MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN: 9781119256830
Author: Amos Gilat
Publisher: John Wiley & Sons Inc
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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**.
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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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