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The Z-score is a standard measure that quantifies the position of a data point relative to the mean of a group of data points in terms of standard deviations. It is a crucial concept for understanding the normal distribution and is calculated with the formula:
\[ Z = \frac{{x - \mu}}{{\sigma}} \]
Where:

• \(x\) represents the value of the data point,
• \(\mu\) is the mean of the distribution,
• \(\sigma\) is the standard deviation.

In our example, to find out how many standard deviations a particular value of red dye (\(x\)) is from the mean (\(\mu\)), we use the above formula. For example, to find the Z-score for 5.0 ml of red dye, we calculate:
\[ Z = \frac{{5.0 - 5}}{{0.2}} = 0 \]
A Z-score of 0 indicates that the value is exactly at the mean. Z-scores also enable us to use standard normal distribution tables to find out probabilities associated with different values of \(x\) as seen in the steps of the provided solution.

Standard deviation (\(\sigma\)) is a statistical measure that describes the spread or dispersion of a set of values within a distribution. A larger standard deviation indicates that the values are more spread out from the mean. In the context of our exercise, a standard deviation of 0.2 ml implies that the amount of red dye in most one-gallon cans is fairly close to the mean. Intuitively, a small standard deviation in this scenario suggests that the paint company maintains consistent quality in their dye mixing process.
Calculating standard deviation is fundamental before you can work out Z-scores and probabilities for the normal distribution. As standard deviation represents consistency, understanding it can help you discern how reliable a set of data can be. When the standard deviation is known, as in our exercise, we can use it to calculate Z-scores, which in turn help us with determining probabilities.

The normal distribution model is a symmetric, bell-shaped curve that represents the spread of a dataset where most of the occurrences take place near the mean and fewer as you move away. This model is used to describe real-life values that cluster around a single mean value, like the amount of red dye in paint cans produced by a company.
The properties of a normal distribution include:

• The mean, median, and mode of the distribution are all equal.
• The curve is symmetric about the mean.
• The area under the curve is 1, implying that the total probability is 100%.
• Approximately 68% of the data falls within one standard deviation of the mean; 95% within two standard deviations; 99.7% within three.

In our exercise, we used the normal distribution model to calculate the probability of finding specific amounts of red dye in the paint cans, which enabled us to understand the quality control of the paint manufacturing process. Understanding the normal distribution model is essential to correctly interpret many natural and industrial processes.