Section 8.1: Basic Properties of Confidence Intervals

1. Determine the margin of error for the sample mean from a Gaussian population with known standard deviation \(\sigma\) at a confidence level \(c\).
2. Recognize the relationship between a confidence level \(c\) and tail value \(\alpha\), and state how these specify the probability in the tails or body of a given distribution.
3. Determine the critical value \(z_{\alpha}\) for a standard Gaussian random variable using both Table A.3 and R’s qnorm.
4. Define interval estimator and confidence interval, and make an analogy between them and point estimators and point estimates.
5. Construct a \(c = 1 - \alpha\) confidence interval for a population mean \(\mu\) using a random sample from a Gaussian population when the population standard deviation \(\sigma\) is known.
6. Sketch the confidence interval from the previous learning objective.

Section 8.3: Intervals Based on a Normal Population Distribution

1. Give a constructive definition of a \(t\)-distributed random variable \(T\) using a random sample from a Gaussian population.
2. Determine the parameter of a \(t\)-distributed random variable \(T\) constructed using a random sample from a Gaussian population.
3. Describe the general properties of the probability density function of a \(t\)-distributed random variable as compared to a standard Gaussian random variable including where each is centered, the symmetry properties of each, and the “fatness” of the tails of each.
4. Determine the critical value \(t_{\alpha, \nu}\) for a random variable \(T\) that is \(t\)-distributed with parameter \(\nu\) using both Table A.5 and R’s qt.
5. Construct a \(c = 1 - \alpha\) confidence interval for a population mean \(\mu\) using a random sample from a Gaussian population when the population standard deviation \(\sigma\) is unknown.
6. Sketch the confidence interval from the previous learning objective.
7. Reason about what confidence interval we have covered, if any, is appropriate for the population mean given a description of a sample from that population.