Question

find the correlation coefficient, r, of the data described below.
a soccer coach is considering a strategy of having defensive players take more shots on goal. he evaluated a sample of his teams recent matches to see how players in defensive roles could contribute offensively.
in each match the coach tracked the number of shots on goal by his defenders, x. he also noted the difference between the number of goals that had been scored by his team and the opposing team, y.
shots on goal by defenders\tgoal difference
5\t-2
5\t-4
7\t4
7\t-5
9\t3
10\t5
round your answer to the nearest thousandth.
r =

Explanation:

Step1: Calculate the means of \(x\) and \(y\)

The formula for the mean \(\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}\), \(\bar{y}=\frac{\sum_{i = 1}^{n}y_{i}}{n}\).
For \(x\): \(x_1 = 5,x_2=5,x_3 = 7,x_4=7,x_5 = 9,x_6=10\), \(\sum_{i=1}^{6}x_{i}=5 + 5+7+7+9+10=43\), \(\bar{x}=\frac{43}{6}\approx7.167\)
For \(y\): \(y_1=-2,y_2 = - 4,y_3=4,y_4=-5,y_5 = 3,y_6=5\), \(\sum_{i=1}^{6}y_{i}=-2-4 + 4-5+3+5=1\), \(\bar{y}=\frac{1}{6}\approx0.167\)

Step2: Calculate the numerator of the correlation coefficient formula

The formula for the numerator \(r_{num}=\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})\)
\((x_1-\bar{x})(y_1-\bar{y})=(5 - 7.167)(-2-0.167)=(-2.167)\times(-2.167) = 4.696\)
\((x_2-\bar{x})(y_2-\bar{y})=(5 - 7.167)(-4 - 0.167)=(-2.167)\times(-4.167)=9.030\)
\((x_3-\bar{x})(y_3-\bar{y})=(7 - 7.167)(4-0.167)=(-0.167)\times(3.833)=-0.640\)
\((x_4-\bar{x})(y_4-\bar{y})=(7 - 7.167)(-5 - 0.167)=(-0.167)\times(-5.167)=0.863\)
\((x_5-\bar{x})(y_5-\bar{y})=(9 - 7.167)(3-0.167)=(1.833)\times(2.833)=5.193\)
\((x_6-\bar{x})(y_6-\bar{y})=(10 - 7.167)(5 - 0.167)=(2.833)\times(4.833)=13.691\)
\(r_{num}=4.696+9.030-0.640 + 0.863+5.193+13.691=32.833\)

Step3: Calculate the denominator of the correlation coefficient formula

The formula for the denominator \(r_{den}=\sqrt{\sum_{i = 1}^{n}(x_{i}-\bar{x})^2\sum_{i = 1}^{n}(y_{i}-\bar{y})^2}\)
\(\sum_{i=1}^{6}(x_{i}-\bar{x})^2=(5 - 7.167)^2+(5 - 7.167)^2+(7 - 7.167)^2+(7 - 7.167)^2+(9 - 7.167)^2+(10 - 7.167)^2\)
\(=(-2.167)^2+(-2.167)^2+(-0.167)^2+(-0.167)^2+(1.833)^2+(2.833)^2\)
\(=4.696+4.696+0.028+0.028+3.360+8.026=20.834\)
\(\sum_{i=1}^{6}(y_{i}-\bar{y})^2=(-2 - 0.167)^2+(-4 - 0.167)^2+(4 - 0.167)^2+(-5 - 0.167)^2+(3 - 0.167)^2+(5 - 0.167)^2\)
\(=(-2.167)^2+(-4.167)^2+(3.833)^2+(-5.167)^2+(2.833)^2+(4.833)^2\)
\(=4.696+17.364+14.692+26.698+8.026+23.368=94.844\)
\(r_{den}=\sqrt{20.834\times94.844}=\sqrt{1975.077}\approx44.442\)

Step4: Calculate the correlation coefficient \(r\)

The formula for the correlation coefficient \(r=\frac{r_{num}}{r_{den}}\)
\(r=\frac{32.833}{44.442}\approx0.739\)

Answer:

\(0.739\)