Question

The fill amount of bottles of a soft drink is normally​ distributed, with a mean of

2.0 liters and a standard deviation of 0.05 liter. Suppose you select a random sample of 25 bottles.

a. What is the probability that the sample mean will be between 1.99 and 2.0 liters​?

b. What is the probability that the sample mean will be below 1.98 liters​?

c. What is the probability that the sample mean will be greater than 2.01 ​liters?

d. The probability is 99​% that the sample mean amount of soft drink will be at least how​ much?

e. The probability is 99​% that the sample mean amount of soft drink will be between which two values​ (symmetrically distributed around the​ mean)?

Answer 1

Part a)

X ~ N ( µ = 2 , σ = 0.05 )
P ( 1.99 < X < 2 )
Standardizing the value
Z = ( X - µ ) / ( σ / √(n))
Z = ( 1.99 - 2 ) / ( 0.05 / √(25))
Z = -1
Z = ( 2 - 2 ) / ( 0.05 / √(25))
Z = 0
P ( -1 < Z < 0 )
P ( 1.99 < X̅ < 2 ) = P ( Z < 0 ) - P ( Z < -1 )
P ( 1.99 < X̅ < 2 ) = 0.5 - 0.1587
P ( 1.99 < X̅ < 2 ) = 0.3413

Part b)

X ~ N ( µ = 2 , σ = 0.05 )
P ( X < 1.98 )
Standardizing the value
Z = ( X - µ ) / (σ/√(n)
Z = ( 1.98 - 2 ) / ( 0.05 / √25 )
Z = -2
P ( ( X - µ ) / ( σ/√(n)) < ( 1.98 - 2 ) / ( 0.05 / √(25) )
P ( X < 1.98 ) = P ( Z < -2 )
P ( X̅ < 1.98 ) = 0.0228

Part c)

X ~ N ( µ = 2 , σ = 0.05 )
P ( X > 2.01 ) = 1 - P ( X < 2.01 )
Standardizing the value
Z = ( X - µ ) / ( σ / √(n))
Z = ( 2.01 - 2 ) / ( 0.05 / √ ( 25 ) )
Z = 1
P ( ( X - µ ) / ( σ / √ (n)) > ( 2.01 - 2 ) / ( 0.05 / √(25) )
P ( Z > 1 )
P ( X̅ > 2.01 ) = 1 - P ( Z < 1 )
P ( X̅ > 2.01 ) = 1 - 0.8413
P ( X̅ > 2.01 ) = 0.1587

Part d)

X ~ N ( µ = 2 , σ = 0.05 )
P ( X > x ) = 1 - P ( X < x ) = 1 - 0.99 = 0.01
To find the value of x
Looking for the probability 0.01 in standard normal table to calculate Z score = -2.3263
Z = ( X - µ ) / ( σ / √(n) )
-2.3263 = ( X - 2 ) / (0.05/√(25))
X = 1.9767
P ( X >= 1.9767 ) = 0.99

Part e)

X ~ N ( µ = 2 , σ = 0.05 )
P ( a < X < b ) = 0.99
Dividing the area 0.99 in two parts we get 0.99/2 = 0.495
since 0.5 area in normal curve is above and below the mean
Area below the mean is a = 0.5 - 0.495
Area above the mean is b = 0.5 + 0.495
Looking for the probability 0.005 in standard normal table to calculate Z score = -2.5758
Looking for the probability 0.995 in standard normal table to calculate Z score = 2.5758
Z = ( X - µ ) / ( σ / √(n) )
-2.5758 = ( X - 2 ) / ( 0.05/√(25) )
a = 1.9742
2.5758 = ( X - 2 ) / ( 0.05/√(25) )
b = 2.0258
P ( 1.9742 < X < 2.0258 ) = 0.99