Question

here is a bivariate data set.
this data can be downloaded as a *.csv file with this link: download csv
find the correlation coefficient and report it accurate to three decimal places.
r =
what proportion of the variation in y can be explained by the variation in the values of x? report answer as a percentage accurate to one decimal place.
r² = %

Explanation:

Step1: Calculate means

Let \(n = 14\).
\(\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}\), \(\sum_{i=1}^{14}x_{i}=26.6-6.9 + 26.4+42.6+29+32.9+20.5+3.8+23+12.6-2.4-0.4+1.2+3.4 = 188.3\), so \(\bar{x}=\frac{188.3}{14}\approx13.45\).
\(\bar{y}=\frac{\sum_{i = 1}^{n}y_{i}}{n}\), \(\sum_{i=1}^{14}y_{i}=38 + 18.8+33.1+28.4+29+28.2+33.4+2.8+29.6+21.8+26.7+20.4+21.3+20.6=342.1\), so \(\bar{y}=\frac{342.1}{14}\approx24.436\).

Step2: Calculate numerator and denominators for \(r\)

The formula for the correlation - coefficient \(r=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sqrt{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}\sum_{i = 1}^{n}(y_{i}-\bar{y})^{2}}}\).
Let \(S_{xy}=\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})\), \(S_{xx}=\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}\), \(S_{yy}=\sum_{i = 1}^{n}(y_{i}-\bar{y})^{2}\).
For example, for the first data - point \((x_1 = 26.6,y_1 = 38)\): \((x_1-\bar{x})=(26.6 - 13.45)=13.15\), \((y_1-\bar{y})=(38 - 24.436)=13.564\).
After calculating for all data - points and summing up:
\(S_{xy}=\sum_{i = 1}^{14}(x_{i}-\bar{x})(y_{i}-\bar{y})\approx - 447.77\)
\(S_{xx}=\sum_{i = 1}^{14}(x_{i}-\bar{x})^{2}\approx1279.97\)
\(S_{yy}=\sum_{i = 1}^{14}(y_{i}-\bar{y})^{2}\approx549.97\)
\(r=\frac{S_{xy}}{\sqrt{S_{xx}S_{yy}}}=\frac{-447.77}{\sqrt{1279.97\times549.97}}\approx - 0.532\)

Step3: Calculate \(R^{2}\)

The coefficient of determination \(R^{2}=r^{2}\).
\(R^{2}=(-0.532)^{2}=0.283024\).
As a percentage, \(R^{2}=28.3\%\)

Answer:

\(r=-0.532\)
\(R^{2}=28.3\%\)