Normal Random Variables

Question

the amount of a certain toxin in the water is normally distributed with mean 72.4 mg and standard deviation 20 mg. The probability that the amount of toxin exceeds the safe limit is .14. What is the safe limit?

Answer 1

This is more of a comment than an answer. If $X$ denotes the amount of toxine. then $Z = \dfrac{X - 72.4}{20}$ is a Standard-Guassian-Bell-Normal(-However you know it) random variable. You can check from tables what is the quantile (the cut) where the probability of the right-hand tail is 0.14 and then make $Z$ equal to that value and solve for $X.$

Answer 2

For a normal distribution, a probability of 0.14 that a certain z-score will be exceeded corresponds to a z-score of 1.08 (this can be found in a table of the standard normal curve or by using the inverse NORM function on a calculator or spreadsheet). This is the same as saying that 86% of the area of the standard normal distribution is below a z-score of 1.08.

The toxin level that corresponds to this z-score is 1.08 standard deviations above the mean, so the safe toxin limit = 72.4 + 1.08(20) = 94.0 mg