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1) The sample mean x̄ is an unbiased estimator of the population mean μ for i.i.d. data.
2) The Central Limit Theorem requires the population itself to be normal.
3) A point estimate (the realized numeric value) is a random variable.
4) For large n and finite variance, the standardized sample mean (x̄−μ)/(σ/√n) is approximately standard normal (CLT).
5) The standard error of x̄ equals the sample standard deviation s, regardless of sample size.
6) When σ is unknown, a CI for μ should use the t distribution with n−1 degrees of freedom (assuming normality or large n).
7) Increasing the confidence level (e.g., 95% → 99%) decreases the critical value.
8) Holding everything else fixed, increasing the sample size n makes a 95% CI for μ narrower.
9) A 99% confidence interval is typically shorter than a 95% interval for the same data.
10) For a sample proportion, the normal approximation is reasonable when both np and n(1−p) are sufficiently large.
11) Bias and variance of an estimator are the same concept.
12) For i.i.d. data with variance σ², the variance of the sample mean is Var(x̄)=σ²/n.
13) The CLT describes the distribution of individual observations for large n.
14) If the population has infinite variance (extremely heavy tails), the usual classical CLT still always works.
15) In simple random sampling without replacement, even when the population size N is at least 10n, dependence cannot be ignored.