Find each probability using the standard normal distribution.


b. P(z < 2.23)

Verified step by step guidance

1. Step 1: Understand the problem. The goal is to find the probability that the standard normal variable z is less than 2.23, denoted as P(z < 2.23). The standard normal distribution has a mean of 0 and a standard deviation of 1.
2. Step 2: Use the standard normal distribution table (z-table) or a statistical software/tool to find the cumulative probability corresponding to z = 2.23. The z-table provides the area under the curve to the left of a given z-value.
3. Step 3: Locate the row in the z-table corresponding to the first two digits of the z-value (2.2 in this case) and the column corresponding to the second decimal place (0.03 for 2.23).
4. Step 4: Read the value from the z-table at the intersection of the row and column identified in Step 3. This value represents P(z < 2.23), the cumulative probability.
5. Step 5: If using statistical software or a calculator, input the z-value (2.23) into the appropriate function (e.g., normalcdf in a graphing calculator or a similar function in software) to directly obtain the cumulative probability.

Key Concepts

Standard Normal Distribution

The standard normal distribution is a special case of the normal distribution where the mean is 0 and the standard deviation is 1. It is represented by the variable 'z', which indicates how many standard deviations an element is from the mean. This distribution is symmetric and bell-shaped, making it essential for calculating probabilities related to normally distributed data.

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, a z-score indicates how far and in what direction a data point deviates from the mean, allowing for the calculation of probabilities.

Cumulative Distribution Function (CDF)

The cumulative distribution function (CDF) of a random variable gives the probability that the variable will take a value less than or equal to a specific value. For the standard normal distribution, the CDF can be used to find probabilities like P(z < 2.23) by looking up the z-score in standard normal distribution tables or using statistical software. This function is crucial for determining probabilities in various statistical analyses.