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A z-table is a tool that helps us find the area under the standard normal distribution curve that corresponds to a particular z-value. This table is organized by possible z-values and their respective probabilities. To use it, locate a specific z-value, and then read the corresponding cumulative probability. Indeed, the table shows the proportion of the data to the left of a given z-value.
In this exercise, we're interested in finding the z-value where 5% of the standard normal curve lies to the right. This means we're interested in the cumulative probability of 0.95. Thus, the z-table acts as a bridge between z-values and cumulative probabilities. By using the table, we found that a z-value of about 1.645 gives us a cumulative probability of 0.95, aligning with the exercise's needs.

Cumulative probability refers to the probability that a random variable will have a value less than or equal to a particular value. When using a z-table, cumulative probability plays a crucial role as it represents the total probability below a z-value on the standard normal distribution curve.

• In any standard normal curve, cumulative probability increases as one moves from left to right.
• It ranges from 0 to 1, with 0 being completely to the left and 1 being completely to the right of the curve.

When solving the problem, you use cumulative probability to find that 95% of the data is below the z-value 1.645. This reveals that the remaining 5% of the data is to the right of this point, which is our area of interest.

A z-value, or z-score, is a numerical measurement that describes a value's relationship to the mean of a group of values. Measured in terms of standard deviations, it tells us how far and in what direction, a given score deviates from the mean. Here’s how to interpret a z-value:

• A z-value of 0 corresponds to the mean of the distribution.
• Positive z-values indicate values above the mean, and negative z-values indicate values below the mean.

In our exercise, a z-value of 1.645 means that this value is 1.645 standard deviations above the mean. When this z-value is found on the standard normal distribution curve, it marks a point where 95% of the data lies to its left, leaving 5% to its right. Understanding the z-value is essential for interpreting position within the distribution.