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The concept of a z-score is an essential part of understanding how a value relates to the mean in a set of data that follows a normal distribution. The z-score tells us how many standard deviations a specific value is from the mean of the dataset. In mathematical terms, the z-score is calculated using the formula: \[ z = \frac{X - \mu}{\sigma} \]where:

• \(X\) is the value of interest,
• \(\mu\) is the mean of the dataset,
• \(\sigma\) is the standard deviation of the dataset.

For example, if a car needs to brake within 40 feet and the mean braking distance is 50 feet with a standard deviation of 8 feet, the calculation would be:\[ z = \frac{40 - 50}{8} = -1.25 \]A negative z-score implies that the value is below the mean.

Probability provides a way to quantify the likelihood of an event. In the context of a normal distribution, probability helps determine the chance that a certain outcome falls within a specified range. After obtaining a z-score, we use it to find the probability of a random variable being less than or equal to that value. For instance, in our exercise, the probability of braking to a stop within 40 feet (a z-score of -1.25) is found to be approximately 21.19% or 0.2119. This probability is derived from the normal distribution table, known as the z-table. Likewise, probabilities help understand and plan for different outcomes, like assessing the risk of collision when braking distance varies.

The z-table is a crucial tool used to find the probability associated with a particular z-score in a standard normal distribution. The table provides the area under the curve to the left of a given z-score. When using a z-table:

• Locate the z-score in the table by looking up its row and column.
• The value found in the table represents the probability that a standard normal variable is less than or equal to that z-score.

For example, a z-score of -1.25 corresponds to a probability of 0.2119, indicating that approximately 21.19% of the data lies below this score. Similarly, a z-score of 1.25 relates to a probability of 0.7881, which is used to assess the likelihood of successfully stopping within 60 feet.

Standard deviation is a measure of the amount of variation or dispersion in a set of values. In the context of normal distribution, it shows how much the individual data points deviate from the mean. A smaller standard deviation means that the data points tend to be closer to the mean, while a larger standard deviation indicates more spread out data. The formula for standard deviation is: \[ \sigma = \sqrt{\frac{\sum (X_i - \mu)^2}{N}} \]where:

• \(X_i\) are the data points,
• \(\mu\) is the mean,
• \(N\) is the number of data points.

In our braking distance scenario, a standard deviation of 8 feet tells us that most braking distances will fall within 8 feet above or below the mean of 50 feet. Understanding standard deviation allows for prediction of how often outcomes will occur within certain ranges.