Question

for the accompanying data set, (a) draw a scatter diagram of the data, (b) compute the correlation coefficient, and (c) determine whether there is a linear relation between x and y. click here to view the data set. click here to view the critical values table. (a) draw a scatter diagram of the data. choose the correct graph. data set

Explanation:

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

The formula for the mean \(\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}\), \(\bar{y}=\frac{\sum_{i = 1}^{n}y_{i}}{n}\).
Given \(x = [2,6,1,7,9]\), \(n = 5\), \(\sum_{i=1}^{5}x_{i}=2 + 6+1 + 7+9=25\), so \(\bar{x}=\frac{25}{5}=5\).
Given \(y = [3,7,6,9,5]\), \(\sum_{i = 1}^{5}y_{i}=3 + 7+6 + 9+5=30\), so \(\bar{y}=\frac{30}{5}=6\).

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

The formula \(S_{xx}=\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}\), \(S_{yy}=\sum_{i = 1}^{n}(y_{i}-\bar{y})^{2}\), \(S_{xy}=\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})\).
\((x_{1}-\bar{x})=(2 - 5)=-3\), \((x_{2}-\bar{x})=(6 - 5)=1\), \((x_{3}-\bar{x})=(1 - 5)=-4\), \((x_{4}-\bar{x})=(7 - 5)=2\), \((x_{5}-\bar{x})=(9 - 5)=4\).
\((y_{1}-\bar{y})=(3 - 6)=-3\), \((y_{2}-\bar{y})=(7 - 6)=1\), \((y_{3}-\bar{y})=(6 - 6)=0\), \((y_{4}-\bar{y})=(9 - 6)=3\), \((y_{5}-\bar{y})=(5 - 6)=-1\).
\(S_{xx}=(-3)^{2}+1^{2}+(-4)^{2}+2^{2}+4^{2}=9 + 1+16+4+16=46\).
\(S_{yy}=(-3)^{2}+1^{2}+0^{2}+3^{2}+(-1)^{2}=9 + 1+0+9+1=20\).
\(S_{xy}=(-3)\times(-3)+1\times1+(-4)\times0+2\times3+4\times(-1)=9 + 1+0+6 - 4=12\).

Step3: Calculate the correlation coefficient \(r\)

The formula \(r=\frac{S_{xy}}{\sqrt{S_{xx}S_{yy}}}\).
Substitute \(S_{xy} = 12\), \(S_{xx}=46\), \(S_{yy}=20\) into the formula: \(r=\frac{12}{\sqrt{46\times20}}=\frac{12}{\sqrt{920}}\approx\frac{12}{30.33}\approx0.396\).

Answer:

The correlation coefficient \(r\approx0.40\)