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When working with normal distributions, the standard normal variable, often denoted as \( z \), plays a crucial role. The standard normal variable is a way to transform a normal distribution with any given mean \( \mu \) and standard deviation \( \sigma \) into a standardized form with a mean of 0 and a standard deviation of 1.
This process involves calculating the \( z \)-score, which is a measure that describes how many standard deviations an element \( x \) is from the mean. The transformation formula is:

• \( z = \frac{x - \mu}{\sigma} \)

This standardized form allows easier calculation of probabilities. It effectively maps our specific distribution to a reference distribution, making it easier to use universal tools like the Z-table.

The \( z \)-score is an essential statistic in probability because it allows comparison between different normal distributions. By converting an individual data point into this standardized form, we can compare different datasets or variables. The \( z \)-score indicates exactly how many standard deviations away from the mean a particular score is.
When you have a positive \( z \)-score, it means the value \( x \) is above the mean. Conversely, a negative \( z \)-score shows the value is below the mean.

• Example: A \( z \)-score of -4 suggests that the observed value is 4 standard deviations below the mean.
• Conversely, if the \( z \)-score were +4, it would mean the value is 4 standard deviations above the mean.

Therefore, calculating the \( z \)-score allows us to address probability questions effectively, by referring these scores to a standardized table or distribution.

Once the normal random variable is converted to a standard normal variable (\( z \)), calculating probabilities becomes significantly easier. The goal of probability calculation in this context is to find the likelihood of an event occurring under the normal distribution.
This is often executed with the help of the standard normal distribution table, commonly known as the Z-table, which provides the probability that a standard normal variable \( Z \) will be less than or equal to a certain value.
When you have the \( z \)-score:

• Locate it in the Z-table to find \( P(z \leq z) \).
• If \( z \) is not found in the table (extreme \( z \)-scores), the probability is often approximated. For example, for \( z \leq -4 \), the probability is effectively 0.

For probabilities like \( P(x \geq a) \), where \( x \) is a normally distributed variable, use the complement rule. If \( P(z < a) \) is found, then \( P(z \geq a) = 1 - P(z < a) \), which helps determine the desired probability efficiently.