(3,4),(4,6),(5,7),(6,6),(7,8),(8,10)
given the coordinates provided, write the equation of the best fit line.

Question

Accepted Answer

# Explanation:
## Step1: Calculate the means of $x$ and $y$ values
Let $x$ values be $x_1 = 3,x_2=4,x_3 = 5,x_4=6,x_5 = 7,x_6=8$ and $y$ values be $y_1 = 4,y_2=6,y_3 = 7,y_4=6,y_5 = 8,y_6=10$
The mean of $x$ values $\bar{x}=\frac{3 + 4+5+6+7+8}{6}=\frac{33}{6}=5.5$
The mean of $y$ values $\bar{y}=\frac{4 + 6+7+6+8+10}{6}=\frac{41}{6}\approx6.83$
## Step2: Calculate the slope $m$
$$
\begin{align*}
m&=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})(y_i-\bar{y})}{\sum_{i = 1}^{n}(x_i-\bar{x})^2}\
(x_1-\bar{x})(y_1-\bar{y})&=(3 - 5.5)(4-6.83)=(- 2.5)(-2.83)=7.075\
(x_2-\bar{x})(y_2-\bar{y})&=(4 - 5.5)(6 - 6.83)=(-1.5)(-0.83)=1.245\
(x_3-\bar{x})(y_3-\bar{y})&=(5 - 5.5)(7 - 6.83)=(-0.5)(0.17)=-0.085\
(x_4-\bar{x})(y_4-\bar{y})&=(6 - 5.5)(6 - 6.83)=(0.5)(-0.83)=-0.415\
(x_5-\bar{x})(y_5-\bar{y})&=(7 - 5.5)(8 - 6.83)=(1.5)(1.17)=1.755\
(x_6-\bar{x})(y_6-\bar{y})&=(8 - 5.5)(10 - 6.83)=(2.5)(3.17)=7.925\
\sum_{i = 1}^{6}(x_i-\bar{x})(y_i-\bar{y})&=7.075+1.245-0.085 - 0.415+1.755+7.925=17.5\
(x_1-\bar{x})^2&=(3 - 5.5)^2=6.25\
(x_2-\bar{x})^2&=(4 - 5.5)^2=2.25\
(x_3-\bar{x})^2&=(5 - 5.5)^2=0.25\
(x_4-\bar{x})^2&=(6 - 5.5)^2=0.25\
(x_5-\bar{x})^2&=(7 - 5.5)^2=2.25\
(x_6-\bar{x})^2&=(8 - 5.5)^2=6.25\
\sum_{i = 1}^{6}(x_i-\bar{x})^2&=6.25+2.25+0.25+0.25+2.25+6.25 = 17.5\
m&=\frac{17.5}{17.5}=1
\end{align*}
$$
## Step3: Calculate the $y$ - intercept $b$
Using the formula $y=mx + b$, substitute $m = 1,x = 5.5,y=6.83$
$6.83=1\times5.5+b$
$b=6.83 - 5.5=1.33$

# Answer:
The equation of the best - fit line is $y=x + 1.33$

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QUESTION IMAGE

Question

Explanation:

Step1: Calculate the means of \(x\) and \(y\) values

Step2: Calculate the slope \(m\)

Step3: Calculate the \(y\) - intercept \(b\)

Answer: