Question

fill in the blank.
finding probabilities associated with distributions that are standard normal distributions is equivalent to
finding probabilities associated with distributions that are standard normal distributions is equivalent to
finding the z - scores associated with the probabilities.
finding the area of the shaded region representing that probability.
finding the mean of the distribution.
finding the variance of the distribution.

Explanation:

In a standard normal distribution, probabilities are represented by areas under the curve. The total area under the standard normal curve is \(1\), and the probability that a random variable \(Z\) (following the standard normal distribution \(Z\sim N(0,1)\)) lies within a certain interval \([a,b]\) is given by \(P(a\leq Z\leq b)=\int_{a}^{b}\frac{1}{\sqrt{2\pi}}e^{-\frac{z^{2}}{2}}dz\), which is equivalent to finding the area of the region between \(z = a\) and \(z = b\) under the standard - normal curve.

• The \(z\) - score is a measure of position, not a probability measure in itself. While \(z\) - scores are used to standardize values for the purpose of finding probabilities (by relating them to the standard normal distribution), the act of finding probabilities is not about finding \(z\) - scores.
• The mean of a standard normal distribution is \(0\), and the variance is \(1\). These are fixed values for the standard normal distribution \(N(0,1)\), and they do not change when calculating different probabilities. Probabilities depend on intervals within the distribution, not on recalculating the mean or variance (which are already known for the standard normal case).

Answer:

finding the area of the shaded region representing that probability.