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Calculating the mean, often denoted as \(\mu\), is the process of finding the average value in a set of numbers. In the context of a normal distribution like our cappuccino example, the mean represents the center of the data distribution. Here, it's crucial because we need to set the mean to ensure a specific outcome - in this case, controlling the amount of cappuccino dispensed by the vending machine so that only 1 in 100 cups overflow.

To calculate the mean when dealing with such a situation:

• Identify the target criteria (here, only 1 out of 100 cups should overflow).
• Use the properties of the normal distribution, such as the standard deviation (\(\sigma\)). In our example, \(\sigma = 0.3\) oz.
• Utilize the Z-score to set the mean formula accordingly.

Once you know these parameters, you can rearrange the mathematical equation to solve for \(\mu\). Here, this process ensures our machine dispenses an optimized average amount, preventing unnecessary spills.

The Z-score is a statistical measure that tells us how many standard deviations an element is from the mean. It's a useful tool in assessing probabilities and is key to understanding data within a normal distribution.

In our task of setting the cappuccino vending machine's mean dispense rate, the Z-score helped us determine how much above or below the average our cap needs to be so that only the 1% of instances see overflow:

• The Z-score was calculated to be approximately 2.33, corresponding to the 99th percentile of a standard normal distribution.
• This means that the point where our cappuccino amount should meet or exceed the cup capacity of 8.5 oz is located 2.33 standard deviations from the desired mean.

Thus, by knowing the typical spread of our normal distribution (given by the standard deviation), we use the Z-score to backtrack and find the exact mean \(\mu\) that achieves our objective. This is especially critical in quality control and consumer satisfaction scenarios.

Percentiles rank data within a distribution and help in understanding where a particular score falls in comparison to others. For example, if a value lies in the 99th percentile, it means it's greater than 99% of the other data points.

In our cappuccino scenario, knowing that only 1 in 100 cups should overflow implies we are interested in the 99th percentile of the cappuccino amounts:

• Percentiles provide a way to set thresholds, such as the overflow limit, ensuring we control variability according to consumer expectations.
• By leveraging the relationship between percentiles and Z-scores, we applied this to a normal distribution to find our target mean \(\mu\).

This approach gives us a robust method to balance between overfill and average fill, optimizing for wastage or inconvenience.