rita is starting a running program. the table shows the total number of miles she runs in different weeks…

Explanation:

Step1: Calculate sums

Let (x) be the week number and (y) be the miles run. (n = 6) (number of data - points). (\sum_{i = 1}^{n}x_{i}=1 + 2+4 + 6+8 + 10=31). (\sum_{i = 1}^{n}y_{i}=5 + 8+13 + 15+19 + 20=80). (\sum_{i = 1}^{n}x_{i}^{2}=1^{2}+2^{2}+4^{2}+6^{2}+8^{2}+10^{2}=1 + 4+16+36+64+100 = 221). (\sum_{i = 1}^{n}x_{i}y_{i}=1\times5+2\times8 + 4\times13+6\times15+8\times19+10\times20=5+16+52+90+152+200 = 515).

Step2: Calculate slope (m)

The formula for the slope (m) of the line of best - fit is (m=\frac{n\sum_{i = 1}^{n}x_{i}y_{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}}). [ \begin{align*} m&=\frac{6\times515-31\times80}{6\times221 - 31^{2}}\ &=\frac{3090-2480}{1326 - 961}\ &=\frac{610}{365}\ &\approx1.671 \end{align*} ]

Step3: Calculate y - intercept (b)

The formula for the y - intercept (b) is (b=\bar{y}-m\bar{x}), where (\bar{x}=\frac{\sum_{i = 1}^{n}x_{i}}{n}=\frac{31}{6}\approx5.167) and (\bar{y}=\frac{\sum_{i = 1}^{n}y_{i}}{n}=\frac{80}{6}\approx13.333). [ \begin{align*} b&=13.333-1.671\times5.167\ &=13.333 - 8.634\ &\approx4.700 \end{align*} ]

Answer:

(y\approx1.671x + 4.700)