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A Z-score is a critical concept in statistics used to express how much a point, observation, or number deviates from the mean of a set of data in terms of standard deviations. The formula to calculate a Z-score is:
\[Z = \frac{(X - \mu)}{\sigma}\]where:

• \(X\) is the value or score you're analyzing.
• \(\mu\) is the mean of the data set.
• \(\sigma\) is the standard deviation.

The Z-score is used to determine the position of a score in a standard normal distribution. A Z-score tells us how many standard deviations away from the mean our value is. For example, a Z-score of -2.05 means that the value is 2.05 standard deviations below the mean.

Cumulative probability in the context of the normal distribution is the total probability that a value falls within a particular range from the mean. When you identify a Z-score, the cumulative probability is the probability that a random variable is less than or equal to that Z-score. For instance, if you have a Z-score of -1.52, the cumulative probability is 0.0643. This means there's a 6.43% chance a value will fall below this Z-score
Cumulative probabilities always range from 0 to 1, or 0% to 100%. They can be found using Z tables or statistical software, providing information about the distribution of data in the context of the standard normal distribution.

Calculating probability with Z-scores involves finding the difference in cumulative probabilities to determine the likelihood of data points falling within a specific range.
This process typically includes:

• Identifying the Z-scores for the boundaries of your range, like -2.05 and 0.00.
• Checking the Z-table for the cumulative probabilities of those Z-scores.
• Subtracting the cumulative probability of the lower boundary from the upper boundary to get the probability of the range.

As an example, if you are calculating the probability for \(-2.05 < z < 0.00\), you take the cumulative probability for 0.00 and subtract the cumulative probability for -2.05 to find the probability of a variable falling in that range. This method allows you to find the probability for any normal distribution range.

A Z Table is a mathematical table associated with Z-scores. It represents the cumulative probabilities of finding a standard normal random variable less than or equal to a given Z-score. When using a Z Table:

• First, identify your Z-score.
• Consult the Z Table to find the corresponding cumulative probability.

Most Z Tables display the area from the mean to a specific Z-score. Some tables show the cumulative probability from the far left of the standard normal distribution to that Z-score.
For values between two Z-scores, identify their cumulative probabilities in the Z Table, then subtract the lower probability from the higher one. Z Tables are integral in solving problems involving the standard normal distribution, guiding you to understand how data can distribute across a population.