Question

based on the data shown below, calculate the regression line (each value to at least two decimal places)
y = x +

x	y
5	47.9
6	46.6
7	42.8
8	42.7
9	37.8
10	36.1
11	33
12	32.7
13	29.8
14	24.6
15	25.9
16	22.8
17	19.2

Explanation:

Step1: Calculate sums

Let \(n = 14\). Calculate \(\sum_{i = 1}^{n}x_i=4 + 5+\cdots+17=147\), \(\sum_{i = 1}^{n}y_i=50.9+47.9+\cdots+19.2 = 458.5\), \(\sum_{i = 1}^{n}x_i^2=4^2 + 5^2+\cdots+17^2=1785\), \(\sum_{i = 1}^{n}x_iy_i=4\times50.9+5\times47.9+\cdots+17\times19.2 = 4307.3\).

Step2: Calculate slope \(m\)

The formula for the slope \(m\) of the regression - line \(y=mx + b\) 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: \(n = 14\), \(\sum_{i = 1}^{n}x_i = 147\), \(\sum_{i = 1}^{n}y_i = 458.5\), \(\sum_{i = 1}^{n}x_i^2 = 1785\), \(\sum_{i = 1}^{n}x_iy_i = 4307.3\) into the formula.
\[

$$\begin{align*} m&=\frac{14\times4307.3-147\times458.5}{14\times1785 - 147^2}\\ &=\frac{60302.2-67399.5}{24990 - 21609}\\ &=\frac{-7097.3}{3381}\\ &\approx - 2.099 \end{align*}$$

\]

Step3: Calculate intercept \(b\)

The formula for the intercept \(b\) is \(b=\overline{y}-m\overline{x}\), where \(\overline{x}=\frac{\sum_{i = 1}^{n}x_i}{n}=\frac{147}{14}=10.5\) and \(\overline{y}=\frac{\sum_{i = 1}^{n}y_i}{n}=\frac{458.5}{14}\approx32.75\).
\[

$$\begin{align*} b&=32.75-(-2.099)\times10.5\\ &=32.75 + 22.0395\\ &\approx54.79 \end{align*}$$

\]

Answer:

\(y=-2.10x + 54.79\)