Question

8.

Determine a line of best fit for the distance, y (in feet), that a three-toed sloth travels after x minutes. [Round all coefficients to the nearest tenth.]
time x 0 1 2 3 4
distances travel y 0 712 19 25




(2 points)
The line of best fit is y =

x +
(round all coefficients to the nearest tenth)

Answer 1

To determine the line of best fit for the given data, we can use the method of linear regression. The data points provided are:

\[
\begin{align*}
(0, 0) \\
(1, 712) \\
(2, 19) \\
(3, 25) \\
(4, 0)
\end{align*}
\]

This gives us the following set of values:

- \(n = 5\) (number of points)
- \( \sum x = 0 + 1 + 2 + 3 + 4 = 10 \)
- \( \sum y = 0 + 712 + 19 + 25 + 0 = 756 \)
- \( \sum xy = (0 \cdot 0) + (1 \cdot 712) + (2 \cdot 19) + (3 \cdot 25) + (4 \cdot 0) = 0 + 712 + 38 + 75 + 0 = 825 \)
- \( \sum x^2 = 0^2 + 1^2 + 2^2 + 3^2 + 4^2 = 0 + 1 + 4 + 9 + 16 = 30 \)

We can use the formulas for the slope \(m\) and the y-intercept \(b\) of the line of best fit:

\[
m = \frac{n(\sum xy) - (\sum x)(\sum y)}{n(\sum x^2) - (\sum x)^2}
\]

\[
b = \frac{\sum y - m(\sum x)}{n}
\]

Now we can compute these values:

1. Calculate \(m\):

\[
m = \frac{5(825) - (10)(756)}{5(30) - (10)^2}
= \frac{4125 - 7560}{150 - 100}
= \frac{-3435}{50}
= -68.7
\]

2. Calculate \(b\):

\[
b = \frac{756 - (-68.7)(10)}{5}
= \frac{756 + 687}{5}
= \frac{1443}{5}
= 288.6
\]

Putting it all together, the equation of the line of best fit is:

\[
y = -68.7x + 288.6
\]

Rounding the coefficients to the nearest tenth, we find:

\[
\boxed{y = -68.7x + 288.6}
\]