Find the area under the standard normal curve within 2.7 standard deviations of the mean. Round the olution to four decimal places, if necessary. Area =
Find the area under the standard normal curve within 2.7 standard deviations of the mean. Round the olution to four decimal places, if necessary. Area =
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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![### Finding the Area Under the Standard Normal Curve
#### Problem Statement:
Find the area under the standard normal curve within 2.7 standard deviations of the mean. Round the solution to four decimal places, if necessary.
**Area = [_______]**
#### Explanation:
To find the area under the standard normal curve within 2.7 standard deviations, you will need to consult the Z-table or use statistical software to find the cumulative probability for \( Z = 2.7 \). This area represents the probability that a value falls between \(-2.7\) and \(2.7\) standard deviations from the mean in a normal distribution.
#### Steps:
1. **Locate the Z-value on the Z-table**:
- Go to the row corresponding to \( Z = 2.7 \) and find the value under it.
2. **Interpret the Z-table value**:
- Z-tables usually give the area to the left of a Z-value. For \( Z = 2.7 \), find the cumulative area to this point.
3. **Calculate the total area**:
- Find the area between \(-2.7\) and \(2.7\) by doubling the cumulative area for \( Z = 2.7 \) and subtracting it from 1.
4. **Round the solution**:
- Round the final area to four decimal places as requested.
Using a Z-table, the cumulative probability for \( Z = 2.7 \) is approximately 0.9965. Therefore, the total area within \( \pm 2.7 \) standard deviations is nearly \( 2 \times 0.9965 - 1 \).
### Detailed Graph/Diagram Explanation:
Although the image does not contain a graph or diagram, one can imagine a bell-shaped curve representing the standard normal distribution. The area under this curve between \( Z = -2.7 \) and \( Z = 2.7 \) covers the central part of the graph, symmetrical around the mean (0).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc94cb5d1-997a-49d0-b0ef-9ec02bf0a187%2Fe39bf322-6e5f-425e-ae2a-bb096f8140a4%2Fqlby5x8_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Finding the Area Under the Standard Normal Curve
#### Problem Statement:
Find the area under the standard normal curve within 2.7 standard deviations of the mean. Round the solution to four decimal places, if necessary.
**Area = [_______]**
#### Explanation:
To find the area under the standard normal curve within 2.7 standard deviations, you will need to consult the Z-table or use statistical software to find the cumulative probability for \( Z = 2.7 \). This area represents the probability that a value falls between \(-2.7\) and \(2.7\) standard deviations from the mean in a normal distribution.
#### Steps:
1. **Locate the Z-value on the Z-table**:
- Go to the row corresponding to \( Z = 2.7 \) and find the value under it.
2. **Interpret the Z-table value**:
- Z-tables usually give the area to the left of a Z-value. For \( Z = 2.7 \), find the cumulative area to this point.
3. **Calculate the total area**:
- Find the area between \(-2.7\) and \(2.7\) by doubling the cumulative area for \( Z = 2.7 \) and subtracting it from 1.
4. **Round the solution**:
- Round the final area to four decimal places as requested.
Using a Z-table, the cumulative probability for \( Z = 2.7 \) is approximately 0.9965. Therefore, the total area within \( \pm 2.7 \) standard deviations is nearly \( 2 \times 0.9965 - 1 \).
### Detailed Graph/Diagram Explanation:
Although the image does not contain a graph or diagram, one can imagine a bell-shaped curve representing the standard normal distribution. The area under this curve between \( Z = -2.7 \) and \( Z = 2.7 \) covers the central part of the graph, symmetrical around the mean (0).
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