Question

A bottling company uses a filling machine to fill plastic bottles with a popular cola. The bottles are supposed to contain 300 milliliters (ml). In fact, the contents vary according to a normal distribution with mean 303 ml and standard deviation is 3 ml.

1. What is the probability that an individual bottle contains less than 301.1 ml?

2. What is the probability that the sample mean contents of a sample of 10 bottles is less than 301.1 ml?

3. What is the probability that the sample mean contents of a sample of 10 bottles is more than 299.8 ml?

Answer 1

1)

X ~ N ( µ = 303 , σ = 3 )  
P ( X < 301.1 )   
Standardizing the value   
Z = ( X - µ ) / σ  
Z = ( 301.1 - 303 ) / 3  
Z = -0.63  
P ( ( X - µ ) / σ ) < ( 301.1 - 303 ) / 3 )  
P ( X < 301.1 ) = P ( Z < -0.63 )   
P ( X < 301.1 ) = 0.2643

2)

P ( X̅ < 301.1 ) = ?
Standardizing the value
Z = ( X - µ ) / (σ/√(n)
Z = ( 301.1 - 303 ) / ( 3 / √10 )
Z = -2
P ( ( X - µ ) / ( σ/√(n)) < ( 301.1 - 303 ) / ( 3 / √(10) )
P ( X̅ < 301.1 ) = P ( Z < -2 )
P ( X̅ < 301.1 ) = 0.0228

3)

P ( X̅ > 299.8 ) = 1 - P ( X̅ < 299.8 )
Standardizing the value
Z = ( X - µ ) / ( σ / √(n))
Z = ( 299.8 - 303 ) / ( 3 / √ ( 10 ) )
Z = -3.37
P ( ( X - µ ) / ( σ / √ (n)) > ( 299.8 - 303 ) / ( 3 / √(10) )
P ( Z > -3.37 )
P ( X̅ > 299.8 ) = 1 - P ( Z < -3.37 )
P ( X̅ > 299.8 ) = 1 - 0.0004
P ( X̅ > 299.8 ) = 0.9996