A manufacturer of industrial solvent guarantees its customers that each drum of

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 103.3 pounds and a standard deviation of 3.93 pounds.

a) What is the probability that a drum meets the guarantee? Give your answer to four decimal places.  

b) What would the standard deviation need to be so that the probability a drum meets the guarantee is 0.99? Give your answer to three decimal places

Explanation / Answer

Solution:

a)Let X be amount of solvent.

X~N(103.3, 3.932)

Mean is 103.3 and std. dev is 3.93 so, variance is 3.932

P (X >100) = P(Z > (100-103.3)/3.93 )
= P(Z > -0.8396)
= 0.7994

b) Now, let c be the std dev such that P(X > 100) = 0.99 then,

let Z=(X-103.3) /s ~N(0,1)

P(Z > z) = 0.99 when z = 0.99
=> z = (100 - 103.3)/s = 0.99
=> -3.3/s = Z0.01 where Zq is the qth quantile of standard normal distribution.
=> -3.3 s = -2.3263

and s = 1.4185

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