Question

Maleia is tracking her running training program. The table gives her 5K run time at the end of each month.


Month Time (minutes)
1 46
2 42
3 40
4 41
5 38
6 36


What is the equation for the line of best fit where x represents the month and y represents the time?
y = −1.74x + 46.6
y = −1.74x + 36.2
y = 1.74x + 46.6
y = 1.74x + 36.2

Answer 1

To find the equation for the line of best fit, we can use the method of linear regression. Given the data:

\[
\begin{align*}
\text{Month} (x) & : \{1, 2, 3, 4, 5, 6\} \\
\text{Time (minutes)} (y) & : \{46, 42, 40, 41, 38, 36\}
\end{align*}
\]

We need to calculate the slope (m) and the y-intercept (b) of the line, which can be done with the formulas for linear regression.

1. Calculate the averages:
- \( \bar{x} = \frac{1+2+3+4+5+6}{6} = 3.5 \)
- \( \bar{y} = \frac{46+42+40+41+38+36}{6} = 40.5 \)

2. Calculate \( m \) (slope):
\[
m = \frac{\sum (x_i - \bar{x})(y_i - \bar{y})}{\sum (x_i - \bar{x})^2}
\]

Compute the necessary sums:
- \( \sum (x_i - \bar{x})(y_i - \bar{y}) = (1-3.5)(46-40.5) + (2-3.5)(42-40.5) + (3-3.5)(40-40.5) + (4-3.5)(41-40.5) + (5-3.5)(38-40.5) + (6-3.5)(36-40.5) \)
- This gives:
- \( = (-2.5)(5.5) + (-1.5)(1.5) + (-0.5)(-0.5) + (0.5)(0.5) + (1.5)(-2.5) + (2.5)(-4.5) \)
- \( = -13.75 - 2.25 + 0.25 + 0.25 - 3.75 - 11.25 = -30.5 \)

- \( \sum (x_i - \bar{x})^2 = (1-3.5)^2 + (2-3.5)^2 + (3-3.5)^2 + (4-3.5)^2 + (5-3.5)^2 + (6-3.5)^2 \)
- This gives:
- \( = 6.25 + 2.25 + 0.25 + 0.25 + 2.25 + 6.25 = 17.5 \)

Then, calculating \( m \):
\[
m = \frac{-30.5}{17.5} \approx -1.743 \approx -1.74
\]

3. Calculate \( b \) (y-intercept):
\[
b = \bar{y} - m \bar{x} = 40.5 - (-1.74)(3.5) \approx 40.5 + 6.09 \approx 46.59 \approx 46.6
\]

Now we have:
\[
y \approx -1.74x + 46.6
\]

Thus, the equation for the line of best fit is:
\[
\boxed{y = -1.74x + 46.6}
\]