Answer to Question #229067 in Statistics and Probability for emily

Question

x1234y56912 Find the gradient of the line of best fit of the data.

Answer 1

$$\def\arraystretch{1.5} \begin{array}{c:c:c} x & y \\ \hline 1 & 5 \\ \hline 2 & 6 \\ \hline 3 & 9 \\ \hdashline 4 & 12 \end{array}$$

$$\bar{x}=\dfrac{\sum_ix_i}{n}=\dfrac{1+2+3+4}{4}=2.5$$

$$\bar{y}=\dfrac{\sum_iy_i}{n}=\dfrac{5+6+9+12}{4}=8$$

$$S_{xx}=\sum_i(x_i-\bar{x})^2=\sum_ix_i^2-n\cdot\bar{x}^2=5$$

$$S_{xy}=\sum_i(x_i-\bar{x})(y_i-\bar{y})=\sum_ix_iy_i-n\cdot\bar{x}\bar{y}=12$$

$$S_{yy}=\sum_i(y_i-\bar{y})^2=\sum_iy_i^2-n\cdot\bar{y}^2=30$$

$$gradient=\dfrac{S_{xy}}{S_{xx}}=\dfrac{12}{5}=2.4$$