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Z-scores are a way to measure how far away a data point is from the mean of a distribution. They provide a way to compare different data points on a level playing field by transforming them into standard deviations from the mean. This transformation allows you to understand where a particular value falls within a normal distribution. You calculate a Z-score using the formula:\[z = \frac{(X - \mu)}{\sigma}\]where:

• \(X\) is the value you're measuring,
• \(\mu\) is the mean of the distribution,
• \(\sigma\) is the standard deviation.

For example, in a standard normal distribution (mean of 0 and standard deviation of 1), a Z-score of 0.75 tells us that the value is 0.75 standard deviations above the mean.
Z-scores are fundamental in statistics because they allow comparison across different datasets. They also help in calculating probabilities, which is crucial in understanding likely outcomes in many fields.

Probability refers to the likelihood of an event happening. In statistics, it is often represented as a value between 0 and 1, where 0 means an event is impossible and 1 indicates certainty.
For normally distributed data, you can use Z-scores to find probabilities. Specifically, Z-scores allow you to determine how likely it is for a value to fall within a certain range. To calculate this, you often reference a Z-table, which tells you the probability (cumulative from the left) up to a specific Z-score. For instance, a Z-score of 2.25 with a corresponding probability of 0.9878 means there is a 98.78% chance a value falls below a Z of 2.25.
In the exercise, you are asked to find the probability that a value falls between two points (0.75 and 2.25). You find this by using the probabilities for each point and subtracting the smaller from the larger, giving you the probability of 0.2144. This means there is a 21.44% probability for a value to fall between 0.75 and 2.25.

The Standard Normal Curve is a bell-shaped curve that represents a normal distribution with a mean of 0 and a standard deviation of 1. It is a graphical representation of how data is distributed across a range of values. This curve is symmetrical around the mean and follows the 68-95-99.7 rule, often known as the empirical rule. This rule states that:

• 68% of data falls within one standard deviation of the mean,
• 95% falls within two standard deviations, and
• 99.7% falls within three standard deviations.

The area under the curve represents probabilities, which makes it useful for determining the likelihood of different outcomes.
For instance, in the exercise mentioned, when calculating the area under the curve between two Z-scores, you're looking for the probability that a value will fall within that range. The standard normal curve serves as a tool to easily calculate these probabilities through the use of Z-scores and accompanying Z-tables.