Given the linear correlation coefficient r and the sample size n, determine the critical values of r and use your finding to state whether or not the given r represents a significant linear correlation. Use a significance level of 0.05.

Help to understand
$e_i=y_i-a{x}_i-b$
$e_i=(y_i-\overline{y})-a(x_i-\overline{x})-(b-\overline{y}+a\overline{x})$

What is a standardized regression coefficient?

set $S$ of coordinates $(x,y)$, and am estimating $f(x)=ax+b$ where $a>0$. I also happen to know that $\mathrm{\forall }x,y((x,y)\in S⟹y<f(x))$.
The question is how I can utilize this knowledge of the upper bound on values to improve the regression result?
My intuition is to run a "normal" linear regression on all coordinates in $S$ giving $g(x)$ and then construct $g^{\prime }(x)=g(x)+c$, with $c$ being the lowest number such that $\mathrm{\forall }x,y((x,y)\in S⟹y\le g^{\prime }(x))$, e.g. such that $g^{\prime }(x)$ lies as high as it can whilst still touching at least point in $S$. I do, however, have absolutely no idea if this is the best way to do it, nor how to devise an algorithm that does this efficiently.

Show that the correlation coeffcient of $x_i-\overline{x}$ and $x_j-\overline{x}$ is $-(n-1{)}^{-1}$

1. In your own words, describe the difference between continuous and discrete data sets.
2. What is the difference between the independent and dependent variable? How do you determine which goes on which axis on your graph?
3. What are the four types of correlation?
4. How do they differ from one another?
5. In your own words, explain the difference between correlation and causation.

Given $X$ and $Y$ sample data as something like
$\left[\begin{array}{cc}1& 2\\ 2& 3\end{array}\right]$
$\left[\begin{array}{c}7\\ 8\end{array}\right]$
In such an arrangement how do I include B3? I would think I would want to add it in as always 1 in $X$. IE
$\left[\begin{array}{cc}1& 2\\ 2& 3\\ 1& 1\end{array}\right]$

Short version, I need to find a regression to this: $a\equiv t(\mathrm{mod}\mathrm{\Delta })$, $a$ and $\mathrm{\Delta }$ are the unknowns constants.
Any idea where I should start looking?
Some context, because I may be wording it in a confusing way: I am trying to find the tempo of time-stamped events $t_i$ for some real time musical analysis. They have a typical interval of $\mathrm{\Delta }$, but there isn't an event at every "tick", so no linear regression, and there may be more than one event for a given "tick". In other words, $t_{n+1}-t_n$ may be $0$ or any $m\mathrm{\Delta }$.