Question

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
5	4
6	6
9	9
10	11
14	12

a. y = 0.894x + 0.535
b. y = - 0.535x + 0.894
c. y = - 0.894x + 0.535
d. y = 0.535x + 0.894

Explanation:

Step1: Calculate the means of x and y

Let $x = [5,6,9,10,14]$, $y = [4,6,9,11,12]$.
$\bar{x}=\frac{5 + 6+9+10+14}{5}=\frac{44}{5}=8.8$
$\bar{y}=\frac{4 + 6+9+11+12}{5}=\frac{42}{5}=8.4$

Step2: Calculate the numerator and denominator for the slope

$numerator=\sum_{i = 1}^{n}(x_i-\bar{x})(y_i - \bar{y})$
$=(5 - 8.8)(4 - 8.4)+(6 - 8.8)(6 - 8.4)+(9 - 8.8)(9 - 8.4)+(10 - 8.8)(11 - 8.4)+(14 - 8.8)(12 - 8.4)$
$=(-3.8)\times(-4.4)+(-2.8)\times(-2.4)+(0.2)\times(0.6)+(1.2)\times(2.6)+(5.2)\times(3.6)$
$=16.72+6.72 + 0.12+3.12+18.72$
$=45.4$
$denominator=\sum_{i = 1}^{n}(x_i-\bar{x})^2$
$=(5 - 8.8)^2+(6 - 8.8)^2+(9 - 8.8)^2+(10 - 8.8)^2+(14 - 8.8)^2$
$=(-3.8)^2+(-2.8)^2+(0.2)^2+(1.2)^2+(5.2)^2$
$=14.44+7.84+0.04+1.44+27.04$
$=50.8$
The slope $m=\frac{numerator}{denominator}=\frac{45.4}{50.8}\approx0.894$

Step3: Calculate the y - intercept

Using the formula $b=\bar{y}-m\bar{x}$, we substitute $m = 0.894$, $\bar{x}=8.8$ and $\bar{y}=8.4$
$b=8.4-0.894\times8.8$
$=8.4 - 7.8672$
$=0.533\approx0.535$
The equation of the line of best - fit is $y = 0.894x+0.535$

Answer:

A. $y = 0.894x+0.535$