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A **z-score** is a numerical measurement that describes a value's relationship to the mean of a group of values, typically in terms of standard deviations. The formula for calculating a z-score is:

• \[ z = \frac{(X - \mu)}{\sigma} \]

where:

• \(X\) is the raw score,
• \(\mu\) is the mean,
• \(\sigma\) is the standard deviation.

A positive z-score indicates the score is above the mean, while a negative score indicates it's below the mean.
A z-score of zero would mean the score is exactly at the mean.When a problem involves standard normal distribution, z-scores help assess how unusual or usual a certain result could be. They are most commonly used to find probabilities under the bell curve, which represents a normal distribution. The information obtained from z-scores allows us to understand how a specific data point compares within a data set.

**Probability** is the measure of how likely an event is to occur. When we discuss probability in the context of normal distributions and z-scores, we are often looking at the probability that a score falls within a certain range of the distribution.To find specific probabilities, we often convert raw data to z-scores, as this allows us to use standard probability tables or calculators. In the case of a standard normal distribution, which has a mean of 0 and a standard deviation of 1, we use z-scores directly to find probabilities linked to the standard normal curve.For example:

• Probability \(P(|z| > 1.68)\) asks for the likelihood of a value falling more than 1.68 standard deviations away from the mean, either above or below.
• Conversely, \(P(|z| < 2.15)\) looks for the chance that a value is less than 2.15 standard deviations from the mean, essentially measuring how common or usual a certain range of results might be.

Probabilities range from 0 to 1, where 0 means it's an impossible event and 1 means it's a certain event.

A **normal distribution table**, also known as a Z-table, is a mathematical tool that is used to find the probability that a statistic is observed below, above, or between values on a standard normal distribution.This table provides key probabilities for the standard normal distribution, which has a mean of 0 and a standard deviation of 1. By using a z-score, one can look up the cumulative probability associated with that score. Understanding how to use this table is essential for finding values such as \(P(z < a)\) or \(P(z > a)\).For example:

• For \(P(z < 1.68)\), you would find the probability directly from the table, which is given as 0.9535.
• For any positive z-score, the table shows the probability to the left of the curve up to that z-value.

If you want the probability to the right \(P(z > a)\), you subtract the given probability from 1, as the total probability under the curve is 1.Normal distribution tables are crucial when you're performing statistical analysis and looking for relationships or phenomena within data sets. They're a great aid in understanding how far a deviation an observed data is from the mean.