Question 4.24: When a pesticide comes into contact with skin, a certain per......
When a pesticide comes into contact with skin, a certain percentage of it is absorbed. The percentage that is absorbed during a given time period is often modeled with a lognormal distribution. Assume that for a given pesticide, the amount that is absorbed (in percent) within two hours of application is lognormally distributed with 𝜇 = 1.5 and 𝜎 = 0.5. Find the probability that more than 5% of the pesticide is absorbed within two hours.
Let Y represent the percentage of pesticide that is absorbed. We need to find P(Y > 5). We cannot use the z table for Y, because Y is not normally distributed. However, ln Y is normally distributed; specifically, ln Y ∼ N(1.5, 0.5²). We express P(Y > 5) as a probability involving ln Y:
P(Y > 5) = P(ln Y > ln 5) = P(ln Y > 1.609)
The z-score of 1.609 is
z = \frac{1.609\ − \ 1.500}{ 0.5}
= 0.22
From the z table, we find that P(ln Y > 1.609) = 0.4129. We conclude that the probability that more than 5% of the pesticide is absorbed is approximately 0.41.