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The Z-score is a statistical measure that explains how many standard deviations a data point is from the mean. It helps us understand the position of an individual data point within a distribution. To calculate the Z-score, we use the formula: \( Z = \frac{X - \mu}{\sigma} \), where \( X \) is the value, \( \mu \) is the mean, and \( \sigma \) is the standard deviation.

The Z-score allows us to transform the normal distribution into a standard normal distribution, which has a mean of 0 and a standard deviation of 1. This transformation is beneficial because it enables us to use the Z-table, a tool that provides the probabilities associated with each Z-score. For our specific problem, we calculated Z-scores to determine the probabilities of earning between certain hourly wages. For example, a wage of \\(24.00 leads to a Z-score of 1.00, meaning it is one standard deviation above the mean wage of \\)20.50.

Probability is the measure of the likelihood of an event happening. In the context of normal distribution, it tells us how likely it is for a certain value of a random variable to occur.

In our exercise, we are looking for the probabilities of different ranges of hourly wages based on a normal distribution. By converting these wages into Z-scores, we can use a Z-table to find the probabilities associated with each Z-score. For example, to find the probability of a random crew member earning between \\(20.50 and \\)24.00, we look up the Z-scores for these amounts. The difference between the probabilities gives us the likelihood of the member earning within this range. This method enables clear understanding and calculation of probabilities in various scenarios.

Standard deviation is a key concept in statistics that measures the amount of variation or dispersion in a set of data values. It is the square root of the variance and gives insight into how spread out the data points are around the mean. In simpler terms, it indicates the typical distance of each data point from the mean.

For a normal distribution, the standard deviation allows us to gauge the spread of the data. In our case, the crew members' hourly wages have a standard deviation of \\(3.50 from the mean of \\)20.50. This information is crucial because it shows how much the wages typically deviate from the average, helping us not only to understand the dataset but also to calculate probabilities using Z-scores. It acts like a measuring stick, showing whether data points are typical or unusual.