In Exercises 21–26, find the indicated probability using the standard normal distribution. If convenient, use technology to find the probability.

P(0.42 < z < 3.15)

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Step 1: Understand the problem. The goal is to find the probability that the standard normal variable z lies between 0.42 and 3.15. This involves using the standard normal distribution, which has a mean of 0 and a standard deviation of 1.

Step 2: Recall that the cumulative distribution function (CDF) of the standard normal distribution, denoted as Φ(z), gives the probability that z is less than or equal to a specific value. For example, Φ(3.15) gives the probability that z ≤ 3.15.

Step 3: Use the property of probabilities for continuous distributions: P(a < z < b) = Φ(b) - Φ(a). In this case, P(0.42 < z < 3.15) = Φ(3.15) - Φ(0.42).

Step 4: Use technology (e.g., a statistical calculator, software like Excel, or a standard normal table) to find the values of Φ(3.15) and Φ(0.42). These values represent the cumulative probabilities up to z = 3.15 and z = 0.42, respectively.

Step 5: Subtract the cumulative probability at z = 0.42 from the cumulative probability at z = 3.15 to find the final probability: P(0.42 < z < 3.15) = Φ(3.15) - Φ(0.42).

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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 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 crucial for calculating probabilities and percentiles in statistics.

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 random variable falling within that range. Tools such as Z-tables or statistical software can be used to determine these probabilities efficiently.

In Exercises 21–26, find the indicated probability using the standard normal distribution. If convenient, use technology to find the probability.P(0.42 < z < 3.15)