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 101.3 pounds and a standard deviation of 3.68 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 0.97? Give your answer to three decimal places.

Answer: 0.6381Step-by-step explanation:Given :  A manufacturer of industrial solvent guarantees its customers that each drum of solvent they ship out contains at least 100 lbs of solvent.[tex]\mu=101.3\text{ pounds}\ \ \sigma=3.68\text{ pounds}[/tex]Let x be a random variable that represents e the amount of solvent in each drum.Since , [tex]z=\dfrac{x-\mu}{\sigma}[/tex]z-score corresponds x = 100 , [tex]z=\dfrac{100-101.3}{3.68}\approx-0.3533[/tex]Required probability : [tex]\text{P-value }: P(x\leq100)=P(z\leq-0.3533)\\\\=1-P(z<-0.3533)\\\\1-(1-P(z<0.3533))\\\\=P(z<0.3533)=0.638068\approx0.6381[/tex][using z-value table.]Hence, the probability that a drum meets the guarantee = 0.6381