Question

Janet is training for a marathon. She decides to track her time per mile, in minutes, for 8 weeks. Which equation best fits the data?

A graph is labeled as Marathon Training. The horizontal axis is labeled as Weeks and the vertical axis is labeled as Time per mile left parenthesis minutes right parenthesis. The values on the horizontal axis range from 0 to 10 in increments of 2 and the values on the vertical axis range from 0 to 10 in increments of 2. Nine points are marked on the graph with their coordinates approximately equal to (0, 10), (1, 9 decimal point 9), (2, 9 decimal point 6), (3, 9 decimal point 5), (4, 9 decimal point 4), (5, 9 decimal point 2), (6, 9), (7, 8 decimal point 7), and (8, 8 decimal point 6).
A.y=0.18x + 10.06
B.y=−0.18x + 10.06
C.y = 2x + 8.6
D.y = −2x + 8.6

Answer 1

Option B.
`y=-0.18x+10.06`

Explanation:

Given: The nine coordinates can be written as
`(0,10), (1,9.9),(2,9.6), (3,9.5), (4,9.4)`
`, (5,9.2), (6,9), (7,8.7), (8,8.6)`

The equation that best fits the data can be found using the formula
`y=ax+b` where a and b can be found using the formula

`a=(\left(N * \sum x y\right)-\left(\sum x * \sum y\right))/(\left(N * \sum x^(2)\right)-\left(\sum x * \sum x\right))` and
`b=(\left(\sum x^(2) * \sum y\right)-\left(\sum x * \sum x y\right))/(\left(N * \sum x^(2)\right)-\left(\sum x * \Sigma x\right))`

Here,
`N=9`

The values of a and b can be found using the values found from the table which is attached below. The values from the table are given by

`\sum x=36`,

`\sum y=83.9`,

`\sum x y=324.9`,

`\sum x^(2)=204`, and

`\sum y^(2)=784.07`

Now, we can find the values of a and b, by substituting these values in the formulas of a and b.

`\begin{aligned}a &=(\left(N * \sum x y\right)-\left(\sum x * \sum y\right))/(\left(N * \sum x^(2)\right)-\left(\sum x * \sum x\right)) \\&=((9 \cdot 324.9)-(36 \cdot 83.9))/((9 \cdot 204)-(36 \cdot 36)) \\&=-0.18\end{aligned}`

and

`\begin{aligned}b &=(\left(\sum x^(2) * \sum y\right)-\left(\sum x * \sum x y\right))/(\left(N * \sum x^(2)\right)-\left(\sum x * \sum x\right)) \\&=((204 \cdot 83.9)-(36 \cdot 315))/((9 \cdot 204)-(36 \cdot 36)) \\&=10.06\end{aligned}`

Thus, the values of a and b are found. Now, substituting the two values in the equation
`y=ax+b` , we get,

`y=-0.18x+10.06`

The correct option is
`y=-0.18x+10.06`