Find the shaded area underneath the Standard Normal curve. Give your answer to four decimal places. -3 -2 -1 0 1 2 3
Find the shaded area underneath the Standard Normal curve. Give your answer to four decimal places. -3 -2 -1 0 1 2 3
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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![### Understanding the Standard Normal Curve
#### Problem
Find the shaded area underneath the Standard Normal curve. Give your answer to four decimal places.
#### Explanation of the Diagram
The image displays a Standard Normal Distribution curve, also known as the bell curve, which is symmetrical about the mean of zero.
- The horizontal axis (x-axis) is marked with values ranging from -3 to 3.
- The area under the curve is divided into two regions:
- The region to the left of \( x = -1 \) is shaded in light blue.
- The region from \( x = -1 \) to \( x = 3 \) is shaded in red.
To find the required area, we have to calculate the area under the curve from \( x = -1 \) to \( x = \infty \) (which encompasses the right side of the curve starting at \( x = -1 \)).
#### Solution Approach
1. Identify the z-score: In this case, \( z = -1 \).
2. Use the standard normal distribution table, calculator, or a statistical software to find the cumulative probability up to \( z = -1 \).
The cumulative area to \( z = -1 \) is approximately 0.1587. Since we need the area to the right:
\[ \text{Area} = 1 - \text{Area to the left of } z \]
\[ \text{Area} = 1 - 0.1587 \]
\[ \text{Area} = 0.8413 \]
#### Final Answer
The area under the Standard Normal curve from \( x = -1 \) to \( x = 3 \) is approximately **0.8413**.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F23f14c8c-5141-4be2-aa93-07a6356450c6%2Fa5ca37c0-366b-487a-b370-64cc19b6824e%2F6w2ia0p_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Understanding the Standard Normal Curve
#### Problem
Find the shaded area underneath the Standard Normal curve. Give your answer to four decimal places.
#### Explanation of the Diagram
The image displays a Standard Normal Distribution curve, also known as the bell curve, which is symmetrical about the mean of zero.
- The horizontal axis (x-axis) is marked with values ranging from -3 to 3.
- The area under the curve is divided into two regions:
- The region to the left of \( x = -1 \) is shaded in light blue.
- The region from \( x = -1 \) to \( x = 3 \) is shaded in red.
To find the required area, we have to calculate the area under the curve from \( x = -1 \) to \( x = \infty \) (which encompasses the right side of the curve starting at \( x = -1 \)).
#### Solution Approach
1. Identify the z-score: In this case, \( z = -1 \).
2. Use the standard normal distribution table, calculator, or a statistical software to find the cumulative probability up to \( z = -1 \).
The cumulative area to \( z = -1 \) is approximately 0.1587. Since we need the area to the right:
\[ \text{Area} = 1 - \text{Area to the left of } z \]
\[ \text{Area} = 1 - 0.1587 \]
\[ \text{Area} = 0.8413 \]
#### Final Answer
The area under the Standard Normal curve from \( x = -1 \) to \( x = 3 \) is approximately **0.8413**.
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