data table\nheight, x\n62 58 67 68 59 65 67 57\niq score, y\n101 97 99 107 90 105 112 120\n(b) calculate the…

Explanation:

Step1: Calculate means

Let (x) be the height - values (x = [57,67,65,59,68,67,58,62]) and (y) be the IQ - score values (y=[120,112,105,90,107,99,97,101]). The mean of (x), (\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}), where (n = 8). (\sum_{i=1}^{8}x_{i}=57 + 67+65 + 59+68+67+58+62=543), so (\bar{x}=\frac{543}{8}=67.875). The mean of (y), (\bar{y}=\frac{\sum_{i = 1}^{n}y_{i}}{n}), (\sum_{i=1}^{8}y_{i}=120 + 112+105+90+107+99+97+101 = 831), so (\bar{y}=\frac{831}{8}=103.875).

Step2: Calculate numerator and denominator components

Calculate ((x_{i}-\bar{x})(y_{i}-\bar{y})), ((x_{i}-\bar{x})^2) and ((y_{i}-\bar{y})^2) for each (i) from (1) to (8). (\sum_{i = 1}^{8}(x_{i}-\bar{x})(y_{i}-\bar{y})=(57 - 67.875)(120 - 103.875)+(67 - 67.875)(112 - 103.875)+\cdots+(62 - 67.875)(101 - 103.875)) (=( - 10.875)\times16.125+( - 0.875)\times8.125+\cdots+( - 5.875)\times( - 2.875)) (=-175.3125-7.03125+\cdots + 16.8828125=-107.875). (\sum_{i = 1}^{8}(x_{i}-\bar{x})^2=(57 - 67.875)^2+(67 - 67.875)^2+\cdots+(62 - 67.875)^2) (=118.265625 + 0.765625+\cdots+34.515625 = 147.875). (\sum_{i = 1}^{8}(y_{i}-\bar{y})^2=(120 - 103.875)^2+(112 - 103.875)^2+\cdots+(101 - 103.875)^2) (=260.015625+66.015625+\cdots + 8.265625 = 798.875).

Step3: Calculate correlation coefficient (r)

The formula for the sample correlation coefficient 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}}) (r=\frac{-107.875}{\sqrt{147.875\times798.875}}=\frac{-107.875}{\sqrt{118085.984375}}) (=\frac{-107.875}{343.63627}\approx - 0.314)

Answer:

(r\approx - 0.314)