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Percentiles are a way to understand the position of a particular data point within a larger set of data. They indicate the percentage of data points that fall below a specific value. For example, if a score is in the 90th percentile, this means it is higher than 90% of the other scores. Percentiles are especially useful in statistics because they help compare different data points across a distribution.
When it comes to the standard normal distribution, converting a percentage to a percentile rank is the first key step in finding the corresponding z-score. By identifying how a specific inclusion percentage relates to the distribution, you can determine what z-score separates that middle percentage from the rest. Understanding percentiles is a fundamental skill for working with data distributions.

The standard normal distribution is a special type of normal distribution that has a mean (\(\mu\)) of 0 and a standard deviation (\(\sigma\)) of 1. This makes it easier to determine probability values and compare data sets. In this distribution, the spread of data forms a bell shape, often referred to as a 'bell curve'.
Using the standard normal distribution is crucial in finding z-scores, as it allows one to interpret and compare how individual scores relate to the average performance. Through z-scores, data can be standardized, meaning different data sets can be compared easily. This standardization is helpful in scenarios where you need to determine which data points fall within certain percentile ranks, such as in the exercise.

Tail values refer to the portions of a probability distribution that lie beyond a specified percentile. In the context of a normal distribution, they usually represent the extreme lower and upper ends of the distribution. Understanding tail values is essential for identifying outliers or extreme results in a data set, particularly in studies that require tail probability analysis.
For example, if you need to separate a middle percentage of data from the distribution, the tail values represent the remainder that lies outside of that central range. When solving for z-scores like \(-z^{*}\) and \(z^{*}\), the tails constitute the extreme percentages outside the middle percentiles, which can provide insights into how frequently extreme values occur.

Extreme values are those that lie in the tails of a distribution. They are distinct from the majority of the data and can signal rare or significant phenomena. Identifying these values is key in understanding the variability and range of a data set.
In the context of a z-score analysis, understanding extreme values involves separating these from the central majority of data. The exercise involves finding critical z-score cutoffs like \(-z^{*}\) and \(z^{*}\), which help to define the range of normal versus extreme outcomes. Recognizing and analyzing extreme values helps in assessing probabilities and making statistical inferences about data distributions.