Question

based on the data shown below, calculate the correlation coefficient (to three decimal places)

x | y
1 | 77
2 | 68.6
3 | 71.8
4 | 70.2
5 | 68.6
6 | 60.2
7 | 61.2
8 | 56.6

r =
enter an integer or decimal number more...

Explanation:

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

The mean of \(x\) values: \(\bar{x}=\frac{1 + 2+3+4+5+6+7+8}{8}=\frac{36}{8} = 4.5\)
The mean of \(y\) values: \(\bar{y}=\frac{77+68.6+71.8+70.2+68.6+60.2+61.2+56.6}{8}=\frac{534.2}{8}=66.775\)

Step2: Calculate the numerator and denominator of the correlation coefficient formula

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

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

• When \(i = 1\): \((1 - 4.5)(77-66.775)=(- 3.5)\times10.225=-35.7875\)
• When \(i = 2\): \((2 - 4.5)(68.6 - 66.775)=(-2.5)\times1.825=-4.5625\)
• When \(i = 3\): \((3 - 4.5)(71.8 - 66.775)=(-1.5)\times5.025=-7.5375\)
• When \(i = 4\): \((4 - 4.5)(70.2 - 66.775)=(-0.5)\times3.425=-1.7125\)
• When \(i = 5\): \((5 - 4.5)(68.6 - 66.775)=(0.5)\times1.825 = 0.9125\)
• When \(i = 6\): \((6 - 4.5)(60.2 - 66.775)=(1.5)\times(-6.575)=-9.8625\)
• When \(i = 7\): \((7 - 4.5)(61.2 - 66.775)=(2.5)\times(-5.575)=-13.9375\)
• When \(i = 8\): \((8 - 4.5)(56.6 - 66.775)=(3.5)\times(-10.175)=-35.6125\)

\(\sum_{i = 1}^{8}(x_{i}-\bar{x})(y_{i}-\bar{y})=-35.7875-4.5625-7.5375-1.7125 + 0.9125-9.8625-13.9375-35.6125=-108.0\)

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

• \((1 - 4.5)^{2}=(-3.5)^{2}=12.25\)
• \((2 - 4.5)^{2}=(-2.5)^{2}=6.25\)
• \((3 - 4.5)^{2}=(-1.5)^{2}=2.25\)
• \((4 - 4.5)^{2}=(-0.5)^{2}=0.25\)
• \((5 - 4.5)^{2}=(0.5)^{2}=0.25\)
• \((6 - 4.5)^{2}=(1.5)^{2}=2.25\)
• \((7 - 4.5)^{2}=(2.5)^{2}=6.25\)
• \((8 - 4.5)^{2}=(3.5)^{2}=12.25\)

\(\sum_{i = 1}^{8}(x_{i}-\bar{x})^{2}=12.25+6.25+2.25+0.25+0.25+2.25+6.25+12.25 = 42\)

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

• \((77 - 66.775)^{2}=(10.225)^{2}=104.550625\)
• \((68.6 - 66.775)^{2}=(1.825)^{2}=3.330625\)
• \((71.8 - 66.775)^{2}=(5.025)^{2}=25.250625\)
• \((70.2 - 66.775)^{2}=(3.425)^{2}=11.730625\)
• \((68.6 - 66.775)^{2}=(1.825)^{2}=3.330625\)
• \((60.2 - 66.775)^{2}=(-6.575)^{2}=43.230625\)
• \((61.2 - 66.775)^{2}=(-5.575)^{2}=31.080625\)
• \((56.6 - 66.775)^{2}=(-10.175)^{2}=103.520625\)

\(\sum_{i = 1}^{8}(y_{i}-\bar{y})^{2}=104.550625+3.330625+25.250625+11.730625+3.330625+43.230625+31.080625+103.520625 = 325.055\)

Step3: Calculate \(r\)

\(r=\frac{-108.0}{\sqrt{42\times325.055}}=\frac{-108.0}{\sqrt{13652.31}}\approx\frac{-108.0}{116.843}\approx - 0.924\)

Answer:

\(-0.924\)