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The standard normal distribution is a special type of normal distribution. It has a mean (c) of 0 and a standard deviation (cc) of 1. This creates a bell-shaped curve that is symmetrical around the mean. The standard normal distribution is used to standardize normal distributions, making it easier to compare different normal distributions or transform data.

In practical terms, any normal distribution can be converted into a standard normal distribution using a transformation formula. This is usually done to simplify calculations or comparisons:

• For any given data point, you calculate its Z-score using the formula: \[Z = \frac{(X - \mu)}{\sigma}\]where \(X\) is the data point, \(\mu\) is the mean of the distribution, and \(\sigma\) is the standard deviation.
• The resulting Z-score tells you how many standard deviations a particular value is from the mean.

A Z-table, also known as a standard normal distribution table, is used to find cumulative probabilities associated with standard normal distribution Z-scores. These tables help us understand the probability that a random variable falls below a certain value in a standard normal distribution.

The Z-table is organized by rows and columns, each corresponding to whole and fractional parts of Z-scores:

• The rows typically represent the leading Z score digit and one decimal place (e.g., -0.7).
• The columns add an extra hundredth place (e.g., 0.03), making it easy to find exact Z-scores such as -0.73.

By finding the intersection of a given row and column in the Z-table, you can easily locate the cumulative probability for any specific Z-score.

Cumulative probability refers to the probability that a random variable takes on a value less than or equal to a specific value. In the context of the standard normal distribution and Z-table, it provides the total area under the curve to the left of a specific Z-score.

For example, if you want to find the cumulative probability for a Z-score of -0.73, you look it up in the Z-table and find the associated value, such as 0.2327. This value means that there's a 23.27% chance that a variable will fall below a Z-score of -0.73.

The cumulative probability for a higher Z-score of 3.12 from the Z-table might be 0.9991, indicating nearly all potential outcomes fall below this point. To compute the probability between two Z-scores, subtract the cumulative probability of the lower score from that of the higher score. This gives the portion of outcomes that lie between them.

The area under the curve in a standard normal distribution graphically represents the cumulative probability. Each part of the curve corresponds to probabilities of specific Z-value ranges.

For example, to find the probability that a variable lies between two Z-scores, you calculate the area under the curve that falls between these scores. This is visualized as a shaded region on the standard normal distribution curve.

• Calculate the cumulative probability for the upper Z-score.
• Calculate the cumulative probability for the lower Z-score.
• Subtract the cumulative probability of the lower Z-score from the upper one to find the area (probability) between them.

In our example, this area between Z = -0.73 and Z = 3.12 represents the probability of 0.7664. This shaded area is the focus when visualizing or interpreting results.