6. Normal Distribution and Continuous Random Variables
Standard Normal Distribution
2:40 minutesProblem 5.1.31
Textbook Question
Finding Area
In Exercises 23–36, find the indicated area under the standard normal curve. If convenient, use technology to find the area.
Between z=0 and z=2.86
Verified step by step guidance1
Step 1: Understand the problem. You are tasked with finding the area under the standard normal curve between z = 0 and z = 2.86. 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: \( P(a \leq Z \leq b) = \Phi(b) - \Phi(a) \), where \( \Phi(z) \) is the cumulative probability up to z. In this case, \( a = 0 \) and \( b = 2.86 \).
Step 4: Look up the cumulative probabilities for \( \Phi(2.86) \) and \( \Phi(0) \) using a z-table or technology (e.g., a calculator or statistical software). Note that \( \Phi(0) \) is always 0.5 because the standard normal curve is symmetric around the mean.
Step 5: Subtract \( \Phi(0) \) from \( \Phi(2.86) \) to find the area between z = 0 and z = 2.86. The result represents the probability or area under the curve for the specified range.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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 comparison of different data sets.
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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 finding areas under the standard normal curve, as they help determine the probability of a value falling within a certain range.
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Area Under the Curve
The area under the curve in a standard normal distribution represents the probability of a random variable falling within a specified 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 crucial for making inferences about data and understanding the likelihood of outcomes.
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