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Cumulative probability is a key concept when dealing with the standard normal distribution. It refers to the probability that a random variable, such as \(Z\), is less than or equal to a specific value. This type of probability helps us to understand how much of the total area under the probability curve is covered up to a certain point.
For example, when we use the \(Z\)-table to find \(P(Z \leq 1)\), we are calculating the cumulative probability up to \(Z = 1\). According to the table, this cumulative probability is approximately \(0.8413\) or \(84.13\%\), indicating that 84.13% of the area under the curve is to the left of \(Z = 1\).
Experienced statisticians often use cumulative probability to determine the likelihood of events occurring within certain boundaries, especially when analyzing data that follows a normal distribution.

The Z-distribution, also known as the standard normal distribution, plays a vital role in statistics, particularly in the context of hypothesis testing and standardization of data. A key feature of the Z-distribution is its symmetry around the mean, which is 0. Furthermore, it has a standard deviation of 1.
What makes the Z-distribution unique is its ability to convert raw scores into standardized scores (or z-scores). This transformation allows us to compare scores from different normal distributions with varied means and standard deviations by expressing them on a common scale.
Additionally, the bell-shaped curve of the Z-distribution represents the density of the variables, where approximately \(68\%\) of the data falls within one standard deviation from the mean, \(95\%\) within two, and \(99.7\%\) within three. Through this, the Z-distribution standardizes data, enabling effective comparisons and analyses.

The Z-table is an essential tool used to find cumulative probabilities associated with the standard normal distribution. It provides a quick reference to understand the cumulative probability for any z-score, allowing users to easily determine probabilities over intervals.
When you look up a value in the Z-table, you will typically find the cumulative probability that \(Z\) is less than or equal to your chosen value. For example, if you want to find \(P(Z \leq -1)\), the Z-table shows approximately \(0.1587\). Similarly, for \(Z = 1\), it shows \(P(Z \leq 1) \approx 0.8413\).
Utilizing the Z-table helps in answering questions regarding the likelihood of a random variable falling within a certain range, as seen in the textbook problem where the probability \(P(-1 \leq x \leq 1)\) was calculated by accessing the cumulative probabilities from the Z-table and subtracting them. This resource is crucial for efficient problem-solving in statistics.