Question

Bottles filled by a certain machine are supposed to contain 12 oz of liquid. In fact the fill volume is random with mean 12.01 oz and standard deviation 0.30 oz.

What is the probability that the mean volume of a random sample of 144 bottles is less than 12 oz?

If the population mean fill volume is increased to 12.03 oz, what is the probability that the mean volume of a sample of size 144 will be less than 12 oz?

Answer 1

a)

X ~ N ( µ = 12.01 , σ = 0.3 )
Standardizing the value
Z = ( X - µ ) / (σ/√(n)
Z = ( 12 - 12.01 ) / ( 0.3 / √144 )
Z = -0.4
P ( ( X - µ ) / ( σ/√(n)) = ( 12 - 12.01 ) / ( 0.3 / √(144) )
P ( X̅ < 12 ) = P ( Z < -0.4 )
= 0.3446

b)

P (X̅ < 12 )
Standardizing the value
Z = ( X - µ ) / (σ/√(n)
Z = ( 12 - 12.03 ) / ( 0.3 / √144 )
Z = -1.2
P ( ( X - µ ) / ( σ/√(n)) = ( 12 - 12.03 ) / ( 0.3 / √(144) )
P ( X̅ < 12 ) = P ( Z < -1.2 )
= 0.1151