Statistical inference
A random sample gives one answer, and another sample would give a different one. What can that one answer honestly say about the whole population?
Where you are 8 inside
What this area is about
Inference reasons from a statistic you know, such as \bar{x} or \hat{p}, to a parameter you do not, such as \mu or p. It does that in two ways: a range of plausible values, or a verdict on a claimed value. Both rest on the sampling distribution of the statistic, so both hold only when the data come from random sampling, random assignment or a random process, and the method’s conditions are checked.
How these fit together
The two ideas. Confidence intervals report \text{estimate} \pm \text{margin of error}. Hypothesis testing assumes a claim and asks how surprising the sample would be under it, and p-values and the significance level measure that surprise and set the bar.
Procedures by parameter. Inference for proportions uses z for one or two shares. Inference for means uses t, because s stands in for an unknown \sigma. Inference for slopes does the same for a regression slope \beta. Chi-square tests compare observed counts in categories with the counts a claim predicts.
Choosing. Choosing the right procedure reads the question and the data design to pick one of these before any arithmetic.
A path through it
Confidence intervals come first. Hypothesis testing and p-values follow and interleave: p-values come after the hypotheses and the test statistic, before the decision. Then proportions and means, in either order. Slopes and chi-square tests follow means. Choosing a procedure comes last, after proportions, means and chi-square tests.
The same reasoning powers comparing two treatments by re-randomizing in experiments.