Question

find the correlation coefficient r of the data below. round your answer to the nearest hundredth. x y 3.8 -6.5 2.3 -0.6 1.1 0 9.6 -2.8 -6.8 8 -4 -3

Explanation:

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

The formula for the mean \(\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}\), \(\bar{y}=\frac{\sum_{i = 1}^{n}y_{i}}{n}\)
\(\sum x=3.8 + 2.3+1.1 + 9.6-6.8-4=5\), \(n = 6\), so \(\bar{x}=\frac{5}{6}\approx0.83\)
\(\sum y=-6.5-0.6 + 0-2.8 + 8-3=-4.9\), \(\bar{y}=\frac{-4.9}{6}\approx - 0.82\)

Step2: Calculate the numerator and denominator of the correlation coefficient formula

The formula for \(r=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sqrt{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}\sum_{i = 1}^{n}(y_{i}-\bar{y})^{2}}}\)
\((x_1-\bar{x})(y_1 - \bar{y})=(3.8 - 0.83)(-6.5+0.82)=2.97\times(-5.68)=-16.87\)
\((x_2-\bar{x})(y_2 - \bar{y})=(2.3 - 0.83)(-0.6 + 0.82)=1.47\times0.22 = 0.32\)
\((x_3-\bar{x})(y_3 - \bar{y})=(1.1 - 0.83)(0 + 0.82)=0.27\times0.82=0.22\)
\((x_4-\bar{x})(y_4 - \bar{y})=(9.6 - 0.83)(-2.8+0.82)=8.77\times(-1.98)=-17.36\)
\((x_5-\bar{x})(y_5 - \bar{y})=(-6.8 - 0.83)(8 + 0.82)=(-7.63)\times8.82=-67.30\)
\((x_6-\bar{x})(y_6 - \bar{y})=(-4 - 0.83)(-3 + 0.82)=(-4.83)\times(-2.18)=10.53\)
\(\sum_{i = 1}^{6}(x_{i}-\bar{x})(y_{i}-\bar{y})=-16.87+0.32 + 0.22-17.36-67.30 + 10.53=-90.46\)

\((x_1-\bar{x})^2=(3.8 - 0.83)^2=8.82\)
\((x_2-\bar{x})^2=(2.3 - 0.83)^2=2.16\)
\((x_3-\bar{x})^2=(1.1 - 0.83)^2=0.07\)
\((x_4-\bar{x})^2=(9.6 - 0.83)^2=76.91\)
\((x_5-\bar{x})^2=(-6.8 - 0.83)^2=58.22\)
\((x_6-\bar{x})^2=(-4 - 0.83)^2=23.33\)
\(\sum_{i = 1}^{6}(x_{i}-\bar{x})^{2}=8.82+2.16+0.07+76.91+58.22+23.33=169.51\)

\((y_1-\bar{y})^2=(-6.5 + 0.82)^2=31.01\)
\((y_2-\bar{y})^2=(-0.6 + 0.82)^2=0.05\)
\((y_3-\bar{y})^2=(0 + 0.82)^2=0.67\)
\((y_4-\bar{y})^2=(-2.8 + 0.82)^2=3.92\)
\((y_5-\bar{y})^2=(8 + 0.82)^2=77.79\)
\((y_6-\bar{y})^2=(-3 + 0.82)^2=4.75\)
\(\sum_{i = 1}^{6}(y_{i}-\bar{y})^{2}=31.01+0.05+0.67+3.92+77.79+4.75=118.19\)

Step3: Calculate \(r\)

\(r=\frac{-90.46}{\sqrt{169.51\times118.19}}=\frac{-90.46}{\sqrt{20033.33}}=\frac{-90.46}{141.54}\approx - 0.64\)

Answer:

\(-0.64\)