Draw a graph for the standard normal distribution. Label the...

Draw a graph for the standard normal distribution. Label the horizontal axis at values of
$-3,-2,-1,0,1,2,$ and $3 .$ Then compute the following probabilities.
$$\begin{array}{l}(a) {P(z \leq 1.5)} \\ (b) {P(z \leq 1)} \\ (c) {P(1 \leq z \leq 1.5)} \\ (d) {P(0< z <2.5)}\end{array}$$

Key Concepts

Standard Normal Distribution

This is a probability distribution that is symmetric around a mean of 0, with a standard deviation of 1. It is commonly depicted as a smooth, bell-shaped curve and is fundamental in statistics for modeling the distribution of standardized data and for performing probability calculations.

Z-Score

A z-score represents how many standard deviations an element is from the mean. In the context of the standard normal distribution, z-scores are used as the horizontal axis values to relate raw scores to the probability outcomes association, facilitating comparisons across different datasets.

Cumulative Distribution Function (CDF)

The CDF for the standard normal distribution calculates the probability that a random variable will take a value less than or equal to a given z-score. This function is pivotal for determining probabilities over intervals and for finding areas under the curve.

Area Under the Curve

This concept represents the probability associated with a segment of the distribution curve. For continuous distributions like the standard normal, probabilities are determined by calculating the area under the curve over a specified interval, which in turn provides the likelihood of a variable falling within that range.

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