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The z-score is a vital concept in statistics, particularly in the realm of normal distribution. It represents the number of standard deviations a data point is from the mean of a distribution. In simpler terms, it helps you understand how a specific value fits within the overall dataset. For example:

• A z-score of 0 indicates the value is precisely at the mean.
• Positive z-scores signify values above the mean.
• Negative z-scores represent values below the mean.

Calculating the z-score involves the formula:\[ z = \frac{(X - \mu)}{\sigma} \]Where:

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

Understanding z-scores allows you to make more intelligent inferences about data, by comparing all observations on a common scale.

Cumulative probability is another fundamental concept in statistics. It refers to the probability that a random variable is less than or equal to a specific value. In other words, it's the "running total" of probabilities from left to right on the graph of the normal distribution. When dealing with a standard normal distribution, cumulative probability helps answer questions like, "What is the likelihood that a value is less than a given threshold?" For instance, if we say there is a 30% cumulative probability at some z-score, it means that 30% of the data falls to the left of that point on the curve. To find cumulative probabilities, we use tools such as probability density functions and integration. Using a standardized normal distribution (with a mean of 0 and a standard deviation of 1) simplifies these calculations.

A Z-table, also known as a standard normal table, is a tool listing the cumulative probabilities of the standard normal distribution from the mean to various z-scores. This table is an essential reference for statisticians as it allows them to find probabilities associated with normal distributions quickly. To use a Z-table:

• Identify the z-score you are interested in.
• Look up this z-score in the table to find the corresponding cumulative probability.

In practice, such as in the original exercise, the Z-table was used to find that the cumulative probability to the left of a z-score of 1.1 is approximately 0.8643. This knowledge helps in solving various problems by identifying other z-scores that correspond to different cumulative probabilities.

In the context of a normal distribution, the "area under the curve" defines probabilities. Since the total area under a normal distribution curve equals 1, any segment of this area reflects the probability of a given z-score range. The concept is central to understanding distributions. For example, an area corresponding to 59% between two z-scores means that there's a 59% chance that a randomly selected data point lies between those scores. Calculating these areas often involves integrating the probability density function, but for standardized distributions, it's much more practical to use the Z-table. Understanding how the area under the curve relates to probability also assists in visualizing statistical problems, making complex data interpretations more intuitive and manageable.