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

Explanation:

Step1: Calculate the means of x and y

Let (x) be the week number and (y) be the miles - run. (n = 6) (\bar{x}=\frac{1 + 2+4 + 6+8 + 10}{6}=\frac{31}{6}\approx5.167) (\bar{y}=\frac{5 + 8+13 + 15+19 + 20}{6}=\frac{80}{6}\approx13.333)

Step2: Calculate the slope (m)

The formula for the slope (m) of the line of best - fit is (m=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}) (\sum_{i = 1}^{6}(x_{i}-\bar{x})(y_{i}-\bar{y})=(1 - 5.167)(5 - 13.333)+(2 - 5.167)(8 - 13.333)+(4 - 5.167)(13 - 13.333)+(6 - 5.167)(15 - 13.333)+(8 - 5.167)(19 - 13.333)+(10 - 5.167)(20 - 13.333)) (=(-4.167)(-8.333)+(-3.167)(-5.333)+(-1.167)(-0.333)+(0.833)(1.667)+(2.833)(5.667)+(4.833)(6.667)) (=34.729+16.895 + 0.389+1.389+16.078+32.229) (=101.719) (\sum_{i = 1}^{6}(x_{i}-\bar{x})^{2}=(1 - 5.167)^{2}+(2 - 5.167)^{2}+(4 - 5.167)^{2}+(6 - 5.167)^{2}+(8 - 5.167)^{2}+(10 - 5.167)^{2}) (=(-4.167)^{2}+(-3.167)^{2}+(-1.167)^{2}+(0.833)^{2}+(2.833)^{2}+(4.833)^{2}) (=17.364+10.029+1.362+0.694+8.026+23.358) (=60.833) (m=\frac{101.719}{60.833}\approx1.672)

Step3: Calculate the y - intercept (b)

We know that (y=mx + b), substituting (\bar{x}) and (\bar{y}) and (m) into the equation: (13.333=1.672\times5.167 + b) (13.333 = 8.638+b) (b=13.333 - 8.638=4.695)

Answer:

(y\approx1.672x + 4.695)