Question

find the correlation coefficient, r, of the data described below.
sally is writing an article for an online music magazine on why some bands last longer than others. while researching the article, she compiled information about how long various bands had lasted without changing their lineups.
one comparison she looked at was between the number of members in each band, x, and the number of years each band had lasted, y.
members years
3 25
5 13
6 12
7 14
8 12
round your answer to the nearest thousandth.
r =

Explanation:

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

The formula for the mean \(\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}\) and \(\bar{y}=\frac{\sum_{i = 1}^{n}y_{i}}{n}\).
For \(x\) values \(x_1 = 3,x_2=5,x_3 = 6,x_4=7,x_5 = 8\), \(\sum_{i=1}^{5}x_{i}=3 + 5+6+7+8=29\), \(\bar{x}=\frac{29}{5}=5.8\).
For \(y\) values \(y_1 = 25,y_2=13,y_3 = 12,y_4=14,y_5 = 12\), \(\sum_{i=1}^{5}y_{i}=25 + 13+12+14+12=76\), \(\bar{y}=\frac{76}{5}=15.2\).

Step2: Calculate the numerator \(\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})\)

\((3 - 5.8)(25-15.2)=(- 2.8)\times9.8=-27.44\)
\((5 - 5.8)(13 - 15.2)=(-0.8)\times(-2.2)=1.76\)
\((6 - 5.8)(12 - 15.2)=(0.2)\times(-3.2)=-0.64\)
\((7 - 5.8)(14 - 15.2)=(1.2)\times(-1.2)=-1.44\)
\((8 - 5.8)(12 - 15.2)=(2.2)\times(-3.2)=-7.04\)
\(\sum_{i = 1}^{5}(x_{i}-\bar{x})(y_{i}-\bar{y})=-27.44 + 1.76-0.64-1.44-7.04=-35.8\)

Step3: Calculate the denominator \(\sqrt{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}\sum_{i = 1}^{n}(y_{i}-\bar{y})^{2}}\)

\(\sum_{i = 1}^{5}(x_{i}-\bar{x})^{2}=(3 - 5.8)^{2}+(5 - 5.8)^{2}+(6 - 5.8)^{2}+(7 - 5.8)^{2}+(8 - 5.8)^{2}\)
\(=(-2.8)^{2}+(-0.8)^{2}+(0.2)^{2}+(1.2)^{2}+(2.2)^{2}=7.84 + 0.64+0.04+1.44+4.84 = 14.8\)
\(\sum_{i = 1}^{5}(y_{i}-\bar{y})^{2}=(25 - 15.2)^{2}+(13 - 15.2)^{2}+(12 - 15.2)^{2}+(14 - 15.2)^{2}+(12 - 15.2)^{2}\)
\(=(9.8)^{2}+(-2.2)^{2}+(-3.2)^{2}+(-1.2)^{2}+(-3.2)^{2}=96.04+4.84 + 10.24+1.44+10.24 = 122.8\)
\(\sqrt{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}\sum_{i = 1}^{n}(y_{i}-\bar{y})^{2}}=\sqrt{14.8\times122.8}=\sqrt{1827.44}\approx42.75\)

Step4: Calculate the correlation coefficient \(r\)

\(r=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sqrt{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}\sum_{i = 1}^{n}(y_{i}-\bar{y})^{2}}}=\frac{-35.8}{42.75}\approx - 0.837\)

Answer:

\(-0.837\)