Question

find the correlation coefficient, ( r ), of the data described below.
rosannes baseball card collection includes sets from many different seasons. her older sets seem to be missing more cards than the newer sets, so she investigated to see whether this was actually the case.
for each set, she recorded the number of years since its release, ( x ), and the number of missing cards, ( y ).

Explanation:

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

The formula for the mean \(\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}\), \(\bar{y}=\frac{\sum_{i = 1}^{n}y_{i}}{n}\)
\(\sum_{i=1}^{5}x_{i}=14 + 21+26+29+30=120\), \(\bar{x}=\frac{120}{5} = 24\)
\(\sum_{i=1}^{5}y_{i}=12 + 27+2+5+38=84\), \(\bar{y}=\frac{84}{5}=16.8\)

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

\((14 - 24)(12-16.8)=(- 10)\times(-4.8)=48\)
\((21 - 24)(27 - 16.8)=(-3)\times10.2=-30.6\)
\((26 - 24)(2 - 16.8)=2\times(-14.8)=-29.6\)
\((29 - 24)(5 - 16.8)=5\times(-11.8)=-59\)
\((30 - 24)(38 - 16.8)=6\times21.2 = 127.2\)
\(\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})=48-30.6-29.6-59+127.2=66\)

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

\(\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}=(14 - 24)^{2}+(21 - 24)^{2}+(26 - 24)^{2}+(29 - 24)^{2}+(30 - 24)^{2}\)
\(=100 + 9+4+25+36=174\)
\(\sum_{i = 1}^{n}(y_{i}-\bar{y})^{2}=(12 - 16.8)^{2}+(27 - 16.8)^{2}+(2 - 16.8)^{2}+(5 - 16.8)^{2}+(38 - 16.8)^{2}\)
\(=23.04+104.04+219.04+139.24+449.44=934.8\)
\(\sqrt{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}\sum_{i = 1}^{n}(y_{i}-\bar{y})^{2}}=\sqrt{174\times934.8}\approx\sqrt{162655.2}\approx403.3\)

Step4: Calculate the correlation coefficient \(r\)

\(r=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sqrt{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}\sum_{i = 1}^{n}(y_{i}-\bar{y})^{2}}}\)
\(r=\frac{66}{403.3}\approx0.16\)

Answer:

\(r\approx0.16\)