Question

use linear regression to find a function that fits the following points: (-2, -1) (3, 9)

Explanation:

Step1: Recall linear regression formula

The linear regression equation is \( y = mx + b \), where \( m=\frac{n\sum xy - \sum x\sum y}{n\sum x^{2}-(\sum x)^{2}} \) and \( b=\frac{\sum y - m\sum x}{n} \), here \( n = 2 \) (number of points).

First, list the values:
For points \((-2,-4)\) and \((3,9)\):
\( x_1=-2,y_1 = - 4 \); \( x_2=3,y_2 = 9 \)
\(\sum x=-2 + 3=1\), \(\sum y=-4 + 9 = 5\)
\(\sum xy=(-2)\times(-4)+3\times9 = 8 + 27=35\)
\(\sum x^{2}=(-2)^{2}+3^{2}=4 + 9 = 13\)

Step2: Calculate slope \( m \)

Substitute into \( m \) formula:
\( m=\frac{2\times35-1\times5}{2\times13-(1)^{2}}=\frac{70 - 5}{26 - 1}=\frac{65}{25}=\frac{13}{5}=2.6 \)

Step3: Calculate intercept \( b \)

Substitute \( m=\frac{13}{5} \), \(\sum x = 1\), \(\sum y=5\), \( n = 2 \) into \( b \) formula:
\( b=\frac{5-\frac{13}{5}\times1}{2}=\frac{\frac{25 - 13}{5}}{2}=\frac{\frac{12}{5}}{2}=\frac{6}{5}=1.2 \)

Step4: Write the linear function

So the linear function is \( y=\frac{13}{5}x+\frac{6}{5} \) or \( y = 2.6x+1.2 \)

Answer:

\( y = \frac{13}{5}x+\frac{6}{5} \) (or \( y = 2.6x + 1.2 \))