Question 1193517
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Refer to this page to refresh your memory on the formulas needed
<a href = "https://www.public.asu.edu/~gwaissi/ASM-e-book/module208.html">https://www.public.asu.edu/~gwaissi/ASM-e-book/module208.html</a>
Specifically check out the section labeled "4. Calculate the sums of squares"


Also, check out the bottom of page 2 on this PDF
<a href = "https://www2.isye.gatech.edu/~hsharp/math3070/section3_4.pdf">https://www2.isye.gatech.edu/~hsharp/math3070/section3_4.pdf</a>
I'm talking about the formula
*[tex \Large SS_{xy} = \sum xy - \frac{(\sum x)(\sum y)}{n}]
Through substitution we can say that
*[tex \Large SS_{yy} = \sum y*y - \frac{(\sum y)(\sum y)}{n}]
*[tex \Large SS_{yy} = \sum y^2 - \frac{(\sum y)^2}{n}]


As a final reference page, check out the top of page 2 of this PDF
<a href = "https://users.stat.ufl.edu/~winner/statnotescomp/regression.pdf">https://users.stat.ufl.edu/~winner/statnotescomp/regression.pdf</a>
where it reads *[tex \large \text{The (unbiased) estimate of the error variance } \sigma^2 \text{ is } s^2 = \text{MSE} = \frac{\text{SSE}}{n-2}]


Now onto the problem at hand.


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I will drop the subscript 'i' to simplify things a bit
So instead of saying something like
*[tex \Large \sum x_i = 422]
I'll simply refer to it as 
*[tex \Large \sum x = 422]


Given info:
*[tex \Large \sum x = 422]
*[tex \Large \sum y = 5313]
*[tex \Large \sum y^2 = 2512988]
*[tex \Large \sum xy = 191328]
*[tex \Large n = 12]
*[tex \Large \hat{\beta_1} = 3.221]


Use some of the given info to say
*[tex \Large SST = SS_{yy}]
*[tex \Large SST = \sum y^2 - \frac{(\sum y)^2}{n}]
*[tex \Large SST = 2512988 - \frac{(5313)^2}{12}]
*[tex \Large SST = 160,657.25]


and furthermore
*[tex \Large SSR = \hat{\beta_1}*SS_{xy}]
*[tex \Large SSR  = \hat{\beta_1}*\left(\sum xy - \frac{(\sum x)(\sum y)}{n}\right)]
*[tex \Large SSR  = 3.221*\left(191328 - \frac{(422)(5313)}{12}\right)]
*[tex \Large SSR  = 14,454.2375]


Now we can compute the SSE
*[tex \Large SSE = SST - SSR]
*[tex \Large SSE = SS_{yy} - \hat{\beta_1}*SS_{xy}]
*[tex \Large SSE = 160,657.25 - 14,454.2375]
*[tex \Large SSE = 146,203.0125]


Finally,
*[tex \Large s^2 = \frac{SSE}{n-2}]
*[tex \Large s^2 = \frac{146,203.0125}{12-2}]
*[tex \Large s^2 = 14,620.30125]
*[tex \Large s^2 = 14,620.30]


Answer: <font color=red>14,620.30</font>
Delete the comma if needed
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