The following table presents the values of X and Y sets of observations:

Question

X 12 15 16 17 14

Y 110 124 134 139 124

Use the information provided to compute Karl Pearson Correlation Coefficient and interpret the result.

Compute the value of coefficient of determination and interpret its value.

Answer 1

Hi! Here's how I would attack this problem:

1. Calculate the averages (xbar and ybar)
2. Calculate the numerator
3. Calculate the denominator
4. Divide to get the coefficient.

I. Calculating xbar and ybar. This is just adding all the Xs together and dividing by 5: (12+15+16+17+14)/5=14.8. You'll then do the same for the Ys: (110+124+134+139+124)/5=126.2.

II. Getting the numerator. First, subtract 14.8 from every value of X and 126.2 from every value of Y.

Now, multiply the last two rows together and add all these resulting numbers together: 45.36 + -0.44 + 9.36 + 28.16 + 1.76 = 84.2.

III. Let's square the values we got for each cell in (X-xbar) to get (X-xbar)^2. Then, we'll do the same for (Y-ybar) to get (Y-ybar)^2.

Then we add the whole (X-xbar)^2 together: 7.84+0.04+1.44+4.84+0.64 = 14.8. And we'll do the same for the (Y-ybar)^2 row: 262.44+4.84+60.84+163.84+4.84 = 496.8. Finally, let's multiply those two numbers together (14.8*496.8 = 7352.64), and take the square root of 7352.64 to be roughly 85.7475.

IV. Finally, we divide the numerator by the denominator:

The Pearson correlation coefficient of this data is 84.2/85.7475 = 0.982