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 0.07 liter. Suppose you select a random sample of 25 bottles.

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

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

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

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

Round to three decimal places as​ needed

Answer 1

Solution :

Given that ,

mean = = 2.0 liters

standard deviation = = 0.07 liters

n = 25

= = 2.0 liters

= / n = 0.07 / 25 = 0.014

a) P( < 1.98 ) = P(( - ) / < (1.98 - 2.0) / 0.014)

= P(z < -1.43)

Using z table

= 0.0764

b) P( > 2.01) = 1 - P( < 2.01 )

= 1 - P[( - ) / < (2.01 - 2.0) / 0.014]

= 1 - P(z < 0.71)

Using z table,    

= 1 - 0.7611

= 0.2389

c) Using standard normal table,

P(Z > z) = 99%

= 1 - P(Z < z) = 0.99  

= P(Z < z ) = 1 - 0.99

= P(Z < z ) = 0.01

= P(Z < -2.326) = 0.01  

z = -2.326

Using z-score formula  

= z * +

= -2.326 * 0.014 + 2.0

= 1.967 liters.

d) Using standard normal table,

P( -z < Z < z) = 99 %

= P(Z < z) - P(Z <-z ) = 0.99

= 2P(Z < z) - 1 = 0.99

= 2P(Z < z) = 1 + 0.99

= P(Z < z) = 1.99/ 2

= P(Z < z) = 0.995

= P(Z < 2.576) = 0.995

= z  ± 2.576

Using z-score formula  

= z * +

= -2.576 * 0.014 + 2.0

= 1.964 liters.

Using z-score formula  

= z * +

= 2.576 * 0.014 + 2.0

= 2.036 liters.

99% two values = 1.964 liters and 2.036 liters