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Understanding the concept of a z-score is crucial when analyzing data that follows a normal distribution. A z-score, also known as a standard score, indicates how many standard deviations an element is from the mean. The formula for calculating the z-score is \[ Z = \frac{X - \mu}{\sigma} \]where

• \( X \) is the data point,
• \( \mu \) represents the mean of the dataset,
• \( \sigma \) is the standard deviation.

In the exercise, calculating the z-score for the piston ring with an inside diameter exceeding 10.075 centimeters allowed us to determine its position relative to the average piston ring. After calculating the z-score, we utilize the z-table to find the probability associated with that score, which is a critical step in problems involving the normal distribution.

The standard deviation is a measure of the amount of variation or dispersion in a set of values. A low standard deviation indicates that the values tend to be close to the mean, whereas a high standard deviation indicates that the values are spread out over a wider range. In a normal distribution, about 68% of values are within one standard deviation of the mean, 95% are within two standard deviations, and 99.7% are within three standard deviations. In the provided exercise, the standard deviation of the piston ring diameters is 0.03 centimeter. This is key to determining the variability of the diameters from the mean diameter, which allows us to calculate the probability of a piston ring having a diameter within a certain range using the standard deviation and the z-score.

Probability, in the context of a normal distribution, represents the likelihood of a random variable falling within a particular range of values. It is often expressed as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty. The area under the curve of a normal distribution function represents the probability of a value being within a certain range. For instance, part (b) of the exercise asks for the probability that a piston ring will have an inside diameter between 9.97 and 10.03 centimeters. By determining the corresponding z-scores and consulting the z-table, we can find the probabilities for these two points and subtract one from another to get the probability within the range. This is the essence of probability in the context of normal distributions.

Percentiles are used to understand and interpret data by indicating the relative standing of a value within a dataset. The nth percentile of a set of values is the value such that n percent of the data is below it. In the realm of the normal distribution, it is associated with the area under the curve to the left of a given z-score. Part (c) of the exercise requires us to find the value below which 15% of piston rings fall, essentially asking for the 15th percentile. To find this, we identify the z-score that corresponds to the cumulative probability of 0.15 and translate that back into the actual value using the inverse z-score formula. Percentiles serve as a useful tool in making decisions based on the distribution of data and measuring statistical outcomes.