91Ó°ÊÓ

A z-score is a statistical measurement that describes a value's relation to the mean of a group of values. It is measured in terms of standard deviations from the mean. For example, if a z-score is 2, it means the value is 2 standard deviations away from the mean. Z-scores can be positive or negative, indicating whether the data point is above or below the mean, respectively. In our exercise, the z-score is 1.22, indicating that the value is 1.22 standard deviations below the mean since it is negative.
Understanding z-scores is crucial because they allow us to compare scores from different distributions and determine probabilities in a standard normal distribution. They help us understand where a specific value lies within a distribution, giving us a sense of how unusual or typical that value is within the dataset.

Cumulative probability refers to the probability that a random variable takes on a value less than or equal to a particular value. In the context of the standard normal distribution, cumulative probability helps determine how much area lies to the left of a specific z-score.
For our particular problem, finding the cumulative probability for the z-score of 1.22 means identifying the area under the curve to the left of this z-score. This area represents the likelihood of a random variable falling below that z-score value. If this cumulative probability is given by the Z-table, one can find the area under the curve up to that specific z-score, useful for various statistical analyses.

A Z-table, also known as a standard normal table, provides the cumulative probabilities of a normal distribution for different z-scores. This table is immensely useful when determining the area under the standard normal curve.
The Z-table displays probabilities to the left of a specified z-score. In our problem, you would use the Z-table to find the cumulative probability for 1.22, helping identify the likelihood or area under the curve for this score. The key is to locate the z-score in the table and read off the corresponding cumulative probability. This value is critical for finding other probabilities, like the area to the right, by performing additional calculations, such as subtraction from 1.

The area under the curve, specifically under the normal curve, represents the probability of a random variable within a certain range. The total area under the standard normal distribution curve equals 1, representing 100% probability.
In statistics, when working with a standard normal distribution, finding areas under the curve helps us determine probabilities related to specific z-scores. For the exercise, identifying the area to the right of a z-score, such as 1.22, involves understanding that the area complements the area found using cumulative probability and the Z-table. By subtracting the cumulative probability from 1, we find the probability of values being greater than the specified z-score, thus determining the area to the right.