Question

1)    An automatic machine in a manufacturing process is operating properly if the lengths of an important component are normally distributed, with mean = 117 cm and std. dev = 5.2 cm.

Find the probability that, if four units are randomly selected, at least two have lengths that exceed 120 cm.

Answer 1

let X denotes the length of the important component in cm.

then X~N(117,5.2²)

let p be the probability that a randomly selected component has length more than 120 cm

then p=P[X>120]=P[(X-117)/5.4>(120-117)/5.4]=P[Z>0.5555556]=0.2892574 where Z~N(0,1)

now let Y denotes the number of components whose lengths are greater than 120 cm out of the 4 randomly selected components

then Y~Bin(4,0.2892574)

so P[at least 2 have lengths that exceed 120 cm]=P[Y>=2]=1-P[Y<2]=1-P[Y=0]-P[Y=1]

=1-⁴C₀0.2892574⁰(1-0.2892574)⁴-⁴C₁0.2892574¹(1-0.2892574)³=1-0.2551816-0.4154144=0.329404 [answer]