Find the area of the shaded region. The graph depicts the standard normal distribution of bone density scores with mean 0 and standard deviation 1.The area of the shaded region is ### Understanding Normal Distribution and Z-ScoresIn this interactive module, we will explore the concept of normal distribution and how z-scores are used to represent specific points within that distribution.#### Normal Distribution CurveThe image illustrates a normal distribution curve, a continuous probability distribution that is symmetric around the mean, depicting that data near the mean are more frequent in occurrence than data far from the mean.**Key Features:**- The curve is bell-shaped and symmetric around the mean.- The total area under the curve is equal to 1.#### Z-ScoreThe z-score represents the number of standard deviations a data point is from the mean. It's a way to quantify the position of a point within the distribution.#### Detailed Explanation of the DiagramThe diagram shows:- The normal distribution curve shaded to the left of a z-score of -0.85.- This shaded area represents the probability that a value falls to the left of z = -0.85, i.e., the cumulative probability up to this z-score.**Interpreting the Diagram:**- The x-axis marks the z-score values.- The specific point, z = -0.85, is highlighted where the shading begins.- The area under the curve to the left of this z-score is shown in blue.Understanding the area under the curve helps in determining probabilities and making statistical inferences about the data.By engaging with this visual representation, students can gain a concrete understanding of how z-scores partition the normal distribution and how areas under the curve relate to probabilities.

Answered: Image /qna-images/answer/ec2e209f-dc81-4107-83c0-74d9c3866445.jpg

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

Big Ideas Math A Bridge To Success Algebra 1: Student Edition 2015

1st Edition

ISBN:9781680331141

Author:HOUGHTON MIFFLIN HARCOURT

Publisher:HOUGHTON MIFFLIN HARCOURT

Chapter11: Data Analysis And Displays

Section11.3: Shapes Of Distributions

Problem 8Q

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

The area of the shaded region is

### Understanding Normal Distribution and Z-Scores

In this interactive module, we will explore the concept of normal distribution and how z-scores are used to represent specific points within that distribution.

#### Normal Distribution Curve
The image illustrates a normal distribution curve, a continuous probability distribution that is symmetric around the mean, depicting that data near the mean are more frequent in occurrence than data far from the mean.

**Key Features:**
- The curve is bell-shaped and symmetric around the mean.
- The total area under the curve is equal to 1.

#### Z-Score
The z-score represents the number of standard deviations a data point is from the mean. It's a way to quantify the position of a point within the distribution.

#### Detailed Explanation of the Diagram
The diagram shows:
- The normal distribution curve shaded to the left of a z-score of -0.85.
- This shaded area represents the probability that a value falls to the left of z = -0.85, i.e., the cumulative probability up to this z-score.

**Interpreting the Diagram:**
- The x-axis marks the z-score values.
- The specific point, z = -0.85, is highlighted where the shading begins.
- The area under the curve to the left of this z-score is shown in blue.

Understanding the area under the curve helps in determining probabilities and making statistical inferences about the data.

By engaging with this visual representation, students can gain a concrete understanding of how z-scores partition the normal distribution and how areas under the curve relate to probabilities.

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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