Answer:
what happend to those four packets of sugar
<h2>the slope is <u>2</u></h2><h2><u /></h2>
<u>i need more characters lol ahjkffhewqjfe</u>
1. B
2. I'm not sure but I think it's D
3. C
Answer:
![y=0.00673(253) +90.190=91.894](https://tex.z-dn.net/?f=y%3D0.00673%28253%29%20%2B90.190%3D91.894)
And the difference is given by:
![r_i =91.894-83=8.894](https://tex.z-dn.net/?f=r_i%20%3D91.894-83%3D8.894)
Step-by-step explanation
We assume that th data is this one:
x: 242-255 -227-251-262-207-140
y: 91- 81 -91 - 92 - 102 - 94 - 91
Find the least-squares line appropriate for this data.
For this case we need to calculate the slope with the following formula:
![m=\frac{S_{xy}}{S_{xx}}](https://tex.z-dn.net/?f=m%3D%5Cfrac%7BS_%7Bxy%7D%7D%7BS_%7Bxx%7D%7D)
Where:
![S_{xy}=\sum_{i=1}^n x_i y_i -\frac{(\sum_{i=1}^n x_i)(\sum_{i=1}^n y_i)}{n}](https://tex.z-dn.net/?f=S_%7Bxy%7D%3D%5Csum_%7Bi%3D1%7D%5En%20x_i%20y_i%20-%5Cfrac%7B%28%5Csum_%7Bi%3D1%7D%5En%20x_i%29%28%5Csum_%7Bi%3D1%7D%5En%20y_i%29%7D%7Bn%7D)
![S_{xx}=\sum_{i=1}^n x^2_i -\frac{(\sum_{i=1}^n x_i)^2}{n}](https://tex.z-dn.net/?f=S_%7Bxx%7D%3D%5Csum_%7Bi%3D1%7D%5En%20x%5E2_i%20-%5Cfrac%7B%28%5Csum_%7Bi%3D1%7D%5En%20x_i%29%5E2%7D%7Bn%7D)
So we can find the sums like this:
![\sum_{i=1}^n x_i =242+255+227+251+262+207+140=1584](https://tex.z-dn.net/?f=%5Csum_%7Bi%3D1%7D%5En%20x_i%20%3D242%2B255%2B227%2B251%2B262%2B207%2B140%3D1584)
![\sum_{i=1}^n y_i =91+ 81 +91 + 92 + 102 + 94 + 91=642](https://tex.z-dn.net/?f=%5Csum_%7Bi%3D1%7D%5En%20y_i%20%3D91%2B%2081%20%2B91%20%2B%2092%20%2B%20102%20%2B%2094%20%2B%2091%3D642)
![\sum_{i=1}^n x^2_i =242^2 +255 ^2 +227^2 +251^2 +262^2 +207^2 +140^2=369212](https://tex.z-dn.net/?f=%5Csum_%7Bi%3D1%7D%5En%20x%5E2_i%20%3D242%5E2%20%2B255%20%5E2%20%2B227%5E2%20%2B251%5E2%20%2B262%5E2%20%2B207%5E2%20%2B140%5E2%3D369212)
![\sum_{i=1}^n y^2_i =91^2 + 81 ^2 +91 ^2 + 92 ^2 + 102 ^2 + 94 ^2 + 91^2=59108](https://tex.z-dn.net/?f=%5Csum_%7Bi%3D1%7D%5En%20y%5E2_i%20%3D91%5E2%20%2B%2081%20%5E2%20%2B91%20%5E2%20%2B%2092%20%5E2%20%2B%20102%20%5E2%20%2B%2094%20%5E2%20%2B%2091%5E2%3D59108)
![\sum_{i=1}^n x_i y_i =242*91 +255*81 +227*91 +251*92 +262*102 +207*94 +140*91=145348](https://tex.z-dn.net/?f=%5Csum_%7Bi%3D1%7D%5En%20x_i%20y_i%20%3D242%2A91%20%2B255%2A81%20%2B227%2A91%20%2B251%2A92%20%2B262%2A102%20%2B207%2A94%20%2B140%2A91%3D145348)
With these we can find the sums:
![S_{xx}=\sum_{i=1}^n x^2_i -\frac{(\sum_{i=1}^n x_i)^2}{n}=369212-\frac{1584^2}{7}=10775.429](https://tex.z-dn.net/?f=S_%7Bxx%7D%3D%5Csum_%7Bi%3D1%7D%5En%20x%5E2_i%20-%5Cfrac%7B%28%5Csum_%7Bi%3D1%7D%5En%20x_i%29%5E2%7D%7Bn%7D%3D369212-%5Cfrac%7B1584%5E2%7D%7B7%7D%3D10775.429)
![S_{xy}=\sum_{i=1}^n x_i y_i -\frac{(\sum_{i=1}^n x_i)(\sum_{i=1}^n y_i)}{n}=145348-\frac{1584*642}{7}=72.571](https://tex.z-dn.net/?f=S_%7Bxy%7D%3D%5Csum_%7Bi%3D1%7D%5En%20x_i%20y_i%20-%5Cfrac%7B%28%5Csum_%7Bi%3D1%7D%5En%20x_i%29%28%5Csum_%7Bi%3D1%7D%5En%20y_i%29%7D%7Bn%7D%3D145348-%5Cfrac%7B1584%2A642%7D%7B7%7D%3D72.571)
And the slope would be:
![m=\frac{72.571}{10775.429}=0.00673](https://tex.z-dn.net/?f=m%3D%5Cfrac%7B72.571%7D%7B10775.429%7D%3D0.00673)
Now we can find the means for x and y like this:
![\bar x= \frac{\sum x_i}{n}=\frac{1584}{7}=226.286](https://tex.z-dn.net/?f=%5Cbar%20x%3D%20%5Cfrac%7B%5Csum%20x_i%7D%7Bn%7D%3D%5Cfrac%7B1584%7D%7B7%7D%3D226.286)
![\bar y= \frac{\sum y_i}{n}=\frac{642}{7}=91.714](https://tex.z-dn.net/?f=%5Cbar%20y%3D%20%5Cfrac%7B%5Csum%20y_i%7D%7Bn%7D%3D%5Cfrac%7B642%7D%7B7%7D%3D91.714)
And we can find the intercept using this:
![b=\bar y -m \bar x=91.714-(0.00673*226.286)=90.190](https://tex.z-dn.net/?f=b%3D%5Cbar%20y%20-m%20%5Cbar%20x%3D91.714-%280.00673%2A226.286%29%3D90.190)
So the line would be given by:
![y=0.00673 x +90.190](https://tex.z-dn.net/?f=y%3D0.00673%20x%20%2B90.190)
The prediction for 253 seconds is:
![y=0.00673(253) +90.190=91.894](https://tex.z-dn.net/?f=y%3D0.00673%28253%29%20%2B90.190%3D91.894)
And the difference is given by:
![r_i =91.894-83=8.894](https://tex.z-dn.net/?f=r_i%20%3D91.894-83%3D8.894)