In Problems 5–12, find the indicated areas. For each problem, be sure to draw a standard normal curve and shade the area that is to be found.
Determine the area under the standard normal curve that lies to the right of
b. z = –0.55

Verified step by step guidance

1. Understand that the problem asks for the area under the standard normal curve to the right of z = -0.55. This means we want the probability P(Z > -0.55) where Z follows a standard normal distribution with mean 0 and standard deviation 1.
2. Recall that the total area under the standard normal curve is 1, and the curve is symmetric about zero. The area to the right of a z-value corresponds to the probability that Z is greater than that z-value.
3. Use the standard normal distribution table (Z-table) or a calculator to find the cumulative area to the left of z = -0.55, which is P(Z < -0.55). This value gives the area under the curve to the left of -0.55.
4. Since the total area under the curve is 1, calculate the area to the right of z = -0.55 by subtracting the cumulative area to the left from 1: \$1 - P(Z < -0.55)$.
5. Draw a standard normal curve, mark the point z = -0.55 on the horizontal axis, and shade the region to the right of this point to visually represent the area you are calculating.

Key Concepts

Standard Normal Distribution

The standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. It is symmetric about the mean, and its shape is the classic bell curve. Values on this curve are represented by z-scores, which indicate how many standard deviations a point is from the mean.

Z-Score and Its Interpretation

A z-score measures the position of a data point relative to the mean of the distribution, expressed in standard deviations. For example, a z-score of –0.55 means the value is 0.55 standard deviations below the mean. Understanding z-scores helps locate areas under the curve corresponding to probabilities.

Area Under the Curve and Probability

The area under the standard normal curve represents probabilities. To find the area to the right of a given z-score, you calculate the probability that a value is greater than that z-score. This area can be found using standard normal tables or technology, and shading the region helps visualize the probability.