In Exercises 6–11, find the indicated area under the standard normal curve. If convenient, use technology to find the area.

Between z = 0 and z = 2.95

검증된 단계별 안내

Step 1: Understand the problem. You are tasked with finding the area under the standard normal curve between z = 0 and z = 2.95. The standard normal curve is a bell-shaped curve with a mean of 0 and a standard deviation of 1.

Step 2: Recall that the area under the standard normal curve represents probabilities. To find the area between two z-scores, you can use the cumulative distribution function (CDF) of the standard normal distribution.

Step 3: Use the formula for the cumulative area under the curve: \( P(a < Z < b) = P(Z < b) - P(Z < a) \). Here, \( a = 0 \) and \( b = 2.95 \). This means you need to calculate \( P(Z < 2.95) \) and \( P(Z < 0) \).

Step 4: Use a z-table or technology (such as a statistical calculator or software) to find the cumulative probabilities. From the z-table or technology, find \( P(Z < 2.95) \), which gives the cumulative area to the left of z = 2.95, and \( P(Z < 0) \), which gives the cumulative area to the left of z = 0.

Step 5: Subtract \( P(Z < 0) \) from \( P(Z < 2.95) \) to find the area between z = 0 and z = 2.95. This difference represents the desired area under the standard normal curve.

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주요 개념

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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 z-score, 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-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 are essential for determining the area under the standard normal curve, as they allow us to find probabilities associated with specific values.

Area Under the Curve

The area under the curve in a standard normal distribution represents the probability of a random variable falling within a particular range. To find the area between two z-scores, one can use statistical tables or technology, such as calculators or software, which provide the cumulative probabilities associated with those z-scores. This area is critical for making inferences about data and understanding the likelihood of certain outcomes.

In Exercises 6–11, find the indicated area under the standard normal curve. If convenient, use technology to find the area.Between z = 0 and z = 2.95