Finding Area
In Exercises 23–36, find the indicated area under the standard normal curve. If convenient, use technology to find the area.


To the left of z=1.365

Verified step by step guidance

1. Step 1: Understand the problem. The goal is to find the area under the standard normal curve to the left of z = 1.365. This area represents the cumulative probability for a standard normal distribution up to the z-score of 1.365.
2. Step 2: Recall that the standard normal distribution is symmetric about the mean (z = 0) and has a total area of 1 under the curve. The cumulative area to the left of a given z-score can be found using a z-table, statistical software, or a calculator with normal distribution functions.
3. Step 3: Use the z-table or technology. Locate the z-score of 1.365 in the z-table. The table provides the cumulative probability (area) to the left of the given z-score. If using technology, input the z-score into the cumulative distribution function (CDF) for the standard normal distribution.
4. Step 4: Interpret the result. The value obtained from the z-table or technology represents the proportion of the data that falls to the left of z = 1.365 in a standard normal distribution.
5. Step 5: If using technology, verify the input. For example, in a calculator or software, use the function for the cumulative probability of a standard normal distribution, such as P(Z ≤ 1.365), to ensure accuracy.

Key Concepts

Standard Normal Distribution

The standard normal distribution is a special normal distribution with a mean of 0 and a standard deviation of 1. It is represented by the variable 'z', which indicates how many standard deviations an element is from the mean. This distribution is crucial for calculating probabilities and areas under the curve, as it allows for the standardization of different normal distributions.

Z-Score

A z-score is a statistical measurement that describes a value's relationship to the mean of a group of values. It is calculated by subtracting the mean from the value and then dividing by the standard deviation. In the context of the standard normal distribution, the z-score helps determine the area under the curve to the left of a specific value, which corresponds to the probability of a random variable being less than that value.

Area Under the Curve

The area under the curve (AUC) in a probability distribution represents the likelihood of a random variable falling within a certain range. For the standard normal distribution, this area can be found using z-scores and standard normal tables or technology. The area to the left of a given z-score indicates the cumulative probability up to that point, which is essential for statistical inference and hypothesis testing.