Question

use a graphing calculator and linear regression to write the equation of a best-fit line for the data in slope-intercept form. round to the nearest hundredth.
x | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8
y | 1.5 | 2.6 | 1.5 | 3.3 | 1.8 | 3.5 | 3.0 | 5.1

Explanation:

Step1: Calculate $\bar{x}$ (mean of x-values)

First, sum all x-values: $1+2+3+4+5+6+7+8=36$
Then divide by the number of data points $n=8$:
$\bar{x}=\frac{36}{8}=4.5$

Step2: Calculate $\bar{y}$ (mean of y-values)

Sum all y-values: $1.5+2.6+1.5+3.3+1.8+3.5+3.0+5.1=22.3$
Divide by $n=8$:
$\bar{y}=\frac{22.3}{8}=2.7875$

Step3: Calculate slope $m$

Use the formula:
$$m=\frac{\sum_{i=1}^{n}(x_i-\bar{x})(y_i-\bar{y})}{\sum_{i=1}^{n}(x_i-\bar{x})^2}$$
First compute numerator terms:
$(1-4.5)(1.5-2.7875)=(-3.5)(-1.2875)=4.50625$
$(2-4.5)(2.6-2.7875)=(-2.5)(-0.1875)=0.46875$
$(3-4.5)(1.5-2.7875)=(-1.5)(-1.2875)=1.93125$
$(4-4.5)(3.3-2.7875)=(-0.5)(0.5125)=-0.25625$
$(5-4.5)(1.8-2.7875)=(0.5)(-0.9875)=-0.49375$
$(6-4.5)(3.5-2.7875)=(1.5)(0.7125)=1.06875$
$(7-4.5)(3.0-2.7875)=(2.5)(0.2125)=0.53125$
$(8-4.5)(5.1-2.7875)=(3.5)(2.3125)=8.09375$
Sum numerator: $4.50625+0.46875+1.93125-0.25625-0.49375+1.06875+0.53125+8.09375=15.85$

Now compute denominator terms:
$(1-4.5)^2=12.25$, $(2-4.5)^2=6.25$, $(3-4.5)^2=2.25$, $(4-4.5)^2=0.25$
$(5-4.5)^2=0.25$, $(6-4.5)^2=2.25$, $(7-4.5)^2=6.25$, $(8-4.5)^2=12.25$
Sum denominator: $12.25+6.25+2.25+0.25+0.25+2.25+6.25+12.25=42$

Calculate slope:
$m=\frac{15.85}{42}\approx0.38$

Step4: Calculate intercept $b$

Use the formula $b=\bar{y}-m\bar{x}$
$b=2.7875 - (0.38)(4.5)$
$b=2.7875 - 1.71=1.0775\approx1.08$

Step5: Write slope-intercept form

Slope-intercept form is $y=mx+b$

Answer:

$y=0.38x+1.08$