91Ó°ÊÓ

In statistics, the Z-value, or Z-score, is a measure of how many standard deviations an element is from the mean of a distribution. In a standard normal distribution, the mean is 0 and the standard deviation is 1. Therefore, the Z-score directly corresponds to a position on the standard normal curve. The formula to calculate a Z-score is:
\[Z = \frac{(X - \mu)}{\sigma}\]where:

• \(X\) is the value being analyzed.
• \(\mu\) is the mean of the distribution.
• \(\sigma\) is the standard deviation of the distribution.

In the context of our exercise, finding the Z-value is crucial for determining the position on a normal distribution curve where exactly 5% of the data falls to the right. The Z-score provides the specific number needed for this alignment. When a Z-value is referenced as having a certain percentage "to the right," it ties directly to the concept of right tail probability.

Cumulative Probability refers to the probability that a random variable will assume a value less than or equal to a specific value. In the standard normal distribution, cumulative probability helps us understand what proportion of the data falls below a certain Z-score.
For example, if a Z-score has a cumulative probability of 0.95, it suggests that 95% of the data lies below or to the left of this Z-score in the standard normal curve.
Cumulative probability is useful when translating between the probability of an event and its corresponding Z-value. In the exercise provided, identifying a Z-score where 95% of the data falls to the left helps us locate the equivalent position on the curve where only 5% remains to the right.

The Normal Distribution Table, also known as the Z-table, lists cumulative probabilities of the standard normal distribution. This tool helps us find the probability that a Z-value is below a particular point. The table typically provides values for cumulative probabilities up to any Z-score.
To use it, you look up the row representing the whole Z-value and the column for the hundredths place. For instance, to find the cumulative probability for a Z-score of 1.64, you would cross-reference 1.6 in the row with 0.04 in the column.
In our exercise, this table is key to identifying the Z-value that corresponds to a cumulative probability of 0.95, giving us the solution that the associated Z-value is approximately 1.645. This understanding can also be supported by using statistical software or a calculator with distribution functions as an alternative method.

Right tail probability is the probability that a standard normal random variable is greater than a specific Z-value. It essentially represents the area under the curve from that Z-value extending infinitely rightward. This concept allows us to discern how much of the data set falls beyond a certain threshold in a distribution.
Illustratively, if we have a Z-value where the right tail probability is 0.05, this tells us that 5% of the observations are expected to be greater than that Z-score. The main point in our exercise is to find the Z-score where this tail contains precisely 5% of the total area under the standard normal curve.
Visualizing this involves sketching a standard normal distribution curve, marking the Z-value (1.645), and shading the area under the curve to the right of this point, representing the 5% right tail probability. This visual interpretation reinforces the mathematical conclusion drawn from the Z-table.