A study by Bechtel et al., 2009, described in the Archives

Chapter 4, Problem 88E

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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?

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.


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