Question

3.3.3 quiz: regression methods what is the equation of the line of best fit for the following data? round the slope and y - intercept of the line to three decimal places. x y 2 2 5 8 7 10 9 11 11 13

Explanation:

Step1: Calculate sums

Let \(n = 5\).
\(\sum_{i = 1}^{n}x_i=2 + 5+7 + 9+11=34\)
\(\sum_{i = 1}^{n}y_i=2 + 8+10 + 11+13=44\)
\(\sum_{i = 1}^{n}x_i^2=2^2+5^2+7^2 + 9^2+11^2=4 + 25+49+81+121 = 280\)
\(\sum_{i = 1}^{n}x_iy_i=2\times2+5\times8+7\times10 + 9\times11+11\times13=4+40+70+99+143 = 356\)

Step2: Calculate slope \(m\)

The formula for the slope \(m\) of the line of best - fit is \(m=\frac{n\sum_{i = 1}^{n}x_iy_i-\sum_{i = 1}^{n}x_i\sum_{i = 1}^{n}y_i}{n\sum_{i = 1}^{n}x_i^2-(\sum_{i = 1}^{n}x_i)^2}\)
Substitute the values:
\[

$$\begin{align*} m&=\frac{5\times356-34\times44}{5\times280 - 34^2}\\ &=\frac{1780-1496}{1400 - 1156}\\ &=\frac{284}{244}\\ &\approx1.164 \end{align*}$$

\]

Step3: Calculate y - intercept \(b\)

The formula for the y - intercept \(b\) is \(b=\frac{\sum_{i = 1}^{n}y_i-m\sum_{i = 1}^{n}x_i}{n}\)
Substitute \(m\approx1.164\), \(\sum_{i = 1}^{n}x_i = 34\), \(\sum_{i = 1}^{n}y_i=44\) and \(n = 5\)
\[

$$\begin{align*} b&=\frac{44-1.164\times34}{5}\\ &=\frac{44 - 39.576}{5}\\ &=\frac{4.424}{5}\\ &=0.885 \end{align*}$$

\]

Step4: Write the equation of the line

The equation of the line of best - fit is \(y=mx + b\), so \(y = 1.164x+0.885\)

Answer:

\(y = 1.164x+0.885\)