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The z-score is a statistical measure that describes the position of a raw score in terms of its distance from the mean, when measured in standard deviation units. To calculate it, you can use the formula:
\[ z = \frac{x-\mu}{\sigma} \]
where \( z \) is the z-score, \( x \) is the value from the data set, \( \mu \) is the mean, and \( \sigma \) is the standard deviation. In the context of the exercise, finding the z-score enables us to transform any given value from the original distribution into a corresponding value on the standard normal distribution, thus simplifying the process of finding probabilities.

Why Calculate a Z-Score?

Understanding the z-score is vital as it allows comparison between different datasets by normalizing the scores. For instance, if you want to compare test scores between two classes with different average performances and spread of scores, the z-score provides a way to determine how a score from one class compares to the average score of the other class.In our paint mixture example, calculating the z-score for a particular amount of red dye helps us understand how unusual (or usual) that amount is in relation to the typical amount expected, as defined by the mean.

The standard normal distribution is a specific normal distribution with a mean of 0 and a standard deviation of 1. It's the foundation for z-score calculations because once a score from any normal distribution is transformed into a z-score, it is expressed in terms of the standard normal distribution, regardless of the original mean or standard deviation.This distribution is critical because it allows us to use standard normal distribution tables (z-tables) to find probabilities associated with specific z-scores, which are otherwise not easily calculated without such tools. Since the properties of the standard normal distribution are well-known and tabulated, it serves as a reference framework for statistical analysis.

Importance of the Standard Normal Distribution

• Universality: Any normal distribution can be converted to a standard normal distribution.
• Comparison: It allows for the comparison of scores from different distributions.
• Predictability: Patterns and probabilities are consistent and easily looked up due to standardization.

Statisticians and researchers highly value the standard normal distribution, often depicted by the bell curve, because of its predictability and the ease it brings to statistical analysis.

A z-table lists the cumulative probabilities of the standard normal distribution up to a given z-score. It's an essential tool for finding the probability associated with a z-score without complex calculations. The table shows the probability that a standard normally distributed variable is less than or equal to a particular value.To use a z-table, you simply find the z-score for the value of interest (as detailed in the exercise solution), and then look up the corresponding probability in the z-table. If the z-score is positive, it refers to the area to the left of that z-score. If the z-score is negative, you must consider the symmetry of the normal distribution, and often you'll use the complement (1 minus the value) to find the probability of being greater than the value on the left side.

Using the Z-Table in the Exercise

For example, if we calculate a z-score for the paint mixture and find that our z-score is 2, we can look up this value in the z-table to find that approximately 97.72% of the data falls below this z-score. Conversely, if we are interested in calculating the probability that the amount of red dye is greater than a certain value, we use the complement of the z-table result, effectively finding the area under the curve to the right of the z-score.