Final answer:
The average volume dispensed by the vending machine is 8 ounces with a standard deviation of approximately 0.463 ounces. The probability of dispensing more than 8.5 ounces is 0.188, for exactly 8.4 ounces is 0, and for less than 8.7 ounces is 0.938.
Step-by-step explanation:
Continuous Uniform Distribution
To address the questions about the coffee dispensing machine, we'll apply principles from statistics regarding the continuous uniform distribution.
(a) The mean of a continuous uniform distribution is calculated as the average of the minimum and maximum values. Hence, the mean (average) is (7.2 ounces + 8.8 ounces) / 2 = 8 ounces.
(b) The standard deviation for a continuous uniform distribution is given by the formula σ = √((b - a)^2 / 12), where 'a' and 'b' are the minimum and maximum values, respectively. So, σ = √((8.8 - 7.2)^2 / 12), which, when rounded, gives us a standard deviation of approximately 0.463 ounces.
(c) Probability of dispensing more than 8.5 ounces: Since we are working with a uniform distribution, the probability of any specific interval is the length of the interval divided by the length of the entire range. So, it would be (8.8 - 8.5) / (8.8 - 7.2) = 0.3 / 1.6 = 0.1875, so the probability rounded to three decimal places is 0.188.
(d) The probability of dispensing exactly 8.4 ounces: In a continuous distribution, the probability of the variable taking on an exact value is 0 because there are an infinite number of possible values.
(e) Probability of dispensing less than 8.7 ounces: This is calculated similarly to part (c). The length of this interval would be (8.7 - 7.2) / 1.6 = 1.5 / 1.6, which gives us a probability of 0.938, rounded to three decimal places.