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Applied Statistics and Probability for Engineers | 6th Edition | ISBN: 9781118539712 | Authors: Douglas C. Montgomery, George C. Runger
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Asbestos fibers in a dust sample are identified by an

Chapter 4, Problem 208SE

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QUESTION:

Problem 208SE

Asbestos fibers in a dust sample are identified by an electron microscope after sample preparation. Suppose that the number of fibers is a Poisson random variable and the mean number of fibers per square centimeter of surface dust is 100. A sample of 800 square centimeters of dust is analyzed. Assume that a particular grid cell under the microscope represents 1/160,000 of the sample.

(a) What is the probability that at least one fiber is visible in the grid cell?

(b) What is the mean of the number of grid cells that need to be viewed to observe 10 that contain fibers?

(c) What is the standard deviation of the number of grid cells that need to be viewed to observe 10 that contain fibers?

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QUESTION:

Problem 208SE

Asbestos fibers in a dust sample are identified by an electron microscope after sample preparation. Suppose that the number of fibers is a Poisson random variable and the mean number of fibers per square centimeter of surface dust is 100. A sample of 800 square centimeters of dust is analyzed. Assume that a particular grid cell under the microscope represents 1/160,000 of the sample.

(a) What is the probability that at least one fiber is visible in the grid cell?

(b) What is the mean of the number of grid cells that need to be viewed to observe 10 that contain fibers?

(c) What is the standard deviation of the number of grid cells that need to be viewed to observe 10 that contain fibers?

ANSWER:

Solution 208SE

Step1 of 4:

Let us consider a random variable X it presents the number of fibres given grid cell. And X follows poisson distribution with parameter  Also we have:

A = Area of sample

    = 800

m = grid cell of microscope

    =  

Here our goal is:

a). We need to find the probability that at least one fiber is visible in the grid cell.

b). We need to find the mean of the number of grid cells that need to be viewed to observe 10 that contain fibers.

c). We need to find the standard deviation of the number of grid cells that need to be viewed to observe 10 that contain fibers.

Step2 of 4:

a).

Let,

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