Section 8.1: Basic Properties of Confidence Intervals

1. Give the correct interpretation of the confidence level \(c\) associated with an interval estimator, and explain what is wrong with some common misinterpretations of confidence intervals.
2. Explain how the width of a confidence interval for the population mean of a Gaussian population with known standard deviation \(\sigma\) varies as you change:
   • the confidence level \(c\);
   • the sample size \(n\); or
   • the population standard deviation \(\sigma\).
3. Construct confidence level \(c\) confidence upper bounds and lower bounds for a population mean \(\mu\) for a Gaussian population with both known and unknown population standard deviation \(\sigma\).
4. Sketch confidence level \(c\) upper bounds and lower bounds for a population mean \(\mu\) for a Gaussian population with both known and unknown population standard deviation \(\sigma\).
5. Relate the form of confidence upper / lower bounds for a population mean to the endpoints of the two-sided confidence interval for the same mean.

Section 8.2: Large Sample Confidence Intervals for a Population Mean and Proportion

1. Identify when a ‘word problem’ asks for a confidence interval for a population proportion and/or a success probability from a binomial experiment.
2. State the standardization of the success proportion \(\widehat{p}_{n} = X/n\) used in constructing the Agresti-Coull confidence interval, and explain why this standardized variable approximately follows a standard Gaussian distribution.
3. Use the binom.agresti.coull function in the R package binom to compute the Agresti-Coull confidence interval for a population proportion.
4. State how the confidence level passed to binom.agresti.coull must be modified to use binom.agresti.coull to construct confidence upper / lower bounds.
5. Sketch confidence level \(c\) confidence intervals, confidence upper bounds, and confidence lower bounds for a population proportion based on the Agresti-Coull confidence interval.
6. Distinguish between ‘word problems’ that call for the use of:
   • \(\bar{x}_{n} \pm z_{\alpha/2} \cdot \sigma / \sqrt{n}\);
   • \(\bar{x}_{n} \pm t_{\alpha/2, n-1} \cdot s / \sqrt{n}\);
   • the Agresti-Coull confidence interval; or
   • none of the above
     
   or their upper / lower bound equivalents.