I came across an example in my book and got confused [see screenshot below]. I know my notation is overkill but I just want to make sure my understanding is clear.
Question 1: Is it correct that $(x_1, Y_1) = (0.5, 15.0)$?
Question 2: Would you ever use the notation $Y_{.5}$ where the input level is in the notation? This seems like a bad idea to me because there could be multiple observations at the $x=.5$ input level. Therefore, regression notation should always be $Y_1, Y_2, ... $
Question 3: In the book problem is it correct that $(x_3, Y_3) = (1.5, 16.5)$? If not, what is the correct way to identify this data pair?
Question 4: In the red box in the screenshot, an example for the first data pair $ (x_1, Y_1) = (.5,15.0) $... let's say that $d=.1$ and therefore one block is $.1$ miles. Then
$$Y = Y_1 = 15.0 = Y_{1(1)} + Y_{1(2)}... + Y_{1\left(\frac{.5}{.1}\right)} $$
$$= Y_{1(1)} + ... + Y_{1(5)} = 15$$
which I know is way overkill on notation but I just want to be clear. There are $5$ blocks to traverse. Is the above correct?
Question 5: Is the following correct?
$$Var(Y) = Var(Y_1) = Var(Y_{1(1)} + ... + Y_{1(5)}) = 5 Var(Y_{1(1)}) = 5 \sigma^2$$
Question $1$ and $3$ are correct. In question 2 one cannot use the notation $Y_.5$ instead one should use $Y_1$. In question $4$ although they have written $Y_i, i = 1,...,x\backslash d$ what they really mean is that one has to go till the last block with $i^{th}$ position such that it is given by $x\backslash d$. Therefore $$Y = Y_1+Y_2+Y_3+Y_4+Y_5$$ as asked in question $4.$ The answer to question $5$ is similar. I hope this helps.