Question

Find the regression equation, letting the first variable be the

Answer 1

Answer:

$y=0.00673(253) +90.190=91.894$

And the difference is given by:

$r_i =91.894-83=8.894$

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:

$m=\frac{S_{xy}}{S_{xx}}$

Where:

$S_{xy}=\sum_{i=1}^n x_i y_i -\frac{(\sum_{i=1}^n x_i)(\sum_{i=1}^n y_i)}{n}$

$S_{xx}=\sum_{i=1}^n x^2_i -\frac{(\sum_{i=1}^n x_i)^2}{n}$

So we can find the sums like this:

$\sum_{i=1}^n x_i =242+255+227+251+262+207+140=1584$

$\sum_{i=1}^n y_i =91+ 81 +91 + 92 + 102 + 94 + 91=642$

$\sum_{i=1}^n x^2_i =242^2 +255 ^2 +227^2 +251^2 +262^2 +207^2 +140^2=369212$

$\sum_{i=1}^n y^2_i =91^2 + 81 ^2 +91 ^2 + 92 ^2 + 102 ^2 + 94 ^2 + 91^2=59108$

$\sum_{i=1}^n x_i y_i =242*91 +255*81 +227*91 +251*92 +262*102 +207*94 +140*91=145348$

With these we can find the sums:

$S_{xx}=\sum_{i=1}^n x^2_i -\frac{(\sum_{i=1}^n x_i)^2}{n}=369212-\frac{1584^2}{7}=10775.429$

$S_{xy}=\sum_{i=1}^n x_i y_i -\frac{(\sum_{i=1}^n x_i)(\sum_{i=1}^n y_i)}{n}=145348-\frac{1584*642}{7}=72.571$

And the slope would be:

$m=\frac{72.571}{10775.429}=0.00673$

Now we can find the means for x and y like this:

$\bar x= \frac{\sum x_i}{n}=\frac{1584}{7}=226.286$

$\bar y= \frac{\sum y_i}{n}=\frac{642}{7}=91.714$

And we can find the intercept using this:

$b=\bar y -m \bar x=91.714-(0.00673*226.286)=90.190$

So the line would be given by:

$y=0.00673 x +90.190$

The prediction for 253 seconds is:

$y=0.00673(253) +90.190=91.894$

And the difference is given by:

$r_i =91.894-83=8.894$