find the pearson correlation coefficient r for the given points. round any intermediate calculations to no…

Explanation:

Step1: Calculate means

Let (x = [1,2,3,4,5,6,7]) and (y=[2,3,9,3,10,9,9]). The mean of (x), (\bar{x}=\frac{1 + 2+3+4+5+6+7}{7}=\frac{28}{7}=4). The mean of (y), (\bar{y}=\frac{2 + 3+9+3+10+9+9}{7}=\frac{45}{7}\approx6.428571).

Step2: Calculate numerator

[ \begin{align*} S_{xy}&=\sum_{i = 1}^{n}(x_i-\bar{x})(y_i - \bar{y})\ &=(1 - 4)(2-\frac{45}{7})+(2 - 4)(3-\frac{45}{7})+(3 - 4)(9-\frac{45}{7})+(4 - 4)(3-\frac{45}{7})+(5 - 4)(10-\frac{45}{7})+(6 - 4)(9-\frac{45}{7})+(7 - 4)(9-\frac{45}{7})\ &=(- 3)(2-\frac{45}{7})+(-2)(3-\frac{45}{7})+(-1)(9-\frac{45}{7})+0+(1)(10-\frac{45}{7})+(2)(9-\frac{45}{7})+(3)(9-\frac{45}{7})\ &=(-3)\times\frac{14 - 45}{7}+(-2)\times\frac{21 - 45}{7}+(-1)\times\frac{63 - 45}{7}+0+(1)\times\frac{70 - 45}{7}+(2)\times\frac{63 - 45}{7}+(3)\times\frac{63 - 45}{7}\ &=(-3)\times\frac{-31}{7}+(-2)\times\frac{-24}{7}+(-1)\times\frac{18}{7}+0+(1)\times\frac{25}{7}+(2)\times\frac{18}{7}+(3)\times\frac{18}{7}\ &=\frac{93}{7}+\frac{48}{7}-\frac{18}{7}+0+\frac{25}{7}+\frac{36}{7}+\frac{54}{7}\ &=\frac{93 + 48-18+0+25+36+54}{7}\ &=\frac{238}{7}=34 \end{align*} ]

Step3: Calculate denominator - part 1

[ \begin{align*} S_{xx}&=\sum_{i = 1}^{n}(x_i-\bar{x})^2\ &=(1 - 4)^2+(2 - 4)^2+(3 - 4)^2+(4 - 4)^2+(5 - 4)^2+(6 - 4)^2+(7 - 4)^2\ &=(-3)^2+(-2)^2+(-1)^2+0^2+1^2+2^2+3^2\ &=9 + 4+1+0+1+4+9\ &=28 \end{align*} ]

Step4: Calculate denominator - part 2

[ \begin{align*} S_{yy}&=\sum_{i = 1}^{n}(y_i-\bar{y})^2\ &=(2-\frac{45}{7})^2+(3-\frac{45}{7})^2+(9-\frac{45}{7})^2+(3-\frac{45}{7})^2+(10-\frac{45}{7})^2+(9-\frac{45}{7})^2+(9-\frac{45}{7})^2\ &=(\frac{14 - 45}{7})^2+(\frac{21 - 45}{7})^2+(\frac{63 - 45}{7})^2+(\frac{21 - 45}{7})^2+(\frac{70 - 45}{7})^2+(\frac{63 - 45}{7})^2+(\frac{63 - 45}{7})^2\ &=(\frac{-31}{7})^2+(\frac{-24}{7})^2+(\frac{18}{7})^2+(\frac{-24}{7})^2+(\frac{25}{7})^2+(\frac{18}{7})^2+(\frac{18}{7})^2\ &=\frac{961}{49}+\frac{576}{49}+\frac{324}{49}+\frac{576}{49}+\frac{625}{49}+\frac{324}{49}+\frac{324}{49}\ &=\frac{961+576 + 324+576+625+324+324}{49}\ &=\frac{3710}{49}\approx75.714286 \end{align*} ]

Step5: Calculate correlation coefficient

The Pearson correlation coefficient (r=\frac{S_{xy}}{\sqrt{S_{xx}S_{yy}}}=\frac{34}{\sqrt{28\times\frac{3710}{49}}}=\frac{34}{\sqrt{2120}}\approx\frac{34}{46.043458}\approx0.738 ]

Answer:

(0.738)