An automated machine dispenses soda into cups at Providence Park during Timbers

Question

An automated machine dispenses soda into cups at Providence Park during Timbers games. The soda dispensed is normally distributed with a mean of mu = 16 ounces and a standard deviation of sigma = 0.8 ounces. a. What is the probability that a randomly selected cup has between 14 and 17 ounces? label the Standard Normal Distribution below and shade appropriately? b. If the cups can hold a maximum of 18.5 ounces, and 2000 sodas are sold on a particular day. how many fans will get sticky fingers (as a result of the soda machine overfilling their cup)?

Explanation / Answer

Solution

Back-up Theory

If a random variable X ~ N(µ, 2), i.e., X has Normal Distribution with mean µ and variance 2, then

Z = (X - µ)/ ~ N(0, 1), i.e., Standard Normal Distribution ………………………..(1)

P(X or t) = P[{(X - µ)/ } or {(t - µ)/ }] = P[Z or {(t - µ)/ }] .……..…(2)

Now, to work out the solution,

Let X = soda dispensed in ounces. We are given X is Normally distributed with µ = 16 and = 0.8

Part (a)

Probability a randomly selected cup has soda between 14 and 17 ounces = P(14 < X < 17)

= P(- 1.25 < Z < 1.25) [vide (2) under Back-up Theory ]

= 0.7887 [using Excel Function] ANSWER

Part (b)

Number of fans with sticky fingers = Total number of fans x Probability the cup overflows

= 2000 x P(X > 18.5) = P(Z > 3.125)[vide (2) under Back-up Theory ]

= 2000 x 0.0009 [using Excel Function]

= 1.8 ~ 2 ANSWER

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