Question

x 1 2 3 4 5 6 7 8 9 10
y 95 91 88 85 70 78 74 70 67 50
correlation coefficient: r =
strength of correlation:
coefficient of determination: r² =
marginal change: a =
y - intercept: b =
regression equation:

Explanation:

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

Let \(x=\{1,2,\cdots,10\}\) and \(y = \{95,91,\cdots,50\}\).
The mean of \(x\), \(\bar{x}=\frac{1 + 2+\cdots+10}{10}=\frac{\sum_{i = 1}^{10}i}{10}=\frac{\frac{10\times(10 + 1)}{2}}{10}=5.5\)
The mean of \(y\), \(\bar{y}=\frac{95+91+\cdots+50}{10}=\frac{778}{10}=77.8\)

Step2: Calculate the numerator and denominator for the correlation - coefficient \(r\)

The formula for the correlation coefficient \(r\) is \(r=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sqrt{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}\sum_{i = 1}^{n}(y_{i}-\bar{y})^{2}}}\)
\(\sum_{i = 1}^{10}(x_{i}-\bar{x})(y_{i}-\bar{y})=(1 - 5.5)(95 - 77.8)+(2 - 5.5)(91 - 77.8)+\cdots+(10 - 5.5)(50 - 77.8)\)
\(=-4.5\times17.2-3.5\times13.2 - 2.5\times10.2-1.5\times7.2+(- 0.5)\times(-7.8)+0.5\times0.2+1.5\times(-3.8)+2.5\times(-7.8)+3.5\times(-10.8)+4.5\times(-27.8)\)
\(=-77.4-46.2-25.5 - 10.8 + 3.9+0.1-5.7-19.5-37.8-125.1=-339\)
\(\sum_{i = 1}^{10}(x_{i}-\bar{x})^{2}=(1 - 5.5)^{2}+(2 - 5.5)^{2}+\cdots+(10 - 5.5)^{2}\)
\(=(-4.5)^{2}+(-3.5)^{2}+(-2.5)^{2}+(-1.5)^{2}+(-0.5)^{2}+0.5^{2}+1.5^{2}+2.5^{2}+3.5^{2}+4.5^{2}\)
\(=20.25 + 12.25+6.25+2.25+0.25+0.25+2.25+6.25+12.25+20.25 = 82.5\)
\(\sum_{i = 1}^{10}(y_{i}-\bar{y})^{2}=(95 - 77.8)^{2}+(91 - 77.8)^{2}+\cdots+(50 - 77.8)^{2}\)
\(=17.2^{2}+13.2^{2}+10.2^{2}+7.2^{2}+(-7.8)^{2}+0.2^{2}+(-3.8)^{2}+(-7.8)^{2}+(-10.8)^{2}+(-27.8)^{2}\)
\(=295.84+174.24+104.04+51.84 + 60.84+0.04+14.44+60.84+116.64+772.84 = 1651.6\)
\(r=\frac{-339}{\sqrt{82.5\times1651.6}}\approx\frac{-339}{\sqrt{136257}}\approx\frac{-339}{369.13}\approx - 0.92\)

Step3: Determine the strength of the correlation

Since \(|r|\approx0.92\), the strength of the correlation is strong negative.

Step4: Calculate the coefficient of determination \(r^{2}\)

\(r^{2}=(-0.92)^{2}=0.8464\)

Step5: Calculate the marginal - change \(a\) (slope) and \(y\) - intercept \(b\) for the regression equation \(y = ax + b\)

The formula for \(a\) is \(a=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}\)
\(a=\frac{-339}{82.5}\approx - 4.11\)
\(b=\bar{y}-a\bar{x}=77.8-(-4.11)\times5.5=77.8 + 22.605=100.405\)
The regression equation is \(y=-4.11x + 100.405\)

Answer:

Correlation Coefficient: \(r\approx - 0.92\)
Strength of Correlation: Strong negative
Coefficient of Determination: \(r^{2}\approx0.8464\)
Marginal Change: \(a\approx - 4.11\)
\(y\) - intercept: \(b\approx100.405\)
Regression Equation: \(y=-4.11x + 100.405\)