A cappuccino vending machine is designed to dispense an average of μ oz per cup. If the ounces per fill are Normally distributed, with a standard deviation of 0.4 oz, at what value should μ be set so that 6 oz cups will overflow only 2 % of the time?

Step-by-step explanation:

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean and standard deviation , the zscore of a measure X is given by:

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this problem, we have that:

At what value should μ be set so that 6 oz cups will overflow only 2 % of the time?

This means that when , Z will have a pvalue of 1-0.02 = 0.98. So when . We have to find .