Artificial silk fibers made from spider silk proteins have high tensile strength

Question

Artificial silk fibers made from spider silk proteins have high tensile strengths. Tensile strength can be modeled by a normal distribution with mean 150 MJ/m3 and standard deviation 10.

(1) Find the probability that a fiber will have tensile strength (a) at least 142; (b) between 141.5 and 159.5 MJ/m3 ?

(2) suppose a potential large contract contains the specification that at most 5% have tensile strength less than 135 MJ/m3 . If the manufacturing process is improved to meet this specification, determine (a) the new mean if the standard deviation is 10 MJ/m3 ; (b) the new standard deviation if the mean is 150 MJ/m3

Explanation / Answer

1 a)P(X>142)=1-P(X<142)=1-P(Z<(142-150)/10)=1-P(Z<-0.8)=1-0.2119=0.7881

b) P(141.5<X<159.5)=P(0.85<Z<0.95)=0.8289-0.1977=0.6313

2)for bottom 5% , z=-1.645

hence mean =135+1.645*10=151.45

b) std deviation=(135-150)/-1.645=9.12

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