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A Z-score is a statistical measurement that describes a value's position relative to the mean of a group of values. It is also an essential concept in the realm of normal distribution. The Z-score tells us how many standard deviations away a specific measurement is from the mean. This can be useful for determining differences between data sets and identifying outliers.

For example, if a data point has a Z-score of 2.33, it means the data point is 2.33 standard deviations above the mean. In this scenario, given the normal distribution of our data—ounces dispensed from the vending machine—a Z-score related to the 99th percentile would show us that only 1% of the data would fall above this point.

By using the Z-score formula:

• \( Z = \frac{X - \mu}{\sigma} \)

Where,

• \(X\) is the data value (such as the 6 oz cup capacity),
• \(\mu\) is the mean,
• \(\sigma\) is the standard deviation.

We can solve for unknown quantities such as the mean \(\mu\), when values for \(X\), \(Z\), and \(\sigma\) are known.

Standard deviation is a measure of the amount of variation or dispersion in a set of values. In simple terms, it tells us how spread out the numbers are in a given data set. A low standard deviation indicates that the values tend to be close to the mean, while a high standard deviation indicates that the values are spread out over a wider range.

In the context of the vending machine exercise, the standard deviation is given as 0.2 oz, meaning this is the typical amount of deviation from the average amount a cup may be dispensed.

Understanding standard deviation helps us know how consistent the machine is at dispensing soft drinks. A smaller standard deviation would mean that most glasses receive an amount of drink close to the average, making it more predictable. This predictability is crucial when trying to ensure a 6 oz glass can be filled (without overflow) 99% of the time.

Percentiles offer a way to understand and interpret data by showing the relative standing of a value within a data set. They partition the data into 100 equal parts, essentially telling us what portion of the data falls below a given value.

For example, the 99th percentile means that 99% of the data points lie below this value, and only 1% lie above. In the exercise, ensuring that a 6 oz glass holds the soft drink 99% of the time indicates that the 99th percentile of ounces dispensed must not exceed 6 oz. This requires calculating the appropriate mean using other known values such as the Z-score and standard deviation.

This is why in the Z-score calculation we refer to the Z-table to translate the 99th percentile into its respective Z-score, which allows us to redefine the mean to balance out this high percentage of confidence for the cup volume being enough to prevent overflow in most cases.