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The Z-score is a crucial concept in understanding normal distribution and probability calculations. It tells us how far away a particular data point is from the mean, expressed in terms of standard deviations. In simple terms, the Z-score allows us to standardize different data points on a common scale, making it easier to compare them.
Let's consider the exercise: for the random variable \( x = 57 \), given a mean \( \mu = 74 \) and standard deviation \( \sigma = 8 \), the Z-score formula is:

• \( Z = \frac{x - \mu}{\sigma} \)
• Substitute the values: \( Z = \frac{57 - 74}{8} = -2.125 \)

A Z-score of \(-2.125\) means our value \( x = 57 \) is about 2.125 standard deviations below the mean of 74. Remember, negative Z-scores mean the observed value is below the mean, while positive scores mean it's above.

Standard deviation is a measure of how spread out the values in a data set are relative to the mean. If you're dealing with a normal distribution, the standard deviation helps describe the distribution's shape and spread.
When the standard deviation is small, data points are close to the mean, resulting in a steeper, narrow distribution. Conversely, large standard deviations yield a flatter, wider distribution. In the exercise, with a standard deviation \( \sigma = 8 \) and mean \( \mu = 74 \), these numbers give us a sense of how much variability exists in the data.
A crucial point is that roughly 68% of data points lie within one standard deviation of the mean in a normal distribution. This expands to about 95% within two standard deviations, and 99.7% within three. Understanding this helps you quickly estimate probabilities and the likelihood of given observations.

Probability calculations in normal distribution utilize Z-scores to find how likely it is for a certain value to occur. After calculating the Z-score, you can refer to a standard normal distribution table, often called a Z-table. This table provides the probability that a standard normal variable is less than your calculated Z-score.
In the step-by-step solution, we calculated a Z-score of \(-2.125\). By looking this up in the Z-table, we find the probability \( P(x < 57) \) is approximately 0.0169, or 1.69%. This value tells us how likely it is for the random variable \( x \) to be less than 57. A probability of 0.0169 indicates a rare occurrence, as the probability is only 1.69%.
Using the Z-table effectively bridges the computed Z-score to a real-world understanding of probabilities. It transforms abstract numbers into meaningful probabilistic insights.