Consider the following model,

- Prior: \(\theta \sim \text{Beta}(\alpha = 5, \beta = 5)\)
- Likelihood: \(X_1, X_1, \ldots, X_n \sim \text{Bern}(\theta)\)
- Posterior: \(\theta \mid X_1, X_1, \ldots, X_n \sim \ ?\)

and observed data with statistics,

- Sample size: \(n = 20\)
- Number of “successes” \(\sum x_i = 15\)

Use the given prior and the observed data to calculate a Bayes’ estimate of \(\theta\). (Use the posterior mean.)

Consider the following model,

- Prior: \(\theta \sim \text{Beta}(\alpha = 50, \beta = 20)\)
- Likelihood: \(X_1, X_1, \ldots, X_n \sim \text{Bern}(\theta)\)
- Posterior: \(\theta \mid X_1, X_1, \ldots, X_n \sim \ ?\)

and observed data with statistics,

- Sample size: \(n = 40\)
- Number of “successes” \(\sum x_i = 32\)

Use the given prior and the observed data to calculate a Bayes’ estimate of \(\theta\). (Use the posterior mean.)

Consider the following model,

- Prior: \(\theta \sim \text{Beta}(\alpha = 10, \beta = 60)\)
- Likelihood: \(X_1, X_1, \ldots, X_n \sim \text{Bern}(\theta)\)
- Posterior: \(\theta \mid X_1, X_1, \ldots, X_n \sim \ ?\)

and observed data with statistics,

- Sample size: \(n = 5\)
- Number of “successes” \(\sum x_i = 2\)

Use the given prior and the observed data to calculate a Bayes’ estimate of \(\theta\). (Use the posterior mean.)

Consider the following model,

- Prior: \(\theta \sim \text{Beta}(\alpha = 10, \beta = 60)\)
- Likelihood: \(X_1, X_1, \ldots, X_n \sim \text{Bern}(\theta)\)
- Posterior: \(\theta \mid X_1, X_1, \ldots, X_n \sim \ ?\)

and observed data with statistics,

- Sample size: \(n = 5\)
- Number of “successes” \(\sum x_i = 2\)

Use the given prior and the observed data to calculate a 99% credible interval

Consider the following model,

- Prior: \(\theta \sim \text{Beta}(\alpha = 4, \beta = 4)\)
- Likelihood: \(X_1, X_1, \ldots, X_n \sim \text{Bern}(\theta)\)
- Posterior: \(\theta \mid X_1, X_1, \ldots, X_n \sim \ ?\)

and observed data with statistics,

- Sample size: \(n = 5\)
- Number of “successes” \(\sum x_i = 2\)

Use the given prior and the observed data to calculate a 90% credible interval

Consider the following model,

- Prior: \(\theta \sim \text{Beta}(\alpha = 10, \beta = 4)\)
- Likelihood: \(X_1, X_1, \ldots, X_n \sim \text{Bern}(\theta)\)
- Posterior: \(\theta \mid X_1, X_1, \ldots, X_n \sim \ ?\)

and observed data with statistics,

- Sample size: \(n = 50\)
- Number of “successes” \(\sum x_i = 2\)

Use the given prior and the observed data to calculate a 95% credible interval

Consider the following model,

- Prior: \(\theta \sim \text{Beta}(\alpha = 10, \beta = 4)\)
- Likelihood: \(X_1, X_1, \ldots, X_n \sim \text{Bern}(\theta)\)
- Posterior: \(\theta \mid X_1, X_1, \ldots, X_n \sim \ ?\)

and observed data with statistics,

- Sample size: \(n = 50\)
- Number of “successes” \(\sum x_i = 20\)

Use the given prior and the observed data to test \(H_0: \theta > 0.50\) vs \(H_1: \theta \leq 0.50\)

Consider the following model,

- Prior: \(\theta \sim \text{Beta}(\alpha = 10, \beta = 10)\)
- Likelihood: \(X_1, X_1, \ldots, X_n \sim \text{Bern}(\theta)\)
- Posterior: \(\theta \mid X_1, X_1, \ldots, X_n \sim \ ?\)

and observed data with statistics,

- Sample size: \(n = 20\)
- Number of “successes” \(\sum x_i = 5\)

Use the given prior and the observed data to test \(H_0: 0.25 < \theta < 0.50\) vs \(H_1: \theta \leq 0.25, \theta \geq 0.50\)

Consider the following model,

- Prior: \(\theta \sim \text{Beta}(\alpha = 3, \beta = 3)\)
- Likelihood: \(X_1, X_1, \ldots, X_n \sim \text{Bern}(\theta)\)
- Posterior: \(\theta \mid X_1, X_1, \ldots, X_n \sim \ ?\)

and observed data with statistics,

- Sample size: \(n = 100\)
- Number of “successes” \(\sum x_i = 70\)

Use the given prior and the observed data to test \(H_0: \theta > 0.80\) vs \(H_1: \theta \leq 0.80\)