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The Z-score is a statistical measurement that describes a value's relationship to the mean of a group of values. It is measured in terms of standard deviations from the mean. If you have a normal random variable like \( X \), with a mean (\( \mu \)) and standard deviation (\( \sigma \)), you calculate the Z-score for a specific value using the formula:\
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\[ Z = \frac{X - \mu}{\sigma} \]\
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\In practical terms, the Z-score lets you translate any normal random variable to the standard normal distribution, which has a mean of 0 and a standard deviation of 1. Knowing the Z-score allows you to determine how many standard deviations an element \( X \) is from its mean. This is crucial in finding probabilities for more complex scenarios not directly available in standard tables.

A standard normal distribution table, also known as the Z-table, is a reference for finding probabilities associated with the standard normal distribution. It shows the cumulative probability up to a certain Z-score.

Since the table gives the area to the LEFT under the standard normal curve, if you're looking for a probability to the RIGHT (as is common with finding \( P[X>c] \)), you'll need to subtract the table value from 1. The Z-table is essential for many statistical analyses because it allows us to state the probability of a random variable falling within a certain interval, or more simply, it tells us what percentage of data falls below a certain Z-score.

Normal random variables have specific properties that define their distribution:

• Their distribution is symmetric around the mean.
• The mean, median, and mode of the distribution are all equal.
• The distribution is determined by two parameters: the mean (\( \mu \)) and variance (\( \sigma^2 \)).
• Approximately 68% of the data falls within one standard deviation of the mean, 95% falls within two standard deviations, and 99.7% falls within three standard deviations (this is known as the empirical rule).

Understanding these properties is fundamental when working with probabilities, as they provide a standardized way to interpret the distribution and its associated probabilities.

The probability \( P[X>c] \) represents the likelihood that a normal random variable \( X \) exceeds a certain value \( c \). It is a right-tail probability because it looks at the area under the normal curve to the right of \( c \).

To calculate this, you first find the corresponding Z-score of \( c \) and then use the standard normal distribution table to find the cumulative probability up to that Z-score. Since the table gives you the area to the LEFT, you'll subtract this from 1 to get the probability to the RIGHT. This probability is key in many statistical analyses, such as hypothesis testing and setting confidence intervals.