Question

An article suggests that substrate concentration (mg/cm3) of influent to a reactor is normally distributed with μ = 0.50 and σ = 0.08. (Round your answers to four decimal places.) (a) What is the probability that the concentration exceeds 0.60?

Answer 1

0.1056 = 10.56% probability that the concentration exceeds 0.60

Explanation:

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the zscore of a measure X is given by:

`Z = (X - \mu)/(\sigma)`

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this problem, we have that:

`\mu = 0.5, \sigma = 0.08`

What is the probability that the concentration exceeds 0.60?

This is 1 subtracted by the pvalue of Z when X = 0.6. So

`Z = (X - \mu)/(\sigma)`

`Z = (0.6 - 0.5)/(0.08)`

`Z = 1.25`

`Z = 1.25` has a pvalue of 0.8944

1 - 0.8944 = 0.1056

0.1056 = 10.56% probability that the concentration exceeds 0.60