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import numpy as np
import matplotlib.pyplot as plt
import scipy.stats as stats

# Parameters for the normal distribution
mu = 65
sigma = 3.06

# Define the range for x values (covering a few standard deviations around the mean)
x = np.linspace(mu - 4*sigma, mu + 4*sigma, 400)
y = stats.norm.pdf(x, mu, sigma)

# Define the bounds for the shaded region
x_shade = np.linspace(58.88, 74.18, 200)
y_shade = stats.norm.pdf(x_shade, mu, sigma)

# Plot the normal distribution
plt.figure(figsize=(8, 5))
plt.plot(x, y, label='Normal Distribution\n$\mu=65$, $\sigma=3.06$')
plt.fill_between(x_shade, y_shade, color='skyblue', alpha=0.5,
                 label='Area between 58.88 and 74.18')

plt.xlabel('Life Span (years)')
plt.ylabel('Probability Density')
plt.title('Normal Distribution of Life Span')
plt.legend()
plt.grid(True)
plt.show()