Question

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?

Answer 1

Answer:

$\mu = 5.178$

Step-by-step explanation:

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

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

$Z = \frac{X - \mu}{\sigma}$

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:

$\sigma = 0.4$

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

This means that when $X = 6$, Z will have a pvalue of 1-0.02 = 0.98. So when $X = 6, Z = 2.055$. We have to find $\mu$.

$Z = \frac{X - \mu}{\sigma}$

$2.055 = \frac{6 - \mu}{0.4}$

$6 - \mu = 2.055*0.4$

$\mu = 6 - 2.055*0.4$

$\mu = 5.178$