Question

question 7 of 10 what is the r - value of the following data to three decimal places? x y 1 20 3 14 5 10 9 6 16 4

Explanation:

Step1: Calculate means

Let \(n = 5\).
\(\bar{x}=\frac{1 + 3+5 + 9+16}{5}=\frac{34}{5}=6.8\)
\(\bar{y}=\frac{20 + 14+10 + 6+4}{5}=\frac{54}{5}=10.8\)

Step2: Calculate numerator and denominators

\[

$$\begin{align*} S_{xy}&=\sum_{i = 1}^{n}(x_i-\bar{x})(y_i - \bar{y})\\ &=(1 - 6.8)(20 - 10.8)+(3 - 6.8)(14 - 10.8)+(5 - 6.8)(10 - 10.8)+(9 - 6.8)(6 - 10.8)+(16 - 6.8)(4 - 10.8)\\ &=(- 5.8)\times9.2+(-3.8)\times3.2+(-1.8)\times(- 0.8)+2.2\times(-4.8)+9.2\times(-6.8)\\ &=-53.36-12.16 + 1.44-10.56-62.56\\ &=-137.2 \end{align*}$$

\]

\[

$$\begin{align*} S_{xx}&=\sum_{i = 1}^{n}(x_i-\bar{x})^2\\ &=(1 - 6.8)^2+(3 - 6.8)^2+(5 - 6.8)^2+(9 - 6.8)^2+(16 - 6.8)^2\\ &=(-5.8)^2+(-3.8)^2+(-1.8)^2+2.2^2+9.2^2\\ &=33.64 + 14.44+3.24+4.84+84.64\\ &=140.8 \end{align*}$$

\]

\[

$$\begin{align*} S_{yy}&=\sum_{i = 1}^{n}(y_i-\bar{y})^2\\ &=(20 - 10.8)^2+(14 - 10.8)^2+(10 - 10.8)^2+(6 - 10.8)^2+(4 - 10.8)^2\\ &=9.2^2+3.2^2+(-0.8)^2+(-4.8)^2+(-6.8)^2\\ &=84.64+10.24 + 0.64+23.04+46.24\\ &=164.8 \end{align*}$$

\]

Step3: Calculate correlation coefficient \(r\)

\[r=\frac{S_{xy}}{\sqrt{S_{xx}S_{yy}}}=\frac{- 137.2}{\sqrt{140.8\times164.8}}=\frac{-137.2}{\sqrt{23203.84}}\approx\frac{-137.2}{152.328}\approx - 0.901\]

Answer:

\(-0.901\)