Question

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

x	y
4	95.2
5	93.4
6	92
7	86.8
8	85.2
9	77.6
10	78.8
11	73.4

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

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

Let \(x = [4,5,6,7,8,9,10,11]\), \(y=[95.2,93.4,92,86.8,85.2,77.6,78.8,73.4]\)
\(\bar{x}=\frac{4 + 5+6+7+8+9+10+11}{8}=\frac{60}{8}=7.5\)
\(\bar{y}=\frac{95.2 + 93.4+92+86.8+85.2+77.6+78.8+73.4}{8}=\frac{692.4}{8}=86.55\)

Step2: Calculate the numerator and denominators of the correlation - coefficient formula

The formula for the correlation coefficient \(r\) is \(r=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sqrt{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}\sum_{i = 1}^{n}(y_{i}-\bar{y})^{2}}}\)
\(\sum_{i = 1}^{8}(x_{i}-\bar{x})(y_{i}-\bar{y})=(4 - 7.5)(95.2-86.55)+(5 - 7.5)(93.4 - 86.55)+(6 - 7.5)(92 - 86.55)+(7 - 7.5)(86.8 - 86.55)+(8 - 7.5)(85.2 - 86.55)+(9 - 7.5)(77.6 - 86.55)+(10 - 7.5)(78.8 - 86.55)+(11 - 7.5)(73.4 - 86.55)\)
\(=(-3.5)\times8.65+(-2.5)\times6.85+(-1.5)\times5.45+(-0.5)\times0.25+0.5\times(-1.35)+1.5\times(-8.95)+2.5\times(-7.75)+3.5\times(-13.15)\)
\(=-30.275-17.125 - 8.175-0.125-0.675-13.425-19.375 - 46.025=-135.2\)
\(\sum_{i = 1}^{8}(x_{i}-\bar{x})^{2}=(4 - 7.5)^{2}+(5 - 7.5)^{2}+(6 - 7.5)^{2}+(7 - 7.5)^{2}+(8 - 7.5)^{2}+(9 - 7.5)^{2}+(10 - 7.5)^{2}+(11 - 7.5)^{2}\)
\(=(-3.5)^{2}+(-2.5)^{2}+(-1.5)^{2}+(-0.5)^{2}+0.5^{2}+1.5^{2}+2.5^{2}+3.5^{2}\)
\(=12.25 + 6.25+2.25+0.25+0.25+2.25+6.25+12.25 = 42\)
\(\sum_{i = 1}^{8}(y_{i}-\bar{y})^{2}=(95.2-86.55)^{2}+(93.4 - 86.55)^{2}+(92 - 86.55)^{2}+(86.8 - 86.55)^{2}+(85.2 - 86.55)^{2}+(77.6 - 86.55)^{2}+(78.8 - 86.55)^{2}+(73.4 - 86.55)^{2}\)
\(=8.65^{2}+6.85^{2}+5.45^{2}+0.25^{2}+(-1.35)^{2}+(-8.95)^{2}+(-7.75)^{2}+(-13.15)^{2}\)
\(=74.8225+46.9225+29.7025+0.0625+1.8225+80.1025+60.0625+173.9225 = 467.42\)

Step3: Calculate the correlation coefficient

\(r=\frac{-135.2}{\sqrt{42\times467.42}}=\frac{-135.2}{\sqrt{19631.64}}=\frac{-135.2}{140.113}\approx - 0.965\)

Answer:

\(-0.965\)