Question

question 8
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linear regression
use a graphing calculator to find the linear regression equation for the line that best fits this data. round slope and y - intercept to two decimal places. write your final answer in a form of an equation y = mx + b
x 1 2 3 4 5 6
y 92 117 139 173 192 209
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Explanation:

Step1: Calculate sums

Let $n = 6$.
$\sum_{i = 1}^{n}x_i=1 + 2+3 + 4+5 + 6=21$
$\sum_{i = 1}^{n}y_i=92 + 117+139+173+192+209 = 922$
$\sum_{i = 1}^{n}x_i^2=1^2 + 2^2+3^2 + 4^2+5^2 + 6^2=1 + 4+9 + 16+25+36 = 91$
$\sum_{i = 1}^{n}x_iy_i=1\times92+2\times117 + 3\times139+4\times173+5\times192+6\times209=92+234+417+692+960+1254 = 3649$

Step2: Calculate slope $m$

The formula for the slope $m$ of the linear - regression line is $m=\frac{n\sum_{i = 1}^{n}x_iy_i-\sum_{i = 1}^{n}x_i\sum_{i = 1}^{n}y_i}{n\sum_{i = 1}^{n}x_i^2-(\sum_{i = 1}^{n}x_i)^2}$
Substitute the values:
\[

$$\begin{align*} m&=\frac{6\times3649-21\times922}{6\times91 - 21^2}\\ &=\frac{21894-19362}{546 - 441}\\ &=\frac{2532}{105}\\ &\approx24.11 \end{align*}$$

\]

Step3: Calculate y - intercept $b$

The formula for the y - intercept $b$ is $b=\frac{\sum_{i = 1}^{n}y_i-m\sum_{i = 1}^{n}x_i}{n}$
Substitute $m\approx24.11$, $\sum_{i = 1}^{n}x_i = 21$ and $\sum_{i = 1}^{n}y_i=922$ and $n = 6$
\[

$$\begin{align*} b&=\frac{922-24.11\times21}{6}\\ &=\frac{922 - 506.31}{6}\\ &=\frac{415.69}{6}\\ &\approx69.28 \end{align*}$$

\]

Answer:

$y = 24.11x+69.28$