There are two primary "engines" for generating estimators.
| Decision | ( H_0 ) True | ( H_0 ) False | |----------|--------------|----------------| | Reject ( H_0 ) | Type I error (prob ( \alpha )) | Correct | | Fail to reject ( H_0 ) | Correct | Type II error (prob ( \beta )) |
Power ( = 1 - \beta = P(\textReject H_0 \mid H_a \text true) ). mathematical statistics lecture
A point estimate like $\hat\theta = 5$ is rarely enough. Is it exactly 5? Probably not. We need a range. This leads to Confidence Intervals.
A $95%$ confidence interval does not mean there is a 95% chance the parameter is in the interval (the parameter is fixed; the interval is random). There are two primary "engines" for generating estimators
The Correct Interpretation: If we repeated the experiment 100 times, calculating a new interval each time, roughly 95 of those intervals would contain the true parameter.
Mathematically, we construct bounds using probability statements: $$P(L \leq \theta \leq U) = 1 - \alpha$$ Mathematical Statistics is the branch of applied mathematics
This accounts for the sampling error. It transforms a single number into a rigorous statement about uncertainty.
Mathematical Statistics is the branch of applied mathematics that provides the theoretical underpinning for data analysis. Unlike descriptive statistics (which simply summarizes data), mathematical statistics develops methods for inference—drawing conclusions about a population based on a sample.
The core question: Given observed data, what can we say about the unknown process that generated it?