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


d. z = 3.45

Verified step by step guidance

1. Recall that the standard normal distribution is a normal distribution with mean 0 and standard deviation 1, and its curve is symmetric about the mean.
2. Draw the standard normal curve, labeling the horizontal axis with z-values, and mark the point z = 3.45 on the right side of the mean.
3. Understand that the area to the right of z = 3.45 represents the probability that a standard normal variable is greater than 3.45.
4. Use the standard normal table (Z-table) or a calculator to find the cumulative area to the left of z = 3.45, which is \(P(Z \leq 3.45)\).
5. Subtract this cumulative area from 1 to find the area to the right: \(P(Z > 3.45) = 1 - P(Z \leq 3.45)\).

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 and bell-shaped, used to model standardized data. Values on this curve are represented by z-scores, which indicate how many standard deviations a point is from the mean.

Z-Score

A z-score measures the position of a data point relative to the mean of a distribution, expressed in standard deviations. For the standard normal curve, z-scores correspond directly to points on the x-axis, helping to find probabilities or areas under the curve for given values.

Area Under the Curve and Probability

The area under the standard normal curve represents probabilities. Finding the area to the right of a specific z-score means calculating the probability that a value is greater than that z-score. This area can be found using standard normal tables or technology, and it corresponds to the shaded region on the curve.