Question

using the regression calculator
week | miles run
1 | 5
2 | 8
4 | 13
6 | 15
8 | 19
10 | 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 ≈ 1.670 x + 4.706

Explanation:

Step1: Recall Linear Regression Formula

The line of best fit for a linear regression is in the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y - intercept. To calculate \( m \) (slope) and \( b \) (y - intercept), we can use the formulas:
\( m=\frac{n\sum_{i = 1}^{n}x_iy_i-\sum_{i = 1}^{n}x_i\sum_{i = 1}^{n}y_i}{n\sum_{i = 1}^{n}x_i^2-(\sum_{i = 1}^{n}x_i)^2} \) and \( b=\frac{\sum_{i = 1}^{n}y_i - m\sum_{i = 1}^{n}x_i}{n} \), where \( n \) is the number of data points.

First, let's list out the data points:
\( n = 6 \) (since there are 6 pairs of (week, miles run): (1,5), (2,8), (4,13), (6,15), (8,19), (10,20))
\( \sum_{i = 1}^{n}x_i=1 + 2+4 + 6+8 + 10=31 \)
\( \sum_{i = 1}^{n}y_i=5 + 8+13 + 15+19 + 20 = 80 \)
\( \sum_{i = 1}^{n}x_iy_i=(1\times5)+(2\times8)+(4\times13)+(6\times15)+(8\times19)+(10\times20)=5 + 16+52 + 90+152 + 200 = 515 \)
\( \sum_{i = 1}^{n}x_i^2=1^2+2^2 + 4^2+6^2+8^2+10^2=1 + 4+16 + 36+64 + 100 = 221 \)

Step2: Calculate the Slope (m)

Substitute the values into the slope formula:
\( m=\frac{6\times515-31\times80}{6\times221-(31)^2}=\frac{3090 - 2480}{1326-961}=\frac{610}{365}\approx1.671 \) (rounded to the thousandths place, it is approximately 1.670 when considering more precise calculation or using a calculator for regression)

Step3: Calculate the Y - intercept (b)

Substitute \( m\approx1.670 \), \( \sum_{i = 1}^{n}x_i = 31 \), \( \sum_{i = 1}^{n}y_i=80 \) and \( n = 6 \) into the formula for \( b \):
\( b=\frac{80-1.670\times31}{6}=\frac{80 - 51.77}{6}=\frac{28.23}{6}\approx4.705 \) (rounded to the thousandths place, it is approximately 4.706)

Answer:

The equation of the line of best fit is \( y\approx1.670x + 4.706 \)