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The concept of a z-score is central to the standard normal distribution. A z-score tells us how far away a particular data point is from the mean of a dataset, measured in standard deviations. For the standard normal distribution, which has a mean of 0 and a standard deviation of 1, the z-score is calculated as follows: \[ z = \frac{X - \mu}{\sigma} \]Where:

• \(X\) is the data point,
• \(\mu\) is the mean (which is 0 for a standard normal distribution),
• \(\sigma\) is the standard deviation (which is 1 for a standard normal distribution).

This means a z-score of 0 indicates a data point right at the mean, while positive z-scores indicate data points above the mean, and negative z-scores indicate data points below the mean. This makes it easier to compare different data points against the same baseline.

Probability in the context of a standard normal distribution refers to the likelihood of a random variable falling within a particular range.
For continuous distributions like the normal distribution, probability is represented by the area under the curve between two z-scores.
For example, finding the probability that a z-score is between 0 and 3.18 involves calculating the area under the curve between these two points. In practical terms:

• A probability of 0 means it is impossible for the event to occur.
• A probability of 1 means the event will certainly occur.
• Probabilities between these values represent the likelihood of occurrence within a specified interval.

The area under the standard normal curve between z=0 and z=3.18 represents this probability, which we find using a z-table.

The cumulative area in a standard normal distribution is the total probability from the leftmost part of the distribution up to a specified z-score.
In other words, it is the area under the curve to the left of a given z-score.
For our task, we needed to find the cumulative area from z=0 to z=3.18. At z=0, the cumulative area is 0.5, which signifies that 50% of the data falls below the mean.
For z=3.18, the cumulative area is approximately 0.9993, meaning 99.93% of all data falls below this z-score. To find the area between z=0 and z=3.18, calculate the difference between these cumulative areas.
This gives a visual understanding of the probability of a random variable falling between these two z-scores.

The z-table is a powerful tool that helps us find the cumulative probabilities associated with different z-scores.
It lists z-scores alongside their corresponding cumulative probabilities (or areas) up to those scores in the standard normal distribution. Here's how to use it:

• Identify the z-score range you're interested in.
• Use the z-table to find the cumulative probabilities at each end of the range.
• Subtract these probabilities to find the area between them.

In our example, the area from z=3.18 is 0.9993 according to the z-table, and the area from z=0 is 0.5.
Thus, the difference, 0.9993 - 0.5 = 0.4993, represents the probability that the random variable is between z=0 and z=3.18.