based on the data shown below, calculate the regression line (each value to at least two decimal places)\ny…

Explanation:

Step1: Calculate sums

Let (n = 14). Calculate (\sum_{i = 1}^{n}x_i=4 + 5+\cdots+17=147), (\sum_{i = 1}^{n}y_i=50.9+47.9+\cdots+19.2 = 458.5), (\sum_{i = 1}^{n}x_i^2=4^2 + 5^2+\cdots+17^2=1785), (\sum_{i = 1}^{n}x_iy_i=4\times50.9+5\times47.9+\cdots+17\times19.2 = 4307.3).

Step2: Calculate slope (m)

The formula for the slope (m) of the regression - line (y=mx + b) is (m=\frac{n\sum_{i = 1}^{n}x_iy_i-\sum_{i = 1}^{n}x_i\sum_{i = 1}^{n}y_i}{n\sum_{i = 1}^{n}x_i^2-(\sum_{i = 1}^{n}x_i)^2}). Substitute the values: (n = 14), (\sum_{i = 1}^{n}x_i = 147), (\sum_{i = 1}^{n}y_i = 458.5), (\sum_{i = 1}^{n}x_i^2 = 1785), (\sum_{i = 1}^{n}x_iy_i = 4307.3) into the formula. [ \begin{align*} m&=\frac{14\times4307.3-147\times458.5}{14\times1785 - 147^2}\ &=\frac{60302.2-67399.5}{24990 - 21609}\ &=\frac{-7097.3}{3381}\ &\approx - 2.099 \end{align*} ]

Step3: Calculate intercept (b)

The formula for the intercept (b) is (b=\overline{y}-m\overline{x}), where (\overline{x}=\frac{\sum_{i = 1}^{n}x_i}{n}=\frac{147}{14}=10.5) and (\overline{y}=\frac{\sum_{i = 1}^{n}y_i}{n}=\frac{458.5}{14}\approx32.75). [ \begin{align*} b&=32.75-(-2.099)\times10.5\ &=32.75 + 22.0395\ &\approx54.79 \end{align*} ]

Answer:

(y=-2.10x + 54.79)