Question

find the correlation coefficient, r, of the data described below. ernesto wants to ensure that the new board game hes designing is accessible to a wide age range. to test this, he had several children of various ages play the board game against each other. each child played the game the same number of times. for each child, ernesto recorded his or her age, x, and the number of games that child had won, y. age games won 7 0 8 2 9 1 10 5 10 9 11 10 round your answer to the nearest thousandth. r =

Explanation:

Step1: Calculate the means

Let \(x = [7,8,9,10,10,11]\) and \(y=[0,2,1,5,9,10]\).
The mean of \(x\), \(\bar{x}=\frac{7 + 8+9+10+10+11}{6}=\frac{55}{6}\approx9.167\).
The mean of \(y\), \(\bar{y}=\frac{0 + 2+1+5+9+10}{6}=\frac{27}{6} = 4.5\).

Step2: Calculate the numerator and denominators

The numerator \(S_{xy}=\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})\)
\((7 - 9.167)(0 - 4.5)+(8 - 9.167)(2 - 4.5)+(9 - 9.167)(1 - 4.5)+(10 - 9.167)(5 - 4.5)+(10 - 9.167)(9 - 4.5)+(11 - 9.167)(10 - 4.5)\)
\(=(- 2.167)\times(-4.5)+(-1.167)\times(-2.5)+(-0.167)\times(-3.5)+(0.833)\times(0.5)+(0.833)\times(4.5)+(1.833)\times(5.5)\)
\(=9.7515 + 2.9175+0.5845+0.4165+3.7485+10.0815\)
\(=27.5\)

The denominator \(S_{xx}=\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}\)
\((7 - 9.167)^{2}+(8 - 9.167)^{2}+(9 - 9.167)^{2}+(10 - 9.167)^{2}+(10 - 9.167)^{2}+(11 - 9.167)^{2}\)
\(=(-2.167)^{2}+(-1.167)^{2}+(-0.167)^{2}+(0.833)^{2}+(0.833)^{2}+(1.833)^{2}\)
\(=4.695 + 1.362+0.028+0.694+0.694+3.36\)
\(=10.833\)

The denominator \(S_{yy}=\sum_{i = 1}^{n}(y_{i}-\bar{y})^{2}\)
\((0 - 4.5)^{2}+(2 - 4.5)^{2}+(1 - 4.5)^{2}+(5 - 4.5)^{2}+(9 - 4.5)^{2}+(10 - 4.5)^{2}\)
\(=(-4.5)^{2}+(-2.5)^{2}+(-3.5)^{2}+(0.5)^{2}+(4.5)^{2}+(5.5)^{2}\)
\(=20.25+6.25 + 12.25+0.25+20.25+30.25\)
\(=89.5\)

The correlation coefficient \(r=\frac{S_{xy}}{\sqrt{S_{xx}S_{yy}}}=\frac{27.5}{\sqrt{10.833\times89.5}}=\frac{27.5}{\sqrt{968.5535}}\approx\frac{27.5}{31.1216}\approx0.884\)

Answer:

\(0.884\)