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A Z-score is a numeric value that represents how many standard deviations a data point is from the mean. In the context of the standard normal distribution, the mean is 0 and the standard deviation is 1. So, if a Z-score is 1, it means the data point is 1 standard deviation away from the mean. Z-scores are useful because they allow us to compare different data points within a distribution. They help normalize different data sets, making them easily comparable.
In our exercise, we're considering Z-scores of 1.42 and 2.17. This tells us where these points lie on the standard normal distribution curve. We can think of Z-scores as specific markers on a number line that lies below the bell-shaped curve, helping us to locate specific intervals.

Probability in the standard normal distribution is found under the curve. The total area under this curve is equal to 1, representing all possible outcomes. Probability thus ranges from 0 to 1, where 0 means an impossible event and 1 means a certain event. For each interval on the normal distribution curve, the area represents the probability of a random variable falling within that interval.
In our scenario, we calculate the probability that a random variable falls between Z-scores of 1.42 and 2.17. Using Z-tables makes it easy to find these probabilities, as they show the cumulative probability from the far left of the curve up to any given Z-score. Thus, finding the probability between two Z-scores involves simple subtraction of these areas. This results in a practical way to measure likelihoods using the normal distribution.

Standard deviation measures how spread out the numbers in a data set are. In a standard normal distribution, this spread is fixed at 1. It acts as a consistent measure that quantifies the amount of variation or dispersion in a set of data points.
A large standard deviation indicates that the data points are spread out over a wider range of values. Conversely, a small standard deviation means that the data points tend to be close to the mean. The concept of standard deviation is essential because it directly impacts the shape and spread of the normal distribution.
For example, knowing the standard deviation in the context of Z-scores helps us understand how far a particular score is from the average in a way that is comparable regardless of other factors, making data analysis more reliable and insightful.

The normal distribution curve, commonly known as the bell curve, is a graphical representation where the mean, median, and mode of the data are all located at the peak. The curve displays a symmetrical distribution of data, with the left half mirroring the right. The curve's tails approach but never quite touch the horizontal axis, which theoretically extends to infinity in both directions. This curve is important in statistics because it models many natural phenomena and serves as the basis for inferential statistics.
In the example exercise, the normal distribution curve helps visualize and calculate the probability between the Z-scores of 1.42 and 2.17. A key property is that approximately 68% of data lies within 1 standard deviation of the mean, approximately 95% within 2 standard deviations, and about 99.7% falls within 3 standard deviations. This distribution ensures that we can predict how data behaves consistently and efficiently.