Finding Probability In Exercises 47–56, find the indicated probability using the standard normal distribution. If convenient, use technology to find the probability.


P(z < - 1.11)

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 -1.11, denoted as P(z < -1.11). The standard normal distribution has a mean of 0 and a standard deviation of 1.
2. Step 2: Recall that the standard normal distribution is symmetric, and probabilities can be found using a Z-table or technology (e.g., statistical software or a calculator). The Z-table provides cumulative probabilities for Z-values.
3. Step 3: Locate the Z-value of -1.11 in the Z-table. The table will give the cumulative probability for Z-values less than -1.11. If using technology, input the Z-value into the software or calculator to find the cumulative probability.
4. Step 4: Interpret the cumulative probability. The value obtained represents the area under the standard normal curve to the left of Z = -1.11.
5. Step 5: If using technology, verify the result by ensuring the input matches the problem (P(z < -1.11)). If using the Z-table, double-check the row and column corresponding to -1.11 to ensure accuracy.

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 allows for the calculation of probabilities and percentiles for any normal distribution by standardizing values. This transformation is crucial for comparing different datasets and finding probabilities associated with specific Z-scores.

Z-Score

A Z-score indicates how many standard deviations an element is from the mean of a distribution. 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 of -1.11 means the value is 1.11 standard deviations below the mean, which is essential for determining the corresponding probability.

Cumulative Probability

Cumulative probability refers to the probability that a random variable takes on a value less than or equal to a specific value. In the context of the standard normal distribution, P(z < -1.11) represents the cumulative probability of Z being less than -1.11. This probability can be found using Z-tables or statistical software, providing insights into the likelihood of observing values in a given range.