Question

question 7
based on the data shown below, calculate the correlation coefficient (rounded to three decimal places)

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 \(x\): \(\sum x=4 + 5+6 + 7+8 + 9+10=49\), \(n = 7\), \(\bar{x}=\frac{49}{7}=7\).
For \(y\): \(\sum y=10.22+12.65 + 12.18+10.41+11.14+7.77+7.1 = 71.47\), \(\bar{y}=\frac{71.47}{7}\approx10.21\).

Step2: Calculate \(S_{xx}\), \(S_{yy}\), and \(S_{xy}\)

The formula \(S_{xx}=\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}\), \(S_{yy}=\sum_{i = 1}^{n}(y_{i}-\bar{y})^{2}\), \(S_{xy}=\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})\).
\((x_{i}-\bar{x})\) values: \(- 3,-2,-1,0,1,2,3\).
\((y_{i}-\bar{y})\) values: \(10.22 - 10.21=0.01\), \(12.65 - 10.21 = 2.44\), \(12.18-10.21 = 1.97\), \(10.41 - 10.21=0.2\), \(11.14 - 10.21 = 0.93\), \(7.77-10.21=-2.44\), \(7.1 - 10.21=-3.11\).
\(S_{xx}=(-3)^{2}+(-2)^{2}+(-1)^{2}+0^{2}+1^{2}+2^{2}+3^{2}=9 + 4+1+0+1+4+9 = 28\).
\(S_{yy}=(0.01)^{2}+(2.44)^{2}+(1.97)^{2}+(0.2)^{2}+(0.93)^{2}+(-2.44)^{2}+(-3.11)^{2}\)
\(=0.0001+5.9536+3.8809+0.04+0.8649+5.9536+9.6721\approx26.3652\).
\(S_{xy}=(-3)\times0.01+(-2)\times2.44+(-1)\times1.97+0\times0.2+1\times0.93+2\times(-2.44)+3\times(-3.11)\)
\(=-0.03-4.88-1.97 + 0+0.93-4.88-9.33=-20.16\).

Step3: Calculate the correlation coefficient \(r\)

The formula \(r=\frac{S_{xy}}{\sqrt{S_{xx}S_{yy}}}\).
\(r=\frac{-20.16}{\sqrt{28\times26.3652}}\approx\frac{-20.16}{\sqrt{738.2256}}\approx\frac{-20.16}{27.17}\approx - 0.742\).

Answer:

\(-0.742\)