Question

find the correlation coefficient r of the data below. round your answer to the nearest hundredth. show your work here enter your answer

Explanation:

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

First, find \(\bar{x}=\frac{-1.3 + 0.4-0.8 + 9.3-15.5 + 9}{6}=\frac{-0.9}{6}=-0.15\)
Then, find \(\bar{y}=\frac{4.8-10 - 0.1-7.1 + 9.8-11.5}{6}=\frac{-14.1}{6}=-2.35\)

Step2: Calculate the numerator and denominators of the correlation formula

The formula for \(r\) is \(r=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sqrt{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}\sum_{i = 1}^{n}(y_{i}-\bar{y})^{2}}}\)

Calculate \((x_{i}-\bar{x})(y_{i}-\bar{y})\) for each pair:

• For \((x=-1.3,y = 4.8)\): \((-1.3+0.15)(4.8 + 2.35)=(-1.15)\times7.15=-8.2225\)
• For \((x = 0.4,y=-10)\): \((0.4 + 0.15)(-10 + 2.35)=0.55\times(-7.65)=-4.2075\)
• For \((x=-0.8,y=-0.1)\): \((-0.8+0.15)(-0.1 + 2.35)=(-0.65)\times2.25=-1.4625\)
• For \((x = 9.3,y=-7.1)\): \((9.3+0.15)(-7.1 + 2.35)=9.45\times(-4.75)=-44.8875\)
• For \((x=-15.5,y = 9.8)\): \((-15.5+0.15)(9.8 + 2.35)=(-15.35)\times12.15=-186.5025\)
• For \((x = 9,y=-11.5)\): \((9+0.15)(-11.5 + 2.35)=9.15\times(-9.15)=-83.7225\)

Sum of \((x_{i}-\bar{x})(y_{i}-\bar{y})\): \(-8.2225-4.2075-1.4625-44.8875-186.5025-83.7225=-329.005\)

Calculate \((x_{i}-\bar{x})^{2}\) for each \(x\):

• \((-1.3 + 0.15)^{2}=(-1.15)^{2}=1.3225\)
• \((0.4+0.15)^{2}=0.3025\)
• \((-0.8 + 0.15)^{2}=0.4225\)
• \((9.3+0.15)^{2}=89.3025\)
• \((-15.5+0.15)^{2}=235.0225\)
• \((9+0.15)^{2}=83.7225\)

Sum of \((x_{i}-\bar{x})^{2}\): \(1.3225+0.3025+0.4225+89.3025+235.0225+83.7225 = 409.095\)

Calculate \((y_{i}-\bar{y})^{2}\) for each \(y\):

• \((4.8+2.35)^{2}=51.1225\)
• \((-10 + 2.35)^{2}=58.5225\)
• \((-0.1+2.35)^{2}=5.0625\)
• \((-7.1+2.35)^{2}=22.5625\)
• \((9.8+2.35)^{2}=147.6225\)
• \((-11.5+2.35)^{2}=83.7225\)

Sum of \((y_{i}-\bar{y})^{2}\): \(51.1225+58.5225+5.0625+22.5625+147.6225+83.7225=368.615\)

Step3: Calculate \(r\)

\(r=\frac{-329.005}{\sqrt{409.095\times368.615}}\)

First, calculate \(409.095\times368.615\approx409.095\times368.615 = 150777.77\)

\(\sqrt{150777.77}\approx388.3\)

\(r=\frac{-329.005}{388.3}\approx - 0.85\)

Answer:

\(-0.85\)