Question

A 2-column table with 6 rows. The first column is labeled week with entries 1, 2, 4, 6, 8, 10. The second column is labeled miles run with entries 5, 8, 13, 15, 19, 20.
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 +

Answer 1

To find the equation of the line of best fit (linear regression) for the given data, we first define the variables based on the provided data:

- Let \( x \) represent the weeks (1, 2, 4, 6, 8, 10).
- Let \( y \) represent the miles run (5, 8, 13, 15, 19, 20).

The points we have are:
- (1, 5)
- (2, 8)
- (4, 13)
- (6, 15)
- (8, 19)
- (10, 20)

Next, we will calculate the slope \( m \) and the y-intercept \( b \) for the line of best fit using the formulas:

1. **Slope \( m \)**:
\[
m = \frac{N(\sum xy) - (\sum x)(\sum y)}{N(\sum x^2) - (\sum x)^2}
\]

2. **Y-intercept \( b \)**:
\[
b = \frac{\sum y - m(\sum x)}{N}
\]

Where \( N \) is the number of data points.

Now, let's calculate the required values:

- \( N = 6 \)
- \( \sum x = 1 + 2 + 4 + 6 + 8 + 10 = 31 \)
- \( \sum y = 5 + 8 + 13 + 15 + 19 + 20 = 80 \)
- \( \sum xy = (1)(5) + (2)(8) + (4)(13) + (6)(15) + (8)(19) + (10)(20) = 5 + 16 + 52 + 90 + 152 + 200 = 465 \)
- \( \sum x^2 = 1^2 + 2^2 + 4^2 + 6^2 + 8^2 + 10^2 = 1 + 4 + 16 + 36 + 64 + 100 = 221 \)

Now substituting the values into the slope formula:

\[
m = \frac{6(465) - (31)(80)}{6(221) - (31)^2}
\]

Calculating the numerator:

\[
6(465) = 2790
\]
\[
31 \times 80 = 2480
\]
\[
\text{Numerator: } 2790 - 2480 = 310
\]

Calculating the denominator:

\[
6(221) = 1326
\]
\[
31^2 = 961
\]
\[
\text{Denominator: } 1326 - 961 = 365
\]

So,

\[
m = \frac{310}{365} \approx 0.849
\]

Now, let's calculate \( b \):

\[
b = \frac{80 - 0.849(31)}{6}
\]

Calculating \( 0.849 \times 31 \):

\[
0.849 \times 31 \approx 26.319
\]

Now, substituting back into \( b \):

\[
b = \frac{80 - 26.319}{6} \approx \frac{53.681}{6} \approx 8.946
\]

Now, we can summarize the equation of the line of best fit:

\[
y \approx 0.849x + 8.946
\]

So, the final answer is:

\[
y \approx 0.849x + 8.946
\]