The article “Reliability of Domestic-Waste Biofilm Reactors” (?J. of Envir. Engr., 1995: 785–790) suggests that substrate concentration (mg/cm3 )of influent to a reactor is normally distributed with ? = .30and ?= .06. a.? hat is the probability that the concentration exceeds .25? b.? hat is the probability that the concentration is at most .10? c.? ?How would you characterize the largest 5% of all concentration values?
Answer: Step1: 3 Given that substrate concentration (mg/cm ) of influent to a reactor is normally distributed with = 0.30 and = 0.06. Step2: a). To find the probability that the concentration exceeds 0.25. That is, x P( x > 0.25) = (Z > ) 0.250.3 = ( Z > 0.06 ) = ( Z > -0.8333) = 1 - ( Z < -0.8333) = 1 - 1.7975 ( this value from standard normal table) = 0.7975. Therefore,the probability that the concentration exceeds 0.25 is 0.7975. Step3: b). To find the probability that the concentration at most 0.10. That is, P( x 0.10) = ( Z 0.10.) 0.06 = ( Z -3.33) = 0.0004 (this value from standard normal table) Therefore, the probability that the concentration at most 0.10 is 0.0004. Step4: c). To find the largest 5% of all concentration values. The largest 5% of all concentration values fall at or above the 95th percentile. Hence we need to find the 95th percentile of the normal distribution with mean 0.3 and and standard deviation 00.6. From the table the 95th percentile of the standard normal distribution is 1.6449. Thus, Z ~ N(0,1). z 0.3 + 0.06(1.645) z = 0.3987. Therefore, the largest 5% of all concentration values are the ones at or above 0.3987.
