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The Z-Score is a statistical measure that tells you how many standard deviations a given data point is from the mean of the distribution. The Z-Score formula is represented as \(Z = \frac{(X - \mu)}{\sigma}\), where \(X\) is a value from the distribution, \(\mu\) is the mean, and \(\sigma\) is the standard deviation.

For our lightbulb example, the Z-Score helps us determine the lifespan at which no more than 20% of the lightbulbs are expected to fail. By using the Z-Score, we can calculate a specific hour value (\(X\)) around which to plan our replacement schedule. Essentially, the Z-Score provides a way to standardize different data points for comparison against a normal distribution.

Standard deviation, denoted by \(\sigma\), measures 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 (average), while a high standard deviation indicates that the values are spread out over a wider range.

In the case of our lightbulbs, a standard deviation of \(50\) hours tells us that the lifetimes of the bulbs are typically within \(50\) hours above or below the average lifetime of \(700\) hours. Understanding standard deviation is crucial for making predictions about a data set, as it allows us to assess the reliability and variability of the measurements.

Percentiles are used in statistics to give you a value below which a given percentage of data points in a dataset falls. The 20th percentile, for example, is the value below which 20% of the observations may be found.

In context with the lightbulb problem, we're interested in the lifespan that corresponds to the 20th percentile. This implies replacing the lightbulbs before 20% of them fail, ensuring continued lighting performance. Percentiles are a key concept in managing quality control and setting benchmarks in various industries.

A normal distribution table, commonly known as the 'Z-table', lists the cumulative probabilities associated with Z-Scores and is used to find the percentage of data that falls within certain Z-Scores in a standard normal distribution. Since the normal distribution is symmetrical, the table provides probabilities for the middle to the extreme end (from 0 to positive/negative infinity).

By using the Z-Score in step 2, the table helped us identify that approximately 20% of the bulbs have lifetimes shorter than the calculated value. This table is an essential tool for statisticians and researchers working with normally distributed data. In our exercise, the normal distribution table directly informed the replacement schedule for the lightbulbs.