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To begin understanding the normal distribution problems, the first step is the calculation of the Z-score. The Z-score helps us understand how far a particular data point is from the average, in terms of standard deviations. It is given by the formula:

• \( Z = \frac{x - \mu}{\sigma} \)

Here, \( x \) is your data point, \( \mu \) is the mean, and \( \sigma \) is the standard deviation.
This score tells us whether the data point falls above or below the mean. For instance, a positive Z-score indicates the data point is above the mean, while a negative Z-score suggests it is below.
In our exercise, say for point \( x = 58 \), the Z-score would be:

• \( Z_a = \frac{58 - 55}{7} = 0.43 \)

A Z-score of 0.43 signifies that the data point is 0.43 standard deviations above the mean. Calculating Z-scores for all given \( x \) values will allow us to move forward with finding probabilities using the standard Z-table.

The standard deviation is a key component when working with normal distributions, as it measures the amount of variation or dispersion in a set of values. In simpler terms, it tells you how much the data points deviate from the mean.
In the context of this problem, knowing the standard deviation is crucial because it facilitates the calculation of the Z-score.

• The formula for the standard deviation is frequently represented as \( \sigma \).

For our normal distribution calculation, understanding that the standard deviation is \( \sigma = 7 \) means that, on average, data points will deviate by 7 units from the average value of 55.
This knowledge provides a foundation not only for calculating Z-scores but also for understanding how tightly the data is clustered around the mean. Less deviation means a sharper peak, while more deviation indicates a wider spread.

Probability finding helps us comprehend which areas under the normal distribution curve correspond to specific circumstances or ranges of values. Once we have calculated the Z-score, the next step is to use a standard Z-table to find the probabilities related to those Z-scores.
The Z-table provides the area (probability) to the left of a Z-score.

• For example, for a Z-score of 0.43, the associated probability is 0.6664. This means that 66.64% of the data falls to the left of this Z-score.

When you require the probability for a score greater than your Z-score (to the right), you subtract the Z-table result from 1. For a Z-score of 0.43, any data point to the right encompasses:

• \( 1 - 0.6664 = 0.3336 \)

Similarly, if you need the probability for a score less than your Z-score (to the left), just use the found value directly from the table. Therefore, solving these exercises requires understanding how to extract meaningful probability insights from the Z-table based on the corresponding Z-scores.