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Understanding the Z-score is fundamental to interpreting data in a normal distribution. The Z-score represents the number of standard deviations a data point is from the mean. In mathematical terms, we calculate it using the formula:
\[ Z = \frac{x - \mu}{\sigma} \]
In this formula, \(x\) is the value of our normal random variable, \(\mu\) is the mean, and \(\sigma\) is the standard deviation. This normalizes different data sets to a standard scale, which allows for comparison, even when the original data have different units or variances.

In the example provided, calculating the Z-score transforms the original scores of \(x\) values into a standard scale where the mean (\(\mu\)) is zero and the standard deviation (\(\sigma\)) is one. This simplification is immensely helpful when trying to determine the probability of certain outcomes. Improving your grasp of Z-score concepts will bolster your confidence in tackling problems involving normal distributions.

The standard normal distribution table, also known as a Z-table, is a reference that shows the area (or probability) to the left of a Z-score in a standard normal distribution. Since the total area under a normal curve equals 1, which corresponds to the probability of all outcomes, the table helps us to find the proportion or probability of observations that fall to the left (or right) of a Z-score.

For a right-tail event such as \(x > 13.5\), we compute \(P(Z > 1.75)\) by subtracting the table value for \(P(Z < 1.75)\) from 1. Such techniques are crucial when handling the tail ends of the distribution, and learning how to navigate the table is key to determining these probabilities effectively. Making ample use of the standard normal distribution table in practice will enhance your ability to compute probabilities for any Z-score.

Probability and statistics are the foundational concepts that allow us to quantify uncertainty and make data-driven predictions. Probability quantifies the likelihood of an outcome or a set of outcomes, while statistics uses this notion to analyze and interpret data. In our normal distribution exercise, understanding probability helps us to calculate, for instance, how likely it is for a random variable to fall between two values.

With the calculated Z-scores, we can look at the cumulative distribution up to those points. By using probability (as read from the standard normal distribution table or calculated by software), we answer questions such as \(P(9.4 < x < 10.6)\), essentially finding the likelihood that a value falls within a specified interval of the mean. Grasping these principles will not only help in academic pursuits but also in real-world applications where decision-making often relies on statistical evidence and probabilistic assessments.