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


b. to the left of z = –1.56 or to the right of z = 2.56

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 = -1.56 or to the right of z = 2.56. This means we are looking for the combined probability of these two tail regions.
2. Recall that the standard normal distribution is symmetric about zero, and the total area under the curve is 1. The area to the left of a z-value corresponds to the cumulative probability up to that z-value.
3. Find the cumulative area to the left of z = -1.56, which is \(P(Z < -1.56)\). This can be found using a standard normal table or a calculator with normal distribution functions.
4. Find the cumulative area to the left of z = 2.56, which is \(P(Z < 2.56)\). Since we want the area to the right of z = 2.56, calculate \(P(Z > 2.56) = 1 - P(Z < 2.56)\).
5. Add the two areas together to get the total area: \(P(Z < -1.56) + P(Z > 2.56)\). This sum represents the combined probability of being in either tail region.

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 Area Under the Curve

A z-score indicates a position on the standard normal curve. The area to the left of a z-score corresponds to the cumulative probability of values less than that z. Similarly, the area to the right is 1 minus the cumulative area to the left.

Calculating Combined Areas

When finding the total area for combined regions (e.g., left of one z and right of another), calculate each area separately and then add them. This approach uses the additive property of probabilities for mutually exclusive events.