Question

asked Feb 17, 2020 151k views

1 Answer

Joel Malone · Answer 1 · 2020-02-22T13:32:30+0000

Answer:

y=0.00673(253) +90.190=91.894

And the difference is given by:

r_i =91.894-83=8.894

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=(S_(xy))/(S_(xx))

Where:

$S_(xy)=\sum_(i=1)^n x_i y_i -((\sum_(i=1)^n x_i)(\sum_(i=1)^n y_i))/(n)$

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

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:

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

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

And the slope would be:

m=(72.571)/(10775.429)=0.00673

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

$\bar x= (\sum x_i)/(n)=(1584)/(7)=226.286$

$\bar y= (\sum y_i)/(n)=(642)/(7)=91.714$

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

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