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Troyanec [42]
3 years ago
6

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?
Mathematics
1 answer:
Alchen [17]3 years ago
8 0

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

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