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Understanding the z-score is fundamental for students navigating the field of probability and statistics. A z-score, also known as a standard score, quantifies the number of standard deviations a data point is from the mean of a distribution. In the given exercise, the z-score is used to transform individual vehicle exit counts into a standardized form that allows comparison on a normal distribution scale. The formula for the z-score is given by:
\[ z = \frac{{x - \mu}}{{\sigma}} \]
where

• \( x \) is the value from the dataset (e.g., the number of cars exiting)
• \( \mu \) represents the mean of the dataset (e.g., 500 vehicles)
• \( \sigma \) is the standard deviation of the dataset (e.g., 75 vehicles)

By employing the z-score calculation, students can easily determine how unusual or typical a particular value is within the data set governed by the normal distribution.

Once a student calculates the z-score, the next phase is to understand what it signifies in terms of probability. This is where the standard normal distribution table, also known as a z-table, comes into play. This table reflects the probability that a standard normal variable will fall between the mean and any number of standard deviations away from the mean (the z-score).
In our textbook example, the z-scores calculated for different scenarios are converted into probabilities using this table. For instance, a positive z-score (such as 2 from part a) allows you to find the area under the curve to the left of that z-score. Conversely, a negative z-score (such as -1.33 from part b) represents an area to the right. The table then provides the cumulative probability for those areas. In problems that deal with ranges (like parts b and c in the exercise), the probabilities of each z-score are subtracted to find the probability that a value falls within that range. Understanding how to read this table is crucial in analyzing and interpreting data under the standard normal distribution.

At its core, the field of probability and statistics is about making sense of data using mathematical tools. Probability is the measure of the likelihood that an event will occur, and it is quantified as a number between 0 and 1, where 0 indicates impossibility and 1 indicates certainty.
Statistics often deals with analyzing a collection of data and drawing conclusions about that data. This sometimes involves normal distribution, which is a continuous probability distribution that is symmetric about the mean, showing that data near the mean are more frequent in occurrence than data far from the mean.
In the exercise provided, calculating the probability of the number of vehicles exiting the highway within a specific range is a typical statistical problem. By identifying that the number of vehicles is normally distributed, we can harness the z-score calculation and the standard normal distribution table to solve for these probabilities, illustrating an intersection of probability and statistics. Mastery of these principles enables students to apply statistical analysis in various real-world situations beyond the textbook, from science experiments to market research.