Section 5.4: Sampling Distributions and Estimators

1. Explain why statistics computed from a simple random sample can be modeled using random variables.
2. Identify the notation \(\bar{X}\) for the sample mean and \(S\) for the sample standard deviation.
3. Explain the difference between the population distribution modeled by a random variable \(X\) and the sampling distributions of statistics such as the sample mean \(\bar{X}\) or sample standard deviation \(S\).

Section 5.5: The Central Limit Theorem

1. Given the population mean \(\mu\) and standard deviation \(\sigma\) of a random variable \(X\), find the mean \(\mu_{\bar{X}}\) and standard deviation \(\sigma_{\bar{X}}\) of the sample mean \(\bar{X}_{n}\) of a simple random sample from the population of size \(n\).
2. Explain, roughly, what the Central Limit Theorem says about the sample mean of a simple random sample from a population.
3. Identify the rule-of-thumb cutoff for the sample size \(n\) when the normal approximation for a sample mean \(\bar{X}_{n}\) applies.
4. Compute probabilities such as \(P(\bar{X}_{n} \leq \bar{x})\), \(P(\bar{X}_{n} \geq \bar{x})\), and \(P(a \leq \bar{X}_{n} \leq b)\) given the population mean \(\mu\), standard deviation \(\sigma\), and sample size \(n\) using the normal approximation for the sample mean.
5. Explain, roughly, why \(X\) from a population and \(\bar{X}_{n}\) from a simple random sample of size \(n\) from the population have different distributions.