# Class Activity 3/5/14

Work problems in teams:

1. Suppose you want to get a feel for what a linear correlation $\displaystyle r=0.3$ (say) looks like. How would you generate a bunch of points in the plane with this value of $\displaystyle r$ ? Try it. Then try for different values of $\displaystyle r$ . As $\displaystyle r$ increases from zero, what is the smallest value where you would subjectively say "if I know one of the variables, I pretty much know the value of the other"?

2. Suppose that points in the $\displaystyle (x,y)$ plane fall roughly on a 45-degree line between the points (0,0) and (10,10), but in a band of about width w (in these same units). What, roughly, is the linear correlation coefficient $\displaystyle r$ ?

3. Prove that $\displaystyle r$ satisfies $\displaystyle -1 \le r \le +1$ . (There are several ways to do this, so, for extra credit, think of two different ways.)

4. Is it possible to have X positively correlated with Y (that is, $\displaystyle r > 0$ ) and Y positively correlated with Z, but Z negatively correlated with X (that is, $\displaystyle r < 0$ ). If so, what is the largest value of $\displaystyle r$ such that we could have $\displaystyle r_{XY} = r_{YZ} = r$ and $\displaystyle r_{ZX} = -r$ . What are the limitations on constructing a set of R.V.s with arbitrarily specified positive and negative pairwise values of $\displaystyle r$ in $\displaystyle [-1,1]$ ?

Or, if you want to work on something completely different (from last year's midterm exam, but here you get to use computers):

5. You hypothesize that the following list of 20 numbers is drawn from a uniform distribution on the interval (0, 1):

0.6816, 0.4633, 0.1646, 0.0985, 0.8236, 0.1750, 0.1636, 0.6660, 0.1640, 0.5166, 0.1638, 0.1536, 0.9535, 0.5409, 0.1637, 0.0366, 0.8092, 0.7486, 0.1202,0.1639