Explain why the margin of error \(z_{\alpha/2}\cdot \sigma_{X} / \sqrt{n}\) is inappropriate when the population standard deviation is unknown.
Compute a probability \(c\) margin of error for the sample mean of a random sample from a Normal population when only the sample standard deviation \(s_{X}\) known.

Distinguish between an interval estimator and an interval estimate (aka “confidence interval”).
Explain what the probability \(c\) is associated with for a confidence level \(c\) confidence interval.
Identify how the width of a confidence interval varies as \(n\), \(c\), and \(s_{X}\) vary.

State the sampling distribution of a \(Z\)-score for a sample mean from a Normal population.
State the sampling distribution of a \(T\)-score for a sample mean from a Normal population.
Identify the degrees of freedom for the \(t\) distribution resulting from a \(T\)-score for a sample mean.
Compare and contrast the density curve for a \(t\)-distributed random variable to the density curve of a standard Normal random variable.
Identify how the shape of the density curve of a \(t\)-distributed random variable changes as its degrees of freedom increase.
Compute a critical value \(t_{\alpha, n-1}\) for a \(t\)-distribution with \(n - 1\) degrees of freedom using qt in R.

Compute a \(t\)-based confidence interval given relevant information about a random sample from a Normal population.
Sketch how the \(t\)-based confidence interval for a population mean is related to the sample mean, critical value, and estimate of the standard error of the sample mean.