associated with a z-score of −1.08 . Since the means are equal and the standard deviation of company B's profit is 2.25 = 1.5 ^0.5 times the standard deviation of company A's profit, a profit that is 1.08 standard deviations below the mean for company A would be1.08/1.5 = 0.72 standard deviations below the mean for company B. From the normal table, a z-score of −0.72 is associated with the 23.6^th percentile.

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