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When dealing with normal distributions, two important values to understand are the mean and standard deviation. The **mean** represents the average value of all data points in the distribution. It is the central point where the data balances out. In the context of the exercise, the mean hourly wage is given as \( \\(20.50 \). This means, on average, a maintenance crew member earns \( \\)20.50 \) per hour.

The **standard deviation** indicates how spread out the data points are around the mean. A smaller standard deviation means the data points are closer to the mean, while a larger standard deviation indicates more variability. Here, the standard deviation is \( \\(3.50 \). This tells us that the wages typically deviate from the mean by \( \\)3.50 \) on either side.

Understanding these two values allows us to grasp how clustered or spread out the wages are in this particular normal distribution. The combination of a mean and standard deviation allows us to describe the overall shape of the wage distribution.

Z-scores allow us to compare individual data points to the mean of the distribution. A **Z-score** is essentially a measure of how many standard deviations an element is from the mean. The formula to calculate a Z-score for a given value \( X \) is:
\[ Z = \frac{X - \mu}{\sigma} \]where \( \mu \) is the mean and \( \sigma \) is the standard deviation.

In our exercise, finding Z-scores helps to standardize various hourly wage values relative to the distribution centered at \( \\(20.50 \). For instance, when determining how far \( \\)24.00 \) is from the mean, we calculate:
\[ Z = \frac{24.00 - 20.50}{3.50} \approx 1.00 \]This Z-score of 1.00 tells us that \( \$24.00 \) is one standard deviation above the mean. By computing Z-scores, we convert different wage estimates to a common scale, enabling us to use Z-tables for finding probabilities associated with those scores.

Understanding Z-scores is key to analyzing individual data in terms of its position in any normal distribution.

Calculating probabilities using the normal distribution involves translating Z-scores into probability values. Once Z-scores are determined, we can use a **Z-table** to find the probability associated with those scores. The Z-table lists probabilities that correspond to standard normal distribution Z-scores.

For example, to find the probability of earning between \( \\(20.50 \) and \( \\)24.00 \), we calculate that the Z-scores are 0 and 1, respectively. The Z-table shows that the probability of a Z-score less than or equal to 1 is 0.8413, and it indicates a probability of 0.5000 for the mean (Z-score of 0). The difference between these probabilities gives us the likelihood that a crew member earns within that wage range, calculated as 0.8413 - 0.5000 = 0.3413.

In another scenario, to determine the likelihood of earning more than \( \\(24.00 \), the complement probability (1 - 0.8413) helps us find a value of 0.1587. This approach reverses the probability of earning less than \( \\)24.00 \). Similarly, by using the Z-table for negative Z-scores, we grasp how often a wage is less than \( \$19.00 \), finding a corresponding probability of 0.3336.

Overall, probability calculations using Z-scores and Z-tables allow us to understand the likelihood of various outcomes within normal distributions.