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The Z-Score Table, often referred to as a z-table, is a crucial tool when working with the standard normal distribution. It's essentially a chart used to find the cumulative probability of a z-score in a standard normal distribution. A z-score represents the number of standard deviations a data point is from the mean, which in the case of a standard normal distribution is 0.

Since the distribution is symmetrical, the table usually provides the area to the left of a given z-value under the curve. Using the z-table:

• Locate the row matching the first decimal place of the z-score.
• Locate the column for the second decimal place of the z-score.
• The intersection gives the cumulative probability.

These probabilities can then be used to calculate the probability of z falling within a given interval, such as between -2.18 and 1.34, by finding the difference between their respective cumulative probabilities.

Cumulative probability is the probability that a random variable takes on a value less than or equal to a certain point. In the case of the standard normal distribution, it tells us the probability that the variable falls to the left of a given z-value.

To calculate cumulative probability using the z-table, you:

• Find the probability for each specific z-value.
• Subtract the cumulative probability of the lower z-value from the higher one to find the probability of falling between two z-scores.

This effectively enables us to interpret how much of the data falls within certain limits on the distribution curve. For example, in the exercise, we calculated the cumulative probability for z = -2.18 and z = 1.34, resulting in a probability of 0.8953 that a value falls between these two points.

The area under the curve in a probability distribution, such as the normal distribution, is a graphical representation of the probabilities. Each point under the curve corresponds to a probability of occurrence, with the entire area under the standard normal curve equaling 1, representing 100% probability.

In solving problems that involve finding the area under the curve, we essentially calculate the probabilities of z-scores falling within a given range. By understanding the area under the curve, we appreciate what portion of the distribution's outcomes are included in our z-score interval. Hence, in the exercise example, the area of 0.8953 under the curve between z = -2.18 and z = 1.34 shows a significant amount of the distribution falls in this range, offering a visual and quantitative understanding of these probabilities.