In my statistics class, we're just beginning to talk about (point) estimation. I understand the basics for the most part, but I have a small question that might actually be due more to notation than anything conceptual:
Say you have a population $P$ of students in a classroom and you want to find the mean height by randomly selecting $n$ students $s_1,...,s_n$ in $S\subseteq P$ to measure. My book calls $X_1,...,X_n$ a random sample of $P$, so do I regard each $X_i$ as a function $X_i: s_i \to\mathbb{R}$ (each $X_i$ has only one point in its domain and $X_i(s_i)$ returns the height of student $s_i$)? And in general, does "random sample" refer to both $s_1,...,s_n$ and their images (heights) $X_1(s_1),...,X_n(s_n)$? Or is this not the right way to think about it? And if this is how I should think of it, then couldn't you just say that since $X_1,...,X_n$ is a random sample, each $X_i$ has the same distribution function, and so we can just call a random sample $Range(X)$, where $X:S\to\mathbb{R}$? So $X_1(s_1)=X(s_1)$, $X_2(s_2)=X(s_2)$,..., $X_n(s_n)=X(s_n)$?
I understand your confusion. What happens is that there is all this talk about the sample space in the first part of a stats class and then they introduce the idea that a random variable "maps" the sample space to a new sample space.
However, either the people, as people, or their associated heights, can be validly called the sample space. However, it happens that one of the sample spaces results in "people" as the output, whilst the other returns numbers. It's really rather irrelevant which is the real sample. Of course, your are actually sampling people, but you might as well just skip a step an say you are sampling heights.
The bottom line is that either can be a sample space or population, it depends on what level of abstraction you are working from.