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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Below is a graph with a mean of -3 and a standard deviation of 3. The shaded region represents the probability of obtaining a value from this distribution that is between 0 and 1.5

### Understanding Normal Distribution and Probability

Below is a graph of a normal distribution with a mean (\( \mu \)) of -3 and a standard deviation (\( \sigma \)) of 3. The shaded region represents the probability of obtaining a value from this distribution that is between 0 and 1.5.

#### Graph 1: Normal Distribution 
- **Mean (\( \mu \))**: -3
- **Standard Deviation (\( \sigma \))**: 3
- **Shaded Region**: The area under the curve between the values 0 and 1.5 on the x-axis.

![Graph 1](description-of-graph)
- The x-axis ranges from -9 to 3.
- The y-axis indicates the probability density function values, ranging from 0 to 0.4.
- The normal distribution curve peaks at -3 and symmetrically tails off towards both ends.
- The shaded region is between the x-values 0 and 1.5, representing the probability of obtaining a value within this range from the given normal distribution.

#### Graph 2: Corresponding Region in Standard Normal Distribution

Shade the corresponding region under the standard normal density curve below.

#### Graph 2: Standard Normal Distribution 
- **Mean (\( \mu \))**: 0
- **Standard Deviation (\( \sigma \))**: 1
- To find the corresponding region, convert the x-values (0 to 1.5) to z-scores using the transformation formula: \( z = \frac{(X - \mu)}{\sigma} \).

**Transformation Process:**
1. For \( X = 0 \):
   \[
   z = \frac{(0 - (-3))}{3} = \frac{3}{3} = 1
   \]
2. For \( X = 1.5 \):
   \[
   z = \frac{(1.5 - (-3))}{3} = \frac{4.5}{3} = 1.5
   \]

Thus, the corresponding shaded region is the area under the standard normal distribution curve between z-scores of 1 and 1.5.

![Graph 2](description-of-graph)
- The z-axis ranges from -7 to 7.
- The y-axis represents the probability density function values, ranging from 0 to
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Transcribed Image Text:### Understanding Normal Distribution and Probability Below is a graph of a normal distribution with a mean (\( \mu \)) of -3 and a standard deviation (\( \sigma \)) of 3. The shaded region represents the probability of obtaining a value from this distribution that is between 0 and 1.5. #### Graph 1: Normal Distribution - **Mean (\( \mu \))**: -3 - **Standard Deviation (\( \sigma \))**: 3 - **Shaded Region**: The area under the curve between the values 0 and 1.5 on the x-axis. ![Graph 1](description-of-graph) - The x-axis ranges from -9 to 3. - The y-axis indicates the probability density function values, ranging from 0 to 0.4. - The normal distribution curve peaks at -3 and symmetrically tails off towards both ends. - The shaded region is between the x-values 0 and 1.5, representing the probability of obtaining a value within this range from the given normal distribution. #### Graph 2: Corresponding Region in Standard Normal Distribution Shade the corresponding region under the standard normal density curve below. #### Graph 2: Standard Normal Distribution - **Mean (\( \mu \))**: 0 - **Standard Deviation (\( \sigma \))**: 1 - To find the corresponding region, convert the x-values (0 to 1.5) to z-scores using the transformation formula: \( z = \frac{(X - \mu)}{\sigma} \). **Transformation Process:** 1. For \( X = 0 \): \[ z = \frac{(0 - (-3))}{3} = \frac{3}{3} = 1 \] 2. For \( X = 1.5 \): \[ z = \frac{(1.5 - (-3))}{3} = \frac{4.5}{3} = 1.5 \] Thus, the corresponding shaded region is the area under the standard normal distribution curve between z-scores of 1 and 1.5. ![Graph 2](description-of-graph) - The z-axis ranges from -7 to 7. - The y-axis represents the probability density function values, ranging from 0 to
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