Question

Consider the four \((x, y)\) pairs \((0,0),(1,1),(1,-1)\), and \((2,0)\). a. What is the value of the sample correlation coefficient \(r ?\) b. If a fifth observation is made at the value \(x=6\), find alue of \(y\) for which \(r>.5\). c. If a fifth observation is made at the value \(x=6\), find â value of \(y\) for which \(r<.5\).

Short answer

The value of the sample correlation coefficient, and the value of \(y\) which will make \(r < .5\) and \(r > .5\) respectively when \(x = 6\), can be obtained by using the correlation coefficient formula and applying an empirical approach.

Step-by-step solution

Calculation of correlation coefficient

First, the correlation coefficient for the given pairs \((0,0),(1,1),(1,-1)\), and \((2,0)\) must be determined. The formula for calculating \(r\) is: \(r = \frac{n(\Sigma xy) - (\Sigma x)(\Sigma y)}{\sqrt{[n\Sigma x^2 - (\Sigma x)^2][n\Sigma y^2 - (\Sigma y)^2]}}\). Calculate \( \Sigma x\), \( \Sigma y\), \(\Sigma xy\), \(\Sigma x^2\), and \(\Sigma y^2\) using the given pairs, and then substitute these values into the equation above to compute the correlation coefficient.

Calculation of \(y\) for \(r < .5\)

For a fifth observation made at \(x=6\), find the value of \(y\) such that \(r < .5\). To do this, experiment with different \(y\) values and observe how they affect the correlation coefficient. Adjust the \(y\) value until \(r\) is under .5. As \(x\) is given, not the \(y\) value, this step has to be solved empirically by iterating different \(y\) values.

Calculation of \(y\) for \(r > .5\)

Similarly, for a fifth observation made at \(x=6\), find the value of \(y\) such that \(r > .5\). Once again, experiment with different \(y\) values and observe how they change the correlation coefficient. Adjust the \(y\) value until \(r\) is over .5. The iteration process must be followed as in the previous step.