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 left of


a. z = –2.45

Verified step by step guidance

1. Understand that the problem asks for the area under the standard normal curve to the left of a given z-score, which represents the cumulative probability up to that z-value.
2. Recall that the standard normal distribution is a normal distribution with mean 0 and standard deviation 1, and its total area under the curve is 1.
3. Draw a standard normal curve, mark the z-score at -2.45 on the horizontal axis, and shade the region to the left of this point to visualize the area you need to find.
4. Use the standard normal distribution table (z-table) or a calculator with normal distribution functions to find the cumulative probability corresponding to z = -2.45. This value gives the area to the left of z = -2.45.
5. Express the area as \(P(Z < -2.45)\), which is the cumulative probability from negative infinity up to -2.45 on the standard normal curve.

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 find probabilities and areas under the curve corresponding to z-scores.

Z-Score

A z-score represents the number of standard deviations a data point is from the mean. It standardizes values, allowing comparison across different normal distributions and enabling the use of standard normal tables.

Area Under the Curve

The area under the standard normal curve to the left of a given z-score represents the cumulative probability of observing a value less than that z-score. This area can be found using z-tables or technology.