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

P(- 0.89 < z < 0)

검증된 단계별 안내

Step 1: Understand the problem. The goal is to find the probability that the standard normal variable z falls between -0.89 and 0. This involves using the standard normal distribution, which has a mean of 0 and a standard deviation of 1.

Step 2: Recall that the standard normal distribution table (or z-table) provides the cumulative probability from the far left of the distribution up to a given z-value. Alternatively, technology such as a graphing calculator or statistical software can be used to find probabilities.

Step 3: Break the problem into two parts. First, find the cumulative probability for z = 0. From the properties of the standard normal distribution, the cumulative probability for z = 0 is 0.5 (since it is the mean of the distribution).

Step 4: Next, find the cumulative probability for z = -0.89. Use a z-table or technology to determine this value. The z-table will give the cumulative probability from the far left of the distribution up to z = -0.89.

Step 5: Subtract the cumulative probability for z = -0.89 from the cumulative probability for z = 0. This difference represents the probability that z falls between -0.89 and 0. Mathematically, this is expressed as P(-0.89 < z < 0) = P(z < 0) - P(z < -0.89).

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주요 개념

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Standard Normal Distribution

The standard normal distribution is a special case of the 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 is often used in statistical analysis to standardize scores.

Z-Scores

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. Z-scores allow for the comparison of scores from different distributions and are essential for finding probabilities in the standard normal distribution.

Probability Calculation

Probability calculation in the context of the standard normal distribution involves finding the area under the curve between two z-scores. This area represents the likelihood of a value falling within that range. Tools such as z-tables or statistical software can be used to determine these probabilities, making it easier to analyze data and draw conclusions.

Finding Probability In Exercises 47–56, find the indicated probability using the standard normal distribution. If convenient, use technology to find the probability.P(- 0.89 < z < 0)