Section 6.1: Statistics and Their Distributions

1. Define a random sample from a population.
2. Explain why a statistic of a random sample is itself a random variable.
3. Define the sampling distribution of a statistic.
4. Given a probability model for how a random sample is generated, construct the sampling distribution of a statistic of that random sample for small sample sizes (\(n = 2\) or \(3\)).

Section 6.2: The Distribution of the Sample Mean

1. Recognize and explain the notation \(X_{1}, X_{2}, \ldots, X_{n} \stackrel{\text{iid}}{\sim} \text{D}\) for a random sample from a population with distribution D.
2. Compute the mean and variance of the sample mean for a random sample \(X_{1}, X_{2}, \ldots, X_{n} \stackrel{\text{iid}}{\sim} \text{D}\) of size \(n\) from a population with mean \(\mu\) and variance \(\sigma^{2}\).
3. Distinguish between the mean and variance of a population and the mean and variance of the sample mean of a random sample from that population.
4. State the premises (conditions) and conclusion of the Central Limit Theorem.
5. Explain in what sense the Central Limit Theorem is an asymptotic result.
6. State under what conditions the Central Limit Theorem-based approximation for the sampling distribution of the sample mean works well for small sample sizes.
7. Use the Central Limit Theorem to approximate the sampling distribution of the sample mean for a random sample from a population with known mean and variance.
8. Use the Central Limit Theorem to approximate the sample total for a random sample from a population with known mean and variance.