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A z-score is a statistical measure that tells us how many standard deviations a data point is from the mean of a data set. It's a way to quantify the position of a score in a distribution. The formula for calculating the z-score of a given value is:\[ z = \frac{{X - \mu}}{{\sigma}} \]where:

• \(X\) is the data point
• \(\mu\) is the mean of the data set
• \(\sigma\) is the standard deviation of the data set

In the context of a standard normal distribution, which has a mean of 0 and a standard deviation of 1, each z-score corresponds to a point on the curve.
Since the standard normal distribution is symmetrical and bell-shaped, a z-score of 0 is precisely at the mean, and positive or negative scores indicate distances to the right or left of the mean, respectively.
The importance of the z-score lies in its ability to help us understand the location and significance of a specific value in relation to the overall distribution. In this example, finding a z-score where 97.5% of the curve lies to the left involves searching for a particular z-score that satisfies this condition.

Cumulative probability refers to the probability that a random variable is less than or equal to a certain value. In the context of the standard normal distribution, it is the probability that a random data point lies within the left-hand side of the curve up to a specific z-score.
Essentially, when we discuss cumulative probability, we're summing up probabilities from the beginning of the range up to the designated value. Mathematically, if you're looking for a cumulative probability of 0.975, it means that 97.5% of the data points are below that particular z-score.
This is crucial because cumulative probability allows us to understand how data points are distributed across intervals on a statistical curve. By analyzing cumulative probabilities, we get valuable insights into how much data lies below (or above) a certain threshold in the total data set.

The Z-table, also known as the standard normal table, is an essential statistical tool for dealing with the standard normal distribution. It provides the cumulative probabilities of z-scores—essentially, it tells you the percentage of data points falling to the left of a particular z-score on a standard normal curve.
To use the Z-table, locate the desired probability value on the table. Each cell in the Z-table represents a cumulative probability. Our task is to find the probability closest to 0.975 in the table, which indicates the proportion of the distribution that is to the left of the target z-score.
Once the closest value is found, trace back to the corresponding z-score value on the table. In our example, the z-score corresponding to 0.975 in the Z-table is 1.96. This means that 97.5% of data points on a standard normal distribution lie to the left of z = 1.96.
The Z-table is a vital reference for statisticians because it facilitates the interpretation and visualization of data in a standardized manner, making complex calculations accessible and understandable.