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The standard normal distribution is a special type of normal distribution with a mean of 0 and a standard deviation of 1. This distribution is frequently used in statistics because it allows us to easily determine probabilities and z-scores, which are helpful in understanding how data points compare to the overall population.
In the context of probability, the standard normal distribution serves as a model that makes it easier to calculate the likelihood that a random variable falls within a certain range. Its symmetric, bell-shaped curve means that probabilities of outcomes can be directly found using statistical tables or software.
The properties of the standard normal distribution make it a useful tool for various statistical applications, particularly when we want to standardize other normal distributions for easier analysis. Because every standard normal distribution adheres to these properties, it provides a universal way of understanding and working with probability.

Z-scores are standardized scores that tell us how many standard deviations a data point is from the mean in a standard normal distribution.
Essentially, a z-score converts individual data points into a common scale, enabling comparison across different sets of data. A positive z-score means the value is above the mean, while a negative z-score indicates it's below the mean. A z-score of 0 denotes that the value is exactly average.
The formula for calculating a z-score is:

• \( z = \frac{X - \mu}{\sigma} \)

where \( X \) is the data value, \( \mu \) is the mean, and \( \sigma \) is the standard deviation. Understanding z-scores helps in interpreting data within the context of the standard normal distribution, providing insights into how typical or atypical a certain value is.

The standard normal curve is a graphical representation of the standard normal distribution, characterized by its characteristic bell shape. This curve is centered at zero and extends infinitely in both directions, though most of the distribution lies within three standard deviations from the mean.
One of the most useful features of the standard normal curve is that it can be used to visually illustrate the probability associated with z-scores. The curve is symmetrical, meaning that both sides mirror each other, with equal probability being assigned to scores further from the mean in either direction.
When solving probability questions like the one given in the exercise, shading regions under the standard normal curve can help you visualize the probability you're calculating. For example, if you are asked to find the probability between two z-values, as in this problem, you would shade the area under the curve that spans those values, representing the total probability encompassed. This shows that probabilities can be presented not only as numerical values but also as graphical areas, making them easier to interpret.