Question

Let x be a random variable representing percentage change in neighborhood population in the past few years, and let y be a random variable representing crime rate (crimes per 1000 population). A random sample of six Denver neighborhoods gave the following information.

x 33 3 11 17 7 6
y 167 39 132 127 69 53


In this setting we have Σx = 77, Σy = 587, Σx2 = 1593, Σy2 = 70,533, and Σxy = 10040.
(a) Find x, y, b, and the equation of the least-squares line.

Answer 1

`m=(2506.833)/(604.833)=4.1447`

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

`\bar x= (\sum x_i)/(n)=(77)/(6)=12.833`

`\bar y= (\sum y_i)/(n)=(587)/(6)=97.833`

And we can find the intercept using this:

`b=\bar y -m \bar x=97.833-(4.1447*12.833)=44.644`

So the line would be given by:

`y=4.1447 x +44.644`

Explanation:

`m=(S_(xy))/(S_(xx))`

Σx = 77, Σy = 587, Σx2 = 1593, Σy2 = 70,533, and Σxy = 10040.

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)`

With these we can find the sums:

`S_(xx)=\sum_(i=1)^n x^2_i -((\sum_(i=1)^n x_i)^2)/(n)=1593-(77^2)/(6)=604.833`

`S_(xy)=\sum_(i=1)^n x_i y_i -\frac{(\sum_(i=1)^n x_i)(\sum_(i=1)^n y_i){n}}=10040-(77*587)/(6)=2506.833`

And the slope would be:

`m=(2506.833)/(604.833)=4.1447`

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

`\bar x= (\sum x_i)/(n)=(77)/(6)=12.833`

`\bar y= (\sum y_i)/(n)=(587)/(6)=97.833`

And we can find the intercept using this:

`b=\bar y -m \bar x=97.833-(4.1447*12.833)=44.644`

So the line would be given by:

`y=4.1447 x +44.644`