When a pesticide comes into contact with skin, a certain percentage of it is abs

Question

When a pesticide comes into contact with skin, a certain percentage of it is absorbed. The
percentage of the pesticide that will be absorbed during a given time period can e modeled with a
lognormal distribution. Assume that for a given pesticide, the amount that is absorbed (in percent) in a
two-hour time period is log-normally distributed with and .
a. Find the mean percentage absorbed.
b. Find the median percentage absorbed.
c. Find the probability that the percent absorbed is more than 10.
d. Find the probability that the percent absorbed is less than 5.
e. During the two-hour period, an intermediate reading reveals that the percent absorbed is 3
percent. What is the probability that the amount absorbed at the end of the two hour period
will be greater than 5 percent?

Explanation / Answer

1) For lognormal
mean = e^(mu + sigma^2/2)= e^(1.5+0.5^2/2)=5.078

2) median = e^mu = e^1.5 = 4.48

3) P(x>10)= 1- P(x<10)

P(x<10) = 1/2 erfc( (mu - log(x))/(sqrt(2)*sigma))

=1/2 erfc( (1.5 - log(10))/(sqrt(2)*.5))=0.946

so P(x>10) =1-0.946 = .054 = 5.4 %

4) P(x<5) = 1/2 erfc( (1.5 - log(5))/(sqrt(2)*.5))=.587=58.7 %

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