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 total area under the standard normal curve


a. to the left of z = –2 or to the right of z = 2

Verified step by step guidance

1. Understand that the problem asks for the total area under the standard normal curve to the left of z = -2 or to the right of z = 2. This means we are looking for the combined probability in the two tails beyond these z-values.
2. Recall that the standard normal distribution is symmetric about zero, so the area to the left of z = -2 is the same as the area to the right of z = 2.
3. Use the standard normal distribution table (z-table) or a calculator to find the area to the left of z = -2. This area represents the probability P(Z < -2).
4. Since the distribution is symmetric, the area to the right of z = 2 is equal to P(Z > 2), which is the same as P(Z < -2). Therefore, the total area is twice the area found in the previous step.
5. Express the total area as: Total area = P(Z < -2) + P(Z > 2) = 2 \times P(Z < -2). This gives the combined probability of being in either tail beyond the specified z-values.

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 total area under the curve equals 1. Z-scores represent the number of standard deviations a value is from the mean.

Z-Score and Its Interpretation

A z-score indicates how many standard deviations a data point is from the mean. Negative z-scores lie to the left of the mean, and positive z-scores lie to the right. Understanding z-scores helps locate areas under the standard normal curve corresponding to specific values.

Area Under the Curve and Probability

The area under the standard normal curve between two points represents the probability of a value falling within that range. To find areas to the left or right of a z-score, one uses standard normal tables or technology. For combined areas (e.g., left of -2 or right of 2), sum the individual tail areas.