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answer all and give me graph usin python code

In this problem, we are given a normal distribution with mean \(\mu = -4\) and standard deviation \(\sigma = 2\). The shaded region on the first graph represents the probability that a value from this distribution is less than \(-8\).

To answer the question:

### Step 1: Convert to Standard Normal Distribution
We need to standardize the given value \(X = -8\) using the z-score formula:

\[
z = \frac{X - \mu}{\sigma}
\]

Substitute the values:

\[
z = \frac{-8 - (-4)}{2} = \frac{-4}{2} = -2
\]

### Step 2: Shade the Corresponding Region
The second graph is a standard normal distribution with mean \(0\) and standard deviation \(1\). We need to shade the region where \(z \leq -2\).

I'll now generate the Python code to plot the graph and shade the corresponding region under the standard normal curve.

#### Python Code for Plotting

```python
import numpy as np
import matplotlib.pyplot as plt
from scipy.stats import norm

# Define the standard normal distribution
mean, std = 0, 1

# Generate data for the normal curve
x = np.linspace(-4, 4, 1000)
y = norm.pdf(x, mean, std)

# Plot the standard normal curve
plt.plot(x, y, label='Standard Normal Distribution')

# Shade the area corresponding to z ≤ -2
x_shade = np.linspace(-4, -2, 500)
y_shade = norm.pdf(x_shade, mean, std)
plt.fill_between(x_shade, y_shade, color='blue', alpha=0.5, label='P(Z ≤ -2)')

# Add labels and title
plt.title('Standard Normal Distribution')
plt.xlabel('Z')
plt.ylabel('Probability Density')

# Display the legend
plt.legend()

# Show the plot
plt.show()
```

### Explanation:
- The Python code plots the standard normal curve.
- It shades the region to the left of \(z = -2\), corresponding to the same area shaded in the original graph.
  
Would you like further details or clarifications?

### Related Questions:
1. How do you find the probability of \(Z \leq -2\) using a z-table or Python?
2. What does the z-score represent in the context of the normal distribution?
3. How do you calculate probabilities for non-standard normal distributions?
4. How would you solve for the shaded area if the mean or standard deviation were different?
5. Can you explain how to interpret the cumulative distribution function (CDF)?

#### Tip:
Always standardize values from a normal distribution using the z-score formula to convert them into standard normal distribution for easier analysis!

Shade the region under a normal distribution with μ = -4 and σ = 2 for X < -8, and shade the corresponding region on a standard normal distribution.

Learn how to calculate and shade regions in a normal distribution with μ = -4 and σ = 2, and find the corresponding area in a standard normal distribution. This step-by-step guide explains the z-score formula and its application.

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