module 3 review (optional)
score: 6/100 answered: 1/15
question 2
linear regression
use linear regression to find the equation for the linear function that best fits this data. round both
numbers to two decimal places. write your final answer in a form of an equation ( y = mx + b )
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Question

Accepted Answer

# Explanation:
## Step1: Calculate the means of $x$ and $y$
The mean of $x$ values: $\bar{x}=\frac{1 + 2+3+4+5+6}{6}=\frac{21}{6}=3.5$
The mean of $y$ values: $\bar{y}=\frac{91 + 107+131+151+181+201}{6}=\frac{862}{6}\approx143.67$
## Step2: Calculate the slope $m$
$$
\begin{align*}
m&=\frac{\sum_{i = 1}^{n}(x_{i}-\bar{x})(y_{i}-\bar{y})}{\sum_{i = 1}^{n}(x_{i}-\bar{x})^{2}}\
\sum_{i = 1}^{6}(x_{i}-\bar{x})(y_{i}-\bar{y})&=(1 - 3.5)(91-143.67)+(2 - 3.5)(107 - 143.67)+(3 - 3.5)(131-143.67)+(4 - 3.5)(151-143.67)+(5 - 3.5)(181-143.67)+(6 - 3.5)(201-143.67)\
&=(- 2.5)(-52.67)+(-1.5)(-36.67)+(-0.5)(-12.67)+(0.5)(7.33)+(1.5)(37.33)+(2.5)(57.33)\
&=131.675 + 55.005+6.335 + 3.665+56.0 + 143.325\
&=396\
\sum_{i = 1}^{6}(x_{i}-\bar{x})^{2}&=(1 - 3.5)^{2}+(2 - 3.5)^{2}+(3 - 3.5)^{2}+(4 - 3.5)^{2}+(5 - 3.5)^{2}+(6 - 3.5)^{2}\
&=(-2.5)^{2}+(-1.5)^{2}+(-0.5)^{2}+(0.5)^{2}+(1.5)^{2}+(2.5)^{2}\
&=6.25+2.25 + 0.25+0.25+2.25+6.25\
&=17.5\
m&=\frac{396}{17.5}\approx22.63
\end{align*}
$$
## Step3: Calculate the $y -$ intercept $b$
Using the formula $b=\bar{y}-m\bar{x}$, substitute $m = 22.63$ and $\bar{x}=3.5$, $\bar{y}=143.67$
$b=143.67-22.63\times3.5=143.67 - 79.205=64.465\approx64.47$

# Answer:
$y = 22.63x+64.47$

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QUESTION IMAGE

Question

Explanation:

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

Step2: Calculate the slope \(m\)

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

Answer: