Question

question #2
determine the r - value for the data set below.

r = 0.769
r = 0.910
r = 0.863
r = 0.811

Explanation:

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

\(\bar{x}=\frac{4 + 1+5+6+6}{5}=\frac{22}{5} = 4.4\)
\(\bar{y}=\frac{12+2+11+14+11}{5}=\frac{50}{5}=10\)

Step2: Calculate the numerator \(\sum(x_i-\bar{x})(y_i - \bar{y})\)

\((4 - 4.4)(12 - 10)+(1 - 4.4)(2 - 10)+(5 - 4.4)(11 - 10)+(6 - 4.4)(14 - 10)+(6 - 4.4)(11 - 10)\)
\(=(- 0.4)\times2+(-3.4)\times(-8)+0.6\times1+1.6\times4+1.6\times1\)
\(=-0.8 + 27.2+0.6+6.4+1.6=35\)

Step3: Calculate the denominator \(\sqrt{\sum(x_i-\bar{x})^2\sum(y_i-\bar{y})^2}\)

\(\sum(x_i-\bar{x})^2=(4 - 4.4)^2+(1 - 4.4)^2+(5 - 4.4)^2+(6 - 4.4)^2+(6 - 4.4)^2\)
\(=(-0.4)^2+(-3.4)^2+(0.6)^2+(1.6)^2+(1.6)^2\)
\(=0.16 + 11.56+0.36+2.56+2.56=17.2\)
\(\sum(y_i-\bar{y})^2=(12 - 10)^2+(2 - 10)^2+(11 - 10)^2+(14 - 10)^2+(11 - 10)^2\)
\(=2^2+(-8)^2+1^2+4^2+1^2\)
\(=4 + 64+1+16+1=86\)
\(\sqrt{\sum(x_i-\bar{x})^2\sum(y_i-\bar{y})^2}=\sqrt{17.2\times86}=\sqrt{1479.2}\approx38.46\)

Step4: Calculate the correlation coefficient \(r\)

\(r=\frac{\sum(x_i-\bar{x})(y_i - \bar{y})}{\sqrt{\sum(x_i-\bar{x})^2\sum(y_i-\bar{y})^2}}=\frac{35}{38.46}\approx0.910\)

Answer:

\(r = 0.910\)