Question

An automatic machine in a manufacturing process is operating properly if the lengths of an important subcomponent are normally distributed with a mean of 118 cm and a standard deviation of 5 cm.

A. Find the probability that one selected subcomponent is longer than 120 cm. Probability =

B. Find the probability that if 3 subcomponents are randomly selected, their mean length exceeds 120 cm. Probability =

C. Find the probability that if 3 are randomly selected, all 3 have lengths that exceed 120 cm. Probability =

Answer #1

A) Let X be the random variable denoting the length of an
important subcomponent. So, X ~ N(118, 5^{2})

Probability that the subcomponent is longer than 120 cm =

P(X > 120)

= 1 - P(X < 120)

= 1 - P(Z < (120-118)/5)

= 1 - P( Z < 0.4)

= 1 - 0.655

= 0.345.

B) The probability that the mean length of the 3 subcomponents exceeds 120 cm will be equal to the probability that one subcomponent is longer than 120 cm, which is equal to 0.345.

C) The probability that if 3 subcomponents are randomly selected, all 3 have lengths that exceed 120 cm =

P(X > 120)^{3}

And we have calculated that P(X > 120) = 0.345

So, P(X > 120)^{3} = 0.345^{3}

= 0.0411.

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