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The z-value, often called a z-score, is a measure that describes a point's position within a standard normal distribution. The standard normal distribution, also known simply as the normal distribution, is a probability distribution that is symmetric about the mean. The mean is located at zero, and the distribution has a standard deviation of one. The z-value essentially tells us how many standard deviations away a point is from the mean.

When you encounter a problem that asks for a specific z-value, it’s asking where to draw a line on a standard normal curve so that a certain amount of data falls between certain points. If you imagine this distribution as a bell-shaped curve, a z-value of 0 sits at the peak, right in the center of the curve. Positive z-values fall to the right of the mean, and negative z-values fall to the left. In the exercise, we needed to find such z-values to ensure that 98% of data is sandwiched between \(-z\) and \(+z\).

To realize this, it's crucial to grasp that the normal curve is symmetric. This means that percentages related to these z-values are mirrored across the mean, which makes calculations predictable once you understand the principle.

Percentiles are rankings used in statistics to indicate how a particular score compares to the overall data set. In the context of a standard normal distribution, a percentile informs us about data positioning. If you're told that a score is at the 95th percentile, it means the score is higher than or equal to 95% of the data points in the distribution.

In the exercise, we needed to focus on two specific percentiles: the 1st percentile and the 99th percentile. Why these ones specifically? Because they help determine the values of \(-z\) and \(+z\) that contain the middle 98% of the data. The 1st percentile related to the lower cutoff (left tail), and the 99th percentile to the upper cutoff (right tail) of the curve. By identifying these values, you can define the range where most (in this case, 98%) data points fall.

This approach is essential in numerous statistical analyses where one needs to set boundaries according to desired coverage or to understand data distribution characteristics.

Cumulative probability is a concept that represents the probability of a variable falling within a certain range in a distribution. In simpler terms, it tells you the likelihood of drawing a score below a particular value in a given distribution. This becomes crucial when dealing with z-values.

When finding z-values for specific percentiles, you’re essentially tackling cumulative probabilities. The cumulative probability for a z-value is the area under the curve up to that point. For instance, if a z-value aligns with a cumulative probability of 0.99, it indicates that 99% of the data falls below that point.

In the original exercise, this understanding was instrumental. By identifying a cumulative probability of 0.99 (i.e., 99th percentile) using a z-table or calculator, we found that the corresponding z-value is approximately 2.33. Thus, \(z = \, \pm 2.33\) ensures that 98% of data lies symmetrically centered around zero, satisfying the conditions given.

Employing cumulative probabilities allows statisticians to make informed predictions and conclusions about where data points typically fall within a standard distribution.