what is the correlation coefficient r for the data set? enter your answer to the nearest hundredth in the…

Answer:

0.94

Explanation:

Step1: Calculate means

Let $x = [12,13,14,14,15,16,16,17,18]$, $y=[110,106,108,109,110,114,116,117,119]$. $\bar{x}=\frac{12 + 13+14+14+15+16+16+17+18}{9}=\frac{135}{9} = 15$ $\bar{y}=\frac{110+106+108+109+110+114+116+117+119}{9}=\frac{999}{9}=111$

Step2: Calculate numerator

$numerator=\sum_{i = 1}^{9}(x_{i}-\bar{x})(y_{i}-\bar{y})$ $(12 - 15)(110 - 111)+(13 - 15)(106 - 111)+(14 - 15)(108 - 111)+(14 - 15)(109 - 111)+(15 - 15)(110 - 111)+(16 - 15)(114 - 111)+(16 - 15)(116 - 111)+(17 - 15)(117 - 111)+(18 - 15)(119 - 111)$ $=(- 3)\times(-1)+(-2)\times(-5)+(-1)\times(-3)+(-1)\times(-2)+0\times(-1)+1\times3+1\times5+2\times6+3\times8$ $=3 + 10+3+2+0+3+5+12+24=62$

Step3: Calculate denominator

$denominator=\sqrt{\sum_{i = 1}^{9}(x_{i}-\bar{x})^{2}\sum_{i = 1}^{9}(y_{i}-\bar{y})^{2}}$ $\sum_{i = 1}^{9}(x_{i}-\bar{x})^{2}=(12 - 15)^{2}+(13 - 15)^{2}+(14 - 15)^{2}+(14 - 15)^{2}+(15 - 15)^{2}+(16 - 15)^{2}+(16 - 15)^{2}+(17 - 15)^{2}+(18 - 15)^{2}$ $=9 + 4+1+1+0+1+1+4+9 = 30$ $\sum_{i = 1}^{9}(y_{i}-\bar{y})^{2}=(110 - 111)^{2}+(106 - 111)^{2}+(108 - 111)^{2}+(109 - 111)^{2}+(110 - 111)^{2}+(114 - 111)^{2}+(116 - 111)^{2}+(117 - 111)^{2}+(119 - 111)^{2}$ $=1+25 + 9+4+1+9+25+36+64 = 174$ $denominator=\sqrt{30\times174}=\sqrt{5220}\approx72.25$

Step4: Calculate correlation coefficient

$r=\frac{numerator}{denominator}=\frac{62}{72.25}\approx0.94$