Question

find the correlation coefficient, r, of the data described below. a bumper sticker company is developing a new line of stickers promoting local sports teams. to estimate demand, the companys marketing department conducted a phone survey of local households. in each call, residents were asked to provide the number of members in their household, x, as well as the number of cars, y. household members cars 1 4 1 5 3 5 3 5 6 6 round your answer to the nearest thousandth. r =

Explanation:

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

Let \(x = [1,1,3,3,6]\) and \(y=[4,5,5,5,6]\).
\(\bar{x}=\frac{1 + 1+3+3+6}{5}=\frac{14}{5} = 2.8\)
\(\bar{y}=\frac{4 + 5+5+5+6}{5}=\frac{25}{5}=5\)

Step2: Calculate the numerator of the correlation - coefficient formula

\[

$$\begin{align*} \sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})&=(1 - 2.8)(4 - 5)+(1 - 2.8)(5 - 5)+(3 - 2.8)(5 - 5)+(3 - 2.8)(5 - 5)+(6 - 2.8)(6 - 5)\\ &=(-1.8)\times(-1)+(-1.8)\times0+(0.2)\times0+(0.2)\times0+(3.2)\times1\\ &=1.8 + 0+0+0 + 3.2\\ &=5 \end{align*}$$

\]

Step3: Calculate the denominator of the correlation - coefficient formula

\[

$$\begin{align*} \sqrt{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}\sqrt{\sum_{i = 1}^{n}(y_{i}-\bar{y})^{2}}&=\sqrt{(1 - 2.8)^{2}+(1 - 2.8)^{2}+(3 - 2.8)^{2}+(3 - 2.8)^{2}+(6 - 2.8)^{2}}\sqrt{(4 - 5)^{2}+(5 - 5)^{2}+(5 - 5)^{2}+(5 - 5)^{2}+(6 - 5)^{2}}\\ &=\sqrt{(-1.8)^{2}+(-1.8)^{2}+(0.2)^{2}+(0.2)^{2}+(3.2)^{2}}\sqrt{(-1)^{2}+0^{2}+0^{2}+0^{2}+1^{2}}\\ &=\sqrt{3.24+3.24 + 0.04+0.04+10.24}\sqrt{1+0+0+0+1}\\ &=\sqrt{16.8}\sqrt{2}\\ &=\sqrt{33.6}\approx5.8 \end{align*}$$

\]

Step4: Calculate the correlation coefficient \(r\)

\(r=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sqrt{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}\sqrt{\sum_{i = 1}^{n}(y_{i}-\bar{y})^{2}}}=\frac{5}{\sqrt{33.6}}\approx0.862\)

Answer:

\(0.862\)