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When working with a normal distribution, percentiles are incredibly helpful in determining where a particular data point stands in relation to the rest of the dataset. A percentile indicates the value below which a given percentage of observations fall. For example, the 90th percentile is the value below which 90% of the observations may be found.

Understanding percentiles helps us ascertain what portion of a dataset falls below a specific threshold. It is significant in the exercise where we need to find the cutoff time for the fastest 10%. This translates to finding the 90th percentile, meaning 90% of the rest of the times will be slower, and thus the remaining 10% are eligible for advanced training.

In general, once you determine the percentile you're interested in, this information can be utilized with a normal distribution table or Z-score formula to find the corresponding cutoff value in your data.

The Z-score is a statistical tool that tells us how many standard deviations an element is from the mean. It is a fundamental part of understanding and working with normal distributions.

To compute a Z-score, you use the formula: \[ z = \frac{x - \mu}{\sigma} \]where:

• \(z\) is the Z-score,
• \(x\) is the value in the dataset,
• \(\mu\) is the mean of the dataset, and
• \(\sigma\) is the standard deviation.

In the case of finding percentiles, instead of starting with a data point to find the Z-score, we often start with a known Z-score corresponding to a desired percentile. For example, a Z-score of around 1.28 represents the 90th percentile in a standard normal distribution.

By rearranging the Z-score formula to solve for \(x\), we can find what value in the dataset corresponds to a particular percentile, helping us determine which applicants will qualify for advanced training.

Standard deviation is essential in statistics as it measures the amount of variation or dispersion in a set of values. When dealing with normal distributions, the standard deviation helps you understand how much the dataset differs from the mean.

If you visualize a normal distribution curve, the standard deviation determines the width of the bell curve. A smaller standard deviation implies data points are closer to the mean, while a larger one indicates more spread out values.

In practical terms, when solving problems related to the normal distribution like the given exercise, the standard deviation aids in calculating the cutoff values when combined with the Z-score. For the task at hand, knowing the standard deviation of 20 seconds allows us to compute the specific task completion time that corresponds to the 90th percentile through the relationship:\[ x = \mu + z * \sigma \]Understanding standard deviation is crucial to interpreting the degree of variability in data such as job applicants' task completion times.