The working lifetime, in years, of a particular model of bread maker is normally distributed with mean 10 and variance 4. Calculate the 12th percentile of the working lifetime, in years.

• 5.30
• 7.65
• 8.41
• 12.35
• 14.70

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

Solution: B

Let [math]X[/math] be normal with mean 10 and variance 4. Let [math]Z[/math] have the standard normal distribution. Let [math]p[/math] = 12^th percentile. Then

[[math]] 0.12 = \operatorname{P}(X \leq p ) = \operatorname{P}\left( \frac{X-10}{2} \leq \frac{p-10}{2} \right) = \operatorname{P}( Z \leq \frac{p-10}{2}). [[/math]]

From the tables, [math]\operatorname{P}( Z ≤ −1.175) =0.12.[/math] Therefore,

[[math]] \frac{p-10}{2} = -1.175; p-10 = -2.35; p = 7.65. [[/math]]

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