Question

A machine dispenses water into a glass. Assuming that the amount of water dispensed follows a continuous uniform distribution from 10 ounces to 16 ounces, a. what is the average amount of water dispensed by the machine? 13 oz. b. what is the standard deviation of the amount of water dispensed? c. What is the probability that 13 or more ounces will be dispensed in a given glass? d. What is the probability that between 12 and 14 ounces will be dispensed in a given glass?

Answer 1

(a) 13 ounces.

(b) 1.732 ounces.

(c) 0.5

(d) 0.33

Explanation:

Given : The amount of water dispensed follows a continuous uniform distribution from 10 ounces to 16 ounces. So, a=10 and b=16.

(a) The average amount of water dispensed by the machine = (a+b)/2 = (10+16)/2 = 13 ounces.

(b) The standard deviation of the amount of water dispensed =
`\int\limits^a_b {((b-a)^(2) )/(12) } \, dx \\` = √36/12 =√3 = 1.732 ounces.

(c) P (13 or more ounces will be dispensed in a given glass) =
`\int\limits^a_b {f(x)} \, dx` =
`\int\limits^b_a {(dx)/(b-a) } \, dx` = 0.5

(d) P (between 12 and 14 ounces will be dispensed in a given glass) = ( here a=12 and b=14.)
`\int\limits^a_b {f(x)} \, dx` =
`\int\limits^b_a {(dx)/(b-a) } \, dx` = 0.33