General Idea:
We need to apply the below formula to find the trend line from the given x and y values.
[tex] (1)\; mean:\; \bar{x}\;=\;\frac{\sum x_i}{n}, \; \;\bar{y}\;=\;\frac{\sum y_i}{n}\\ \\ (2) \;trend\; line: \; y=A+Bx, \;B=\frac{S_{xy}}{S_{xx}}, \;\;A=\bar{y}-B\bar{x}\\ \\ Where:\\ \\ S_{xx}=\frac{\sum x_i^2}{n}-\bar{x}^2\\ \\S_{xy}=\frac{\sum x_iy_i}{n}-\bar{x} \cdot \bar{y} [/tex]
Applying the concept:
In the graph given the trend line passes through two points (1, 8) and (7, 1).
We need to make use of the two point formula to find the equation of trend line
[tex] m=\frac{y_2-y_1}{x_2-x_1} [/tex]
Substituting the points (1, 8) and (7, 1) in the above formula to find slope we get...
[tex] m = \frac{1-8}{7-1}=\frac{-7}{6} [/tex]
The point slope form of an equation which is used to write the equation of line when we know a point and slope is given below:
[tex] y-y_1=m(x-x_1) [/tex]
Substituting the values in the above formula we get ...
[tex] y-8=\frac{-7}{6}(x-1)\\Distributing \; -7/6\; in\; the\; right\; side\; of\; the\; equation \\ \\ y-8=\frac{-7}{6}x+\frac{7}{6}\\ Adding\; 8\; on\; both\; sides\\ \\ y-8+8=\frac{-7}{6} x+\frac{7}{6} +8\\ Combining \; like \; terms\\ \\ y=\frac{-7}{6}x+\frac{7}{6}+\frac{8 \cdot 6}{1 \cdot 6}\\ \\ y=\frac{-7}{6}x+\frac{7}{6}+\frac{48}{6}\\ \\ y=\frac{-7}{6} x+\frac{55}{6} [/tex]
Conclusion:
The equation best represents a trend line for the scatter plot is Option B.
B. y=−7/6x+55/6
