91Ó°ÊÓ

A Z-score, also known as a standard score, measures how many standard deviations an element is from the mean. It is calculated using the formula:
\[ Z = \frac{X - \mu}{\sigma} \]
where \(X\) is the value, \(\mu\) is the mean of the population, and \(\sigma\) is the standard deviation.
The Z-score helps standardize different datasets for comparison. For a standard normal distribution, the mean is 0, and the standard deviation is 1. This means that a Z-score directly indicates the number of standard deviations a value is from the mean 0.
Higher Z-scores (positive) indicate values above the mean, while lower Z-scores (negative) indicate values below the mean.

The complement rule is an important concept in probability. It states that the probability of an event occurring is 1 minus the probability of the event not occurring.
For the standard normal distribution, the total area under the curve is 1. Therefore, the area to the right of a Z-score is the complement of the area to the left. If the area to the left of a Z-score is known, the area to the right can be found by subtracting the left area from 1.
This is given by: \[ P(Z > z) = 1 - P(Z \leq z) \]
In our problem, the area to the left of \(Z = 1.20\) is 0.8849. Using the complement rule, we find the area to the right: \[P(Z > 1.20) = 1 - P(Z \leq 1.20) = 1 - 0.8849 = 0.1151\]
Thus, the area to the right of Z = 1.20 is 0.1151.

Probability is a measure of the likelihood that an event will occur. In the context of the standard normal distribution, probabilities can be visualized as areas under the curve.
The total probability is always 1, representing the entire distribution. Probabilities for specific Z-scores represent areas under the curve to the left or right of the Z-score.
In our problem, we look for the area under the curve to the right of Z = 1.20. This area represents the probability of obtaining a value greater than 1.20 in a standard normal distribution.
Once we know the area to the left of Z = 1.20, we use the complement rule to find the area to the right. With the area to the left given as 0.8849, the probability or area to the right is calculated as: \[1 - 0.8849 = 0.1151 \]
This indicates a probability of approximately 11.51% for values greater than Z = 1.20.