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 5.7 cm.

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

Probability =

B. Find the probability that if 4 subcomponents are randomly selected, their mean length exceeds 120 cm.

Probability =

Answer 1

Part a)

X ~ N ( µ = 118 , σ = 5.7 )
P ( X > 120 ) = 1 - P ( X < 120 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 120 - 118 ) / 5.7
Z = 0.3509
P ( ( X - µ ) / σ ) > ( 120 - 118 ) / 5.7 )
P ( Z > 0.3509 )
P ( X > 120 ) = 1 - P ( Z < 0.3509 )
P ( X > 120 ) = 1 - 0.6372
P ( X > 120 ) = 0.3628

Part b)

X ~ N ( µ = 118 , σ = 5.7 )
P ( X̅ > 120 ) = 1 - P ( X < 120 )
Standardizing the value
Z = ( X - µ ) / ( σ / √(n))
Z = ( 120 - 118 ) / ( 5.7 / √ ( 4 ) )
Z = 0.7018
P ( ( X - µ ) / ( σ / √ (n)) > ( 120 - 118 ) / ( 5.7 / √(4) )
P ( Z > 0.7 )
P ( X̅ > 120 ) = 1 - P ( Z < 0.7 )
P ( X̅ > 120 ) = 1 - 0.7586
P ( X̅ > 120 ) = 0.2414