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2 weeks ago
A company's manufacturing process uses 500 gallons of water at a time. A "scrubbing" machine then removes most of a chemical pollutant before pumping the water into a nearby lake. Legally the treated water should contain no more than 80 parts per million of the chemical, but the machine isn't perfect and it is costly to operate. Since there's a fine if the discharged water exceeds the legal maximum, the company sets the machine to attain an average of 75 ppm for the batches of water treated. They believe the machine's output can be described by a Normal model with standard deviation 4.2 ppm. (SHOW WORK.)


a. What percent of the batches of water discharged exceed the 80ppm standard?

b. The company's lawyers insist that they have not more than 2% of the water over the limit.  To what  mean value should the company set the scrubbing machine? Assume the standard deviation does not change.
c. Because achieving a mean that low would raise the costs too much, they decide to leave the mean set at 75 ppm and try to reduce the standard deviation to achieve the "only 2% over" goal. Find the new standard deviation needed.

d. Explain what achieving a smaller standard deviation means in this context.

Textbook 

Stats: Modeling the World


Edition: 4th
Authors:
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Posts: 236
2 weeks ago
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a. z = = 1.19
   P(z ≥ 1.19) ≈ 0.117

According to the normal model, we expect about 11.7% of the batches to exceed the 80ppm standard.
b. z = 2.054
    = 2.054
    μ = 80 - 2.054(4.2) = 71.37

According to the Normal model, a mean of about 71.37ppm would need to be achieved.
c.   = 2.054
    σ ≈ 2.434


According to the Normal model, the new standard deviation would need to be at most 2.43ppm.
d. The scrubber must be more consistent in its performance from batch to batch.
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