The Environmental Protection Agency (EPA) Requires Cities ✓ Solved
The Environmental Protection Agency Epa Requires That Cities Monitor
The Environmental Protection Agency (EPA) requires that cities monitor over 80 contaminants in their drinking water. Samples from the Lake Huron Water Treatment Plant provided results showing the observed ranges of contaminant levels, all below the allowable maximums. For each substance, estimate the standard deviation assuming both uniform and normal distributions, based on the given ranges and statutory limits. Round your answers to four decimal places.
Sample Paper For Above instruction
Introduction
Monitoring water quality is essential for safeguarding public health, ensuring regulatory compliance, and maintaining ecological integrity. The Environmental Protection Agency (EPA) mandates that municipal water systems test for over 80 contaminants, setting maximum permissible levels for each substance. This paper focuses on estimating the standard deviation for three specific water contaminants—chromium, barium, and fluoride—based on observed sample ranges from the Lake Huron Water Treatment Plant, and assuming different probability distribution models. Accurate estimation of variability helps interpret the reliability of the contaminant levels relative to regulatory standards and informs risk assessments.
Methodology
The analysis utilizes sample ranges provided for each substance and incorporates assumptions based on uniform and normal distribution models. The core formulas for estimating standard deviation (σ) for each distribution are:
- Uniform Distribution:
\[
\sigma = \frac{\text{Range}}{\sqrt{12}}
\]
because the range of a uniform distribution over a finite interval \([a, b]\) is \(\text{Range} = b - a\), and the standard deviation relates to the range via \(\sigma = \frac{b - a}{\sqrt{12}}\).
- Normal Distribution:
\[
\sigma \approx \frac{\text{Range}}{4}
\]
based on the empirical rule, which states that approximately 95% of data lies within two standard deviations on either side of the mean, so the total range can be approximated as \(\sim 4\sigma\).
The provided ranges and statutory maximum limits are as follows:
- Chromium: Range 0.41 to 0.60; Allowable maximum 100
- Barium: Range 0.006 to 0.020; Allowable maximum 2
- Fluoride: Range 1.09 to 1.15; Allowable maximum 4
Using these, the estimated standard deviations are calculated as described.
Results
Chromium:
- Range: 0.60 - 0.41 = 0.19
- Uniform distribution:
\[
\sigma_{u} = \frac{0.19}{\sqrt{12}} \approx \frac{0.19}{3.4641} \approx 0.0549
\]
- Normal distribution:
\[
\sigma_{n} = \frac{0.19}{4} = 0.0475
\]
Barium:
- Range: 0.020 - 0.006 = 0.014
- Uniform distribution:
\[
\sigma_{u} = \frac{0.014}{\sqrt{12}} \approx \frac{0.014}{3.4641} \approx 0.0040
\]
- Normal distribution:
\[
\sigma_{n} = \frac{0.014}{4} = 0.0035
\]
Fluoride:
- Range: 1.15 - 1.09 = 0.06
- Uniform distribution:
\[
\sigma_{u} = \frac{0.06}{\sqrt{12}} \approx \frac{0.06}{3.4641} \approx 0.0173
\]
- Normal distribution:
\[
\sigma_{n} = \frac{0.06}{4} = 0.0150
\]
All estimated standard deviations are rounded to four decimal places as required.
Discussion
Estimating the variability in contaminant levels through these methods facilitates a better understanding of their distribution characteristics. The uniform distribution assumptions tend to yield slightly higher estimates of variability compared to the normal distribution model, reflecting a more conservative view of possible fluctuations in observed data. These estimations are critical for regulatory agencies to assess whether measurements are consistent with safety standards or if further scrutiny is necessary. Such statistical approaches contribute to proactive water quality management and risk mitigation strategies.
Conclusion
Estimating the standard deviation for water contaminant levels enables us to gauge the potential variability inherent in sampling data. Using the provided ranges, the uniform distribution yields estimated standard deviations of approximately 0.0549 (Cr), 0.0040 (Ba), and 0.0173 (F), whereas the normal distribution estimates are slightly lower at approximately 0.0475, 0.0035, and 0.0150 respectively. These calculations underscore the importance of understanding data distribution assumptions when interpreting water quality data and emphasize ongoing monitoring to ensure compliance and public safety.
References
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- Gordon, L. (2019). Water Quality Monitoring and Management. Routledge.
- American Water Works Association. (2018). Water Quality and Treatment. McGraw-Hill.
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