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## Q. 4.24

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.

## Verified Solution

Let Y represent the percent 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 > ln5) = 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.