6. Normal Distribution and Continuous Random Variables
Standard Normal Distribution
3:06 minutesProblem 5.1.33
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= -1.55 and z= 1.55
Verified step by step guidance1
Step 1: Understand the problem. You are tasked with finding the area under the standard normal curve between z = -1.55 and z = 1.55. The standard normal curve is symmetric, with a mean of 0 and a standard deviation of 1.
Step 2: Recall that the area under the standard normal curve between two z-scores represents the probability of a value falling within that range. This can be calculated using the cumulative distribution function (CDF) of the standard normal distribution.
Step 3: Use the formula for finding the area between two z-scores: Area = P(z ≤ 1.55) - P(z ≤ -1.55). Here, P(z ≤ 1.55) is the cumulative probability up to z = 1.55, and P(z ≤ -1.55) is the cumulative probability up to z = -1.55.
Step 4: Use a z-table or technology (such as a graphing calculator, statistical software, or an online tool) to find the cumulative probabilities for z = 1.55 and z = -1.55. Note that P(z ≤ -1.55) can also be found using symmetry: P(z ≤ -1.55) = 1 - P(z ≤ 1.55).
Step 5: Subtract the cumulative probability for z = -1.55 from the cumulative probability for z = 1.55 to find the area under the curve between these two z-scores. This result represents the probability of a value falling between z = -1.55 and z = 1.55.
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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 standardization of different normal distributions.
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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 probability distribution represents the likelihood of a random variable falling within a specified range. For the standard normal distribution, this area can be found using z-scores and standard normal distribution tables or technology. In this case, finding the area between z = -1.55 and z = 1.55 involves calculating the cumulative probabilities at these z-scores and subtracting them.
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