[' # Define priors\n    alpha = pm.Normal(\'alpha\', mu=0, sigma=10)\n    beta = pm.Normal(\'beta\', mu=0, sigma=10)\n    sigma = pm.HalfNormal(\'sigma\', sigma=10)\n\n    # Define likelihood\n    likelihood = pm.Binomial(\'likelihood\', n=n, p=pm.math.invlogit(alpha + beta * r), observed=r)\n\n    # Sample posterior\n    trace = pm.sample(1000, tune=1000, chains=4, return_inferencedata=True, idata_kwargs={"log_likelihood": True})\n\n# Posterior distributions\naz.plot_posterior(trace, var_names=[\'alpha\', \'beta\', \'sigma\'], ref_val=0, hdi_prob=0.95)\n\n# Plot the posterior distributions\nplt.show()\n\n# 2. The data below is the number of hours a group of 10 students studied for an exam and their corresponding exam scores. Use PyMC to fit a Bayesian linear regression model to the data. You must define the model within the \'with pm.Model() as m:\' block below. Your output must define a complete Bayesian model with appropriate priors, likelihood, and then sample the posterior using, `pm.sample(1000, tune=1000, chains=4, return_inferencedata=True, idata_kwargs={"log_likelihood": True})`. Do not include any extra commentary or text outside the code. Follow best practices for expert-level Bayesian modeling.\n    \n# Description: The number of hours a group of 10 students studied for an exam and their corresponding exam scores.\n\n    \nimport pymc as pm\nimport numpy as np\nimport arviz as az\nimport matplotlib.pyplot as plt\n\n# Given Data\nhours = np.array([2, 3, 4, 5, 6, 7, 8, 9, 1']