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A study by Bechtel et al., 2009, described in the Archives
Chapter 4, Problem 88E(choose chapter or problem)
Problem 88E
A study by Bechtel et al., 2009, described in the Archives of Environmental & Occupational Health considered polycyclic aromatic hydrocarbons and immune system function in beef cattle. Some cattle were near major oil- and gas-producing areas of western Canada. The mean monthly exposure to PM1.0 (particulate matter that is ◊◊1◊min diameter) was approximately 7.1 ◊g/m3 with standard deviation 1.5. Assume that the monthly exposure is normally distributed.
(a) What is the probability of a monthly exposure greater than 9 ◊g/m3
(b) What is the probability of a monthly exposure between 3 and 8 ◊g/m3?
(c) What is the monthly exposure level that is exceeded with probability 0.05?
(d) What value of mean monthly exposure is needed so that the probability of a monthly exposure more than 9 ◊g/m3 is 0.01?
Questions & Answers
QUESTION:
Problem 88E
A study by Bechtel et al., 2009, described in the Archives of Environmental & Occupational Health considered polycyclic aromatic hydrocarbons and immune system function in beef cattle. Some cattle were near major oil- and gas-producing areas of western Canada. The mean monthly exposure to PM1.0 (particulate matter that is ◊◊1◊min diameter) was approximately 7.1 ◊g/m3 with standard deviation 1.5. Assume that the monthly exposure is normally distributed.
(a) What is the probability of a monthly exposure greater than 9 ◊g/m3
(b) What is the probability of a monthly exposure between 3 and 8 ◊g/m3?
(c) What is the monthly exposure level that is exceeded with probability 0.05?
(d) What value of mean monthly exposure is needed so that the probability of a monthly exposure more than 9 ◊g/m3 is 0.01?
ANSWER:
Solution :
Step 1 of 4:
Given the random variable X is the monthly exposure.
Then X is a normal random variable with mean and .
Our goal is:
a). We need to find the probability of a monthly exposure greater than 9 .
b). We need to find the probability of a monthly exposure between 3 and 8 .
c). We need to find the monthly exposure level that is exceeded with probability 0.05.
d). We need to find the value of mean monthly exposure is needed.
a). The probability of a monthly exposure greater than 9.
Now we need to find P(X>9).
P(X>9) = P
P(X>9) = P
P(X>9) = P
P(X>9) = P
P(X>9) = 1-P
Using areas under the normal probability table,
P(X>9) = 1-0.8962
P(X>9) = 0.1038
Therefore, the probability of a monthly exposure greater than 9 is 0.1038.