Question

A. A manufacturer knows that their items have a normally distributed length, with a mean of 11.7 inches, and standard deviation of 1.4 inches. If one item is chosen at random, what is the probability that it is less than 9 inches long?

B. A manufacturer knows that their items have a normally distributed lifespan, with a mean of 8.6 years, and standard deviation of 2.6 years. If you randomly purchase one item, what is the probability it will last longer than 15 years? Use the normal table and round answer to four decimal places

Answer 1

Part a)
X ~ N ( µ = 11.7 , σ = 1.4 )
P ( X < 9 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 9 - 11.7 ) / 1.4
Z = -1.9286
P ( ( X - µ ) / σ ) < ( 9 - 11.7 ) / 1.4 )
P ( X < 9 ) = P ( Z < -1.9286 )
P ( X < 9 ) = 0.0269

Part b)
X ~ N ( µ = 8.6 , σ = 2.6 )
P ( X > 15 ) = 1 - P ( X < 15 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 15 - 8.6 ) / 2.6
Z = 2.4615
P ( ( X - µ ) / σ ) > ( 15 - 8.6 ) / 2.6 )
P ( Z > 2.4615 )
P ( X > 15 ) = 1 - P ( Z < 2.4615 )
P ( X > 15 ) = 1 - 0.9931
P ( X > 15 ) = 0.0069