Question

In: Statistics and Probability

3. A mix beverage machine releases a certain amount of syrup into a chamber where it...

3. A mix beverage machine releases a certain amount of syrup into a chamber where it is mixed with carbonated water. The amount of syrup follows a normal distribution with a mean of 1.29 fl.oz. and a standard deviation 0.016 fl.oz. a. Find the probability that a syrup amount does not exceed 1.33 fl.oz.

b. Find the probability that a syrup amount is less than 1.29 fl.oz.

c. Find the probability that a syrup amount exceeds the mean value by more than two standard deviations.

d. Find the syrup amount so that the probability that it is exceeded is 5%.

Solutions

Expert Solution

Solution :

Given that ,

mean = = 1.29

standard deviation = = 0.016

a) P( x< 1.33)  = P[(x - ) / < (1.33 - 1.29) / 0.016 ]

= P(z < 2.50)

Using z table,

= 0.9938

b) P( x< 1.29)  = P[(x - ) / < (1.29 - 1.29) / 0.016 ]

= P(z < 0)

Using z table,

= 0.5000

c) P(x > 1.322) = 1 - p( x< 1.322)

=1- p P[(x - ) / < (1.322 - 1.29) / 0.016]

=1- P(z < 2.00)

Using z table,

= 1 - 0.9772

= 0.0228

d) Using standard normal table,

P(Z > z) = 5%

= 1 - P(Z < z) = 0.05  

= P(Z < z) = 1 - 0.05

= P(Z < z ) = 0.95

= P(Z < 1.645 ) = 0.95  

z = 1.645

Using z-score formula,

x = z * +

x = 1.645 * 0.016 + 1.29

x = 1.32 fl.oz


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