Chlorine Concentration In A Municipal Water Supply Is Unifor

Chlorine Concentration In A Municipal Water Supply Is A Uniformly Dist

Chlorine concentration in a municipal water supply is a uniformly distributed random variable that ranges between 0.67 ppm and 0.97 ppm. Determine the mean, standard deviation, and the probabilities for specific concentration ranges, including the likelihood that the concentration exceeds 0.92 ppm, falls below 0.74 ppm, or lies between 0.82 ppm and 0.88 ppm. Round all numerical answers to the specified decimal places and provide explanations based on the properties of uniform distributions.

Paper For Above instruction

The task involves analyzing a uniform distribution representing the chlorine concentration in a municipal water supply. The distribution ranges from 0.67 ppm to 0.97 ppm. Given the properties of a uniform distribution, we can derive formulas to compute the mean, standard deviation, and specific probabilities associated with the chlorine concentration levels.

Introduction

Uniform distributions are a fundamental concept in probability theory, characterized by constant probability density over a specified interval. In the context of environmental monitoring and public health, understanding the distribution of contaminants such as chlorine in water supplies is vital. Chlorine is commonly used to disinfect water, but excessively high or low levels can pose health risks; hence, assessing the statistical properties of its concentrations is imperative for regulatory compliance and safety management.

Properties of Uniform Distribution

Let X be a uniformly distributed random variable over the interval [a, b]. The key properties include:

  • Mean (Expected Value) E[X] = (a + b) / 2
  • Standard deviation σ = (b - a) / √12
  • Probability that X falls within a certain subinterval c to d (where a ≤ c

Calculating the Mean Chlorine Concentration

Given the interval [0.67, 0.97], the mean (average) concentration is calculated as:

Mean = (a + b) / 2 = (0.67 + 0.97) / 2 = 1.64 / 2 = 0.82 ppm

Thus, the mean chlorine concentration is 0.82 ppm, rounded to two decimal places.

Calculating the Standard Deviation

The standard deviation for a uniform distribution is given as:

Standard deviation = (b - a) / √12

= (0.97 - 0.67) / √12 = 0.30 / 3.4641 ≈ 0.0865

Therefore, the standard deviation of the chlorine concentration is approximately 0.0865 ppm, rounded to four decimal places.

Calculating Probabilities

  • Probability that concentration exceeds 0.92 ppm:
  • Since the distribution is uniform, P(X > 0.92) = (b - 0.92) / (b - a) = (0.97 - 0.92) / 0.30 = 0.05 / 0.30 ≈ 0.1667
  • Rounded to four decimal places: 0.1667
  • Probability that concentration is under 0.74 ppm:
  • P(X
  • Rounded to four decimal places: 0.2333
  • Probability that concentration is between 0.82 ppm and 0.88 ppm:
  • P(0.82 ≤ X ≤ 0.88) = (0.88 - 0.82) / 0.30 = 0.06 / 0.30 = 0.2
  • Rounded to four decimal places: 0.2000

Conclusion

Through the analysis of the uniform distribution modeling chlorine concentrations, we established its mean at 0.82 ppm, with a standard deviation of approximately 0.0865 ppm. The probability calculations underscore the likelihoods of specific concentration ranges, which are crucial for environmental monitoring and ensuring water safety standards are maintained. This statistical assessment provides a clear framework for water quality management and helps in making informed decisions in public health efforts.

References

  • Devore, J. L. (2015). Probability and Statistics for Engineering and the Sciences. Cengage Learning.
  • Mendenhall, W., Beaver, R. J., & Beaver, B. M. (2012). Introduction to Probability and Statistics. Cengage Learning.
  • Grinstead, C. M., & Snell, J. L. (1997). Introduction to Probability. American Mathematical Society.
  • Ross, S. M. (2014). Introduction to Probability Models. Academic Press.
  • Blitzstein, J., & Hwang, J. (2014). Introduction to Probability. CRC Press.
  • Sheskin, D. J. (2011). Handbook of Parametric and Nonparametric Statistical Procedures. Chapman and Hall/CRC.
  • Walpole, R. E., Myers, R. H., Myers, S. L., & Ye, K. (2012). Probability & Statistics for Engineers & Scientists. Pearson.
  • Freund, J. E. (2010). Modern Elementary Statistics. Pearson.
  • Kendall, M., & Stuart, A. (1973). The Advanced Theory of Statistics. Griffin.
  • Casella, G., & Berger, R. L. (2002). Statistical Inference. Duxbury.

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