Textbook Question
Using and Interpreting Concepts
Finding Area In Exercises 17–22, find the area of the shaded region under the standard normal curve. If convenient, use technology to find the area.
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6. Normal Distribution and Continuous Random Variables
Standard Normal Distribution
2:56 minutesProblem 5.1.29
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.
To the right of z= -0.355
Verified step by step guidance1
Step 1: Understand the problem. The goal is to find the area under the standard normal curve to the right of z = -0.355. The standard normal curve is symmetric, with a mean of 0 and a standard deviation of 1.
Step 2: Recall that the total area under the standard normal curve is 1. The area to the right of a given z-score can be found using the cumulative distribution function (CDF) of the standard normal distribution.
Step 3: Use the formula for the cumulative area to the left of z, which is given by P(Z ≤ z). For z = -0.355, find the cumulative area to the left of this z-score using a z-table or statistical software.
Step 4: Subtract the cumulative area to the left of z = -0.355 from 1 to find the area to the right. Mathematically, this is expressed as P(Z > -0.355) = 1 - P(Z ≤ -0.355).
Step 5: If using technology, input z = -0.355 into a statistical calculator or software to directly compute the area to the right. Alternatively, use a z-table to find P(Z ≤ -0.355), then subtract it from 1 as described in Step 4.
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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 used to describe how data is distributed in a standardized way, allowing for comparison across different datasets. The z-score represents the number of standard deviations a data point is from the mean, facilitating the calculation of probabilities and areas under the curve.
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Z-Score
A z-score indicates how many standard deviations an element is from the mean of a distribution. It is calculated by subtracting the mean from the value and then dividing by the standard deviation. In the context of the standard normal distribution, a z-score of -0.355 means the value is 0.355 standard deviations below the mean, which is essential for determining the area to the right of this z-score.
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Area Under the Curve
The area under the curve of a probability distribution represents the likelihood of a random variable falling within a particular range. For the standard normal distribution, this area can be found using z-tables or technology, such as statistical software or calculators. In this case, finding the area to the right of z = -0.355 involves calculating the probability that a value is greater than this z-score.
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