y = 1.61x - 17.70
Explanations:To calculate the line of best fit from a table, follow the following steps:
Step 1:Caculate the means of x and y
[tex]\begin{gathered} \bar{x}=\text{ }\frac{\sum x}{n} \\ \bar{x}=\text{ }\frac{5+6+14+15+21+23+24}{7} \\ \bar{x}\text{ = }\frac{108}{7} \\ \bar{x}\text{ = }15.43 \end{gathered}[/tex][tex]\begin{gathered} \bar{y}\text{ = }\frac{\sum y}{n} \\ \bar{y}\text{ = }\frac{-14-4+6+7+15+17+23}{7} \\ \bar{y}\text{ = }\frac{50}{7} \\ \bar{y}\text{ = }7.14 \end{gathered}[/tex]Step 2: Calculate the slope of the line of best fit
[tex]m\text{ = }\frac{\sum (x_i-\bar{x})(y_i-\bar{y})}{\sum (x_i-\bar{x})^2}[/tex][tex]m\text{ = }\frac{(5-15.43)(-14-7.14)+(6-15.43)(-4-7.14)+(14-15.43)(6-7.14)+(15-15.43)(7-7.14)+(21-15.43)(15-7.14)+(23-15.43)(17-7.14)+(24-15.43)(23-7.14)}{(5-15.43)^2+(6-15.43)^2+(14-15.43)^2+(15-15.43)^2+(21-15.43)^2+(23-15.43)^2+(24-15.43)^2}[/tex][tex]\begin{gathered} m\text{ = }\frac{581.57}{361.71} \\ m\text{ = }1.61 \end{gathered}[/tex]Step 3: Calculate the y-intercept, c, using the formula below:
[tex]\begin{gathered} c\text{ = }\bar{y}-m\bar{x} \\ c\text{ = 7.14-1.61(15.43)} \\ c\text{ = }-17.70 \end{gathered}[/tex]Step 4: Substitute the values of m and c into the slope - intercept form of the equation of a line:
y = mx + c
y = 1.61x - 17.70
The equation of the line of best fit is y = 1.61x - 17.70
