Inference for means
A sample has an average. What does it say about the average of the whole population, or about what a treatment did to it?
Where you are 7 inside
What this area is about
A sample mean \bar{x} is known; the population mean \mu is not, and another sample would give another \bar{x}. This area turns sample means into an interval for \mu or a test of a claimed value. The population standard deviation \sigma is almost never known, so the sample’s s_x stands in for it, and every procedure here uses t, not z.
How these fit together
Foundations. The t distribution is the curve that \displaystyle t = \frac{\bar{x} - \mu}{s_x/\sqrt{n}} follows for samples from a normal population, with n - 1 degrees of freedom. The conditions for one mean say when that can be trusted: a random sample, or a randomized experiment using random assignment; n \le 0.10N when sampling without replacement; and a population close to normal, or n \ge 30, or a smaller sample whose plot shows no strong skew and no outliers.
One mean. The one-sample interval estimates \mu and sizes a sample; the one-sample test checks a claimed value \mu_0.
Two means. The interval and the test for a difference of means compare two independent groups, each checked against the conditions.
Paired data. Paired data reduces linked measurements to one column of differences and runs one-sample inference on it.
A path through it
The t distribution and the conditions first, then the one-sample interval and test. Next the two-sample interval, then the two-sample test, and paired data last.
Next, inference for slopes reuses this t reasoning, and choosing an inference procedure picks among them.