Question

rita is starting a running program. the table shows the total number of miles she runs in different weeks. what is the equation of the line of best fit for the data? state each number to the thousandths place. y = x +
week\tmiles run
1\t5
2\t8
4\t13
6\t15
8\t19
10\t20

Explanation:

Step1: Calculate sums

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

Step2: Calculate slope \(m\)

The formula for the slope \(m\) of the line of best - fit is \(m=\frac{n\sum_{i = 1}^{n}x_{i}y_{i}-\sum_{i = 1}^{n}x_{i}\sum_{i = 1}^{n}y_{i}}{n\sum_{i = 1}^{n}x_{i}^{2}-(\sum_{i = 1}^{n}x_{i})^{2}}\).
Substitute the values:
\[

$$\begin{align*} m&=\frac{6\times515-31\times80}{6\times221 - 31^{2}}\\ &=\frac{3090-2480}{1326 - 961}\\ &=\frac{610}{365}\\ &\approx1.6712 \end{align*}$$

\]

Step3: Calculate y - intercept \(b\)

The formula for the y - intercept \(b\) is \(b=\frac{\sum_{i = 1}^{n}y_{i}-m\sum_{i = 1}^{n}x_{i}}{n}\).
Substitute \(m\approx1.6712\), \(\sum_{i = 1}^{n}x_{i}=31\), \(\sum_{i = 1}^{n}y_{i}=80\) and \(n = 6\):
\[

$$\begin{align*} b&=\frac{80-1.6712\times31}{6}\\ &=\frac{80 - 51.8072}{6}\\ &=\frac{28.1928}{6}\\ &\approx4.6988 \end{align*}$$

\]

Answer:

\(y = 1.6712x+4.6988\)