Construct a scatter plot, find the value of the linear correlation coefficient r

Question

Listed below are baseball team statistics, consisting of

the proportions of wins and the result of this difference: Difference (number of runs scored)-(number of runs allowed). The statistics are from a recent year, and the teams are NY—Yankees, Toronto, Boston, Cleveland, Texas, Houston, SanFrancisco, and Kansas City.
Difference 163, 55 –5, 88, 51, 16, -214
Wins 0.599, 0.537, 0.531, 0.481, 0.494, 0.506, 0.383

Construct a scatter plot, find the value of the linear correlation coefficient r, and find the critical values of r from Table VI, Appendix A, p. A-14, of your textbook Elementary Statistics Use α = 0.05.is there sufficient evidence to conclude that there is a linear correlation between the proportion of wins and the above difference?

Answer 1

Hello Glennis,

 

The formula for correlation coefficient is:

 

r=(nΣ(xy)-ΣxΣy)/√((nΣ(x²)-(Σx)²)(nΣ(y²)-(Σy)²)

 

This looks like a big frightening equation, but let's break it down one piece at a time:

 

Σx = the sum of the first set

Σy = the sum of the second set

Σ(xy) = the sum of the products of each pair of #s

Σ(x²) = the sum of the squares of the first set

Σ(y²) = the sum of the squares of the second set

n = the number of (x,y) pairs

 

I'm going to make a table to make it easy to calculate these values.  The first columns are the given values, and at the bottom is the sum of each column.

 

difference (x)     wins (y)         x²                  y²                  xy

163                   0.599          26569           .358801           91.117

55                     0.537          3025            .288369           29.535

-5                     0.531          25                .281961          -2.655

88                     0.481          7744            .231361           42.328

51                     0.494          2601            .244036           25.194

16                     0.506           256             .256036           8.096

214                   0.383          45796          .146689           81.962

Σx=592      Σy=3.531     Σ(x²)=86016    Σ(y²)=1.807253    Σ(xy)=275.577

n=7

 

(go to full screen or reduce your text size if it is not coming out right.

 

Now that we have these 6 values, we can plug them into our formula

r = (nΣ(xy)-ΣxΣy)/√((nΣ(x²)-(Σx)²)(nΣ(y²)-(Σy)²)

r = (7(275.577)-(592)(3.531))/√((7(86016)-(592²))(7(1.807253)-(3.531²)

r = (1929.039-2090.352)/√((602112-350464)(12.650771-12.467961)

r = -161.313/√(251648*.18281)

r = -161.313/√46003.77088

r = -161.313/214.4848966

r = -.7521

 

This reveals a fairly strong negative correlation of r=-.7521.

To determine if this is significant at α=.05, you can consult the appendix in your book for critical r values.  check under df=6 because df=n-1.  My table shows a critical r of .707.  Since the calculated r exceeds the critical r, we can conclude that there is a linear correlation between the two variables.  (We only care about the absolute value of our r for this part so .7521>.707)