Inference

Goal

- To calculate estimators of population parameters, and to quantify the accuracy of these estimators with probabilities.

Interested in parameters in a population.
Calculate a statistic from a sample:
- $Statistics \overset{Estimate}{\to} Parameters$
Quantify how confident we are about a statistic estimating its corresponding parameter.

Unit
- Objects we are interested in, and which measurements are recorded.
Population
- A large group of units that we are interested in studying.
Sample
- A subset of the population
- Similar to the Weak Law of Large Numbers, if you sample more and more, eventually you just get the population or get closer to the mean of the population.
Parameter
- A numerical measure of a population.
- Usually not known, it's hard to take measurements on every unit in a population.
- Parameters of interest:
  - $μ$ : popl. mean
  - $σ$ : popl. std. deviation
  - $p$ : popl. proportion
Statistic
- A numerical measure of a sample
- Known, it's easier to take measurements on a sample and calculate Statistics.
- Statistics of interest:
  - $\bar{x}$ : sample mean
  - $s$ : sample std. deviation
  - $\hat{p}$ : sample proportion
- A statistic is also an Estimator
The idea is to have a good sample to estimate the population.