Question

A manufacturer of industrial solvent guarantees its customers that each drum of solvent they ship out contains at least 100 lbs of solvent. Suppose the amount of solvent in each drum is normally distributed with a mean of 101.8 pounds and a standard deviation of 3.76 pounds. a) What is the probability that a drum meets the guarantee

Answer 1

0.6844 = 68.44% probability that a drum meets the guarantee.

Explanation:

Normal Probability Distribution:

Problems of normal distributions can be solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the z-score 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 p-value, we get the probability that the value of the measure is greater than X.

Mean of 101.8 pounds and a standard deviation of 3.76 pounds.

This means that
`\mu = 101.8, \sigma = 3.76`

What is the probability that a drum meets the guarantee?

Probability of at least 100 lbs of solvent, which is 1 subtracted by the p-value of Z when X = 100. So

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

`Z = (100 - 101.8)/(3.76)`

`Z = -0.48`

`Z = -0.48` has a p-value of 0.3156.

1 - 0.3156 = 0.6844

0.6844 = 68.44% probability that a drum meets the guarantee.