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


d. z = 3.49

Verified step by step guidance

1. Recall that the standard normal curve represents the distribution of a standard normal variable \( Z \) with mean 0 and standard deviation 1.
2. Identify that the problem asks for the area to the left of \( z = 3.49 \), which corresponds to the cumulative probability \( P(Z < 3.49) \).
3. Draw a standard normal curve, mark the point \( z = 3.49 \) 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 \( P(Z < 3.49) \). This value represents the area under the curve to the left of \( z = 3.49 \).
5. Interpret the result as the proportion of values in a standard normal distribution that fall below \( z = 3.49 \).

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. Positive z-scores lie above the mean, while negative ones lie below. In this problem, z = 3.49 indicates a value far to the right on the standard normal curve.

Area Under the Curve (Probability)

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 standard normal tables or software, and it corresponds to the probability that a randomly selected value is below the specified z.