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Z-scores are a way of describing how far a particular point in a data set is from the average, measured in standard deviations. Imagine you are looking at a bell-shaped curve, which is typically associated with a normal distribution. A z-score tells you how many standard deviations a specific number or value is away from the mean (average) of the data set. Why are z-scores useful? They help us standardize values from different data sets, allowing for comparisons that would otherwise be impossible. For instance, if you want to know how good a student's test score is compared to the average test score, you can compare their z-score to the z-scores of others.Calculation of a z-score is quite simple. The formula is: \[ z = \frac{x - \mu}{\sigma} \]Where:

• \(x\) is the data point you're examining.
• \(\mu\) is the mean (average) of the data set.
• \(\sigma\) is the standard deviation of the data set.

A z-score can be positive or negative, indicating whether the data point is above or below the mean. This becomes particularly useful when using the z-table to find probabilities.

In the context of the standard normal distribution, probabilities tell us the likelihood that a random variable will fall within a particular range. These probabilities correspond to the areas under the curve of a standard normal distribution graph.A probability is a number, usually between 0 and 1, that reflects the chance of a particular event occurring. A probability of 1 suggests that an event will definitely happen, and a probability of 0 means it definitely will not happen.When working with the normal distribution, we often compute the probability that a variable falls between two z-scores, much like determining the area between those points on the graph. In our exercise, we were asked to determine the probability that the variable lies between \(z = 0.24\) and \(z = -1.12\). To do this, we look up each z-score in a Z-table and find their corresponding probabilities, then calculate the difference.

The Z-table, also known as the standard normal distribution table, is a mathematical table that illustrates the probability that a standard normal random variable is less than or equal to a given value. It is crucial for finding probabilities associated with z-scores in a standard normal distribution. By using the Z-table, you can learn the cumulative probability up to any given z-score.The table is structured so that:

• The rows represent the integer and first decimal place of the z-score (like 0.2 for 0.24).
• The columns represent the second decimal place (like 0.04 to gain the exact 0.24).

For instance, if you look for a z-score of 0.24, you align the row and column in the table to find a probability, which indicates the area under the curve to the left of \(z = 0.24\). Similarly, you find the probability for \(z = -1.12\), and this information then helps compute the area between these z-scores, as we did by subtracting the smaller cumulative probability from the larger one to get the final probability value of interest.