6. Find the area of the shaded region. The graph depicts the standard normal distribution of bone density scores with a mean of 0 and a standard deviation of 1. ### Understanding Standard Normal Distribution CurveThe image depicts a standard normal distribution curve, also known as the bell curve, which is used in statistics to represent data distribution that is symmetrically distributed around the mean.#### Key Feature: Shaded Area- **Shaded Region**: The graph features a shaded area under the curve to represent the probability or proportion of outcomes within a specific range.- **Z-Score**: The left boundary of the shaded area is labeled with a Z-score of -0.88. The Z-score measures how many standard deviations an element is from the mean.#### Explanation of Components:- **Horizontal Axis (X-axis)**: Represents the possible values of the variable, or Z-scores, that have been standardized to fit the normal distribution.- **Vertical Axis (Y-axis)**: Represents the probability density, or the likelihood of each value occurring.- **Curve**: The bell-shaped curve is higher in the middle, indicating that values around the mean are the most common. It tapers off toward the ends, showing that extremely high or low values are rare.- **Shaded Area**: The specific area under the curve between the Z-score of -0.88 and the mean (0 in the case of a standard normal distribution). This area represents the cumulative probability from the left tail up to Z = -0.88.#### Importance of the Distribution:- **Symmetry**: The curve is symmetrical around the mean, indicating that values to the left and right of the mean are equally distributed.- **Empirical Rule**: States that approximately 68% of data lies within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations.- **Applications**: Used extensively in various fields such as social sciences, natural sciences, and finance to describe real-world phenomena, make predictions, and test hypotheses.Understanding a normal distribution curve is crucial for comprehending how data behaves in a variety of contexts, making it a foundational concept in statistical analysis.

Given Data: Z follows standard normal distribution Mean=0 SD=1

Answered: 6. Find the area of the shaded region.…

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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6. Find the area of the shaded region. The graph depicts the standard normal distribution of bone density scores with a mean of 0 and a standard deviation of 1.

### Understanding Standard Normal Distribution Curve

The image depicts a standard normal distribution curve, also known as the bell curve, which is used in statistics to represent data distribution that is symmetrically distributed around the mean.

#### Key Feature: Shaded Area
- **Shaded Region**: The graph features a shaded area under the curve to represent the probability or proportion of outcomes within a specific range.
- **Z-Score**: The left boundary of the shaded area is labeled with a Z-score of -0.88. The Z-score measures how many standard deviations an element is from the mean.

#### Explanation of Components:
- **Horizontal Axis (X-axis)**: Represents the possible values of the variable, or Z-scores, that have been standardized to fit the normal distribution.
- **Vertical Axis (Y-axis)**: Represents the probability density, or the likelihood of each value occurring.
- **Curve**: The bell-shaped curve is higher in the middle, indicating that values around the mean are the most common. It tapers off toward the ends, showing that extremely high or low values are rare.
- **Shaded Area**: The specific area under the curve between the Z-score of -0.88 and the mean (0 in the case of a standard normal distribution). This area represents the cumulative probability from the left tail up to Z = -0.88.

#### Importance of the Distribution:
- **Symmetry**: The curve is symmetrical around the mean, indicating that values to the left and right of the mean are equally distributed.
- **Empirical Rule**: States that approximately 68% of data lies within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations.
- **Applications**: Used extensively in various fields such as social sciences, natural sciences, and finance to describe real-world phenomena, make predictions, and test hypotheses.

Understanding a normal distribution curve is crucial for comprehending how data behaves in a variety of contexts, making it a foundational concept in statistical analysis.

Features Features Normal distribution is characterized by two parameters, mean (µ) and standard deviation (σ). When graphed, the mean represents the center of the bell curve and the graph is perfectly symmetric about the center. The mean, median, and mode are all equal for a normal distribution. The standard deviation measures the data's spread from the center. The higher the standard deviation, the more the data is spread out and the flatter the bell curve looks. Variance is another commonly used measure of the spread of the distribution and is equal to the square of the standard deviation.

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