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.

Definition Definition Measure of central tendency that is the average of a given data set. The mean value is evaluated as the quotient of the sum of all observations by the sample size. The mean, in contrast to a median, is affected by extreme values. Very large or very small values can distract the mean from the center of the data. Arithmetic mean: The most common type of mean is the arithmetic mean. It is evaluated using the formula: μ = 1 N ∑ i = 1 N x i Other types of means are the geometric mean, logarithmic mean, and harmonic mean. Geometric mean: The nth root of the product of n observations from a data set is defined as the geometric mean of the set: G = x 1 x 2 ... x n n Logarithmic mean: The difference of the natural logarithms of the two numbers, divided by the difference between the numbers is the logarithmic mean of the two numbers. The logarithmic mean is used particularly in heat transfer and mass transfer. ln x 2 − ln x 1 x 2 − x 1 Harmonic mean: The inverse of the arithmetic mean of the inverses of all the numbers in a data set is the harmonic mean of the data. 1 1 x 1 + 1 x 2 + ...

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