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The sampling distribution is the probability distribution of a statistic (like \(\bar{x}\)) over repeated random samples of the same size from the same population. It describes how the statistic varies due to sampling randomness.
Standard error (SE) is the standard deviation of a statistic’s sampling distribution (e.g., \(\mathrm{SE}(\bar{x}) = \frac{\sigma}{\sqrt{n}}\), often estimated by \(\frac{s}{\sqrt{n}}\)). It measures typical sampling variability of the estimate.
A 95% confidence interval means: if you repeated the same sampling process many times and built an interval each time, about 95% of those intervals would contain the true parameter. It does not mean there is a 95% probability the parameter is in this particular interval (frequentist interpretation).
The Central Limit Theorem says that for large n, many properly standardized sums/means are approximately normal, even if the population isn’t. This justifies normal-based confidence intervals and tests (z/t-style) under large-sample conditions.
Likelihood treats the observed data as fixed and views the parameter as variable: \(L(\theta \mid \mathrm{data}) \propto P(\mathrm{data} \mid \theta)\). It measures how plausible different parameter values are given the observed data.
MLE chooses the parameter value θ̂ that maximizes the likelihood (or log-likelihood): \(\hat{\theta} = \arg\max_{\theta} L(\theta \mid \mathrm{data})\). Intuition: pick parameters that make the observed data most probable under the model.