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

Answer:

$m=\frac{2506.833}{604.833}=4.1447$  

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

$\bar x= \frac{\sum x_i}{n}=\frac{77}{6}=12.833$  

$\bar y= \frac{\sum y_i}{n}=\frac{587}{6}=97.833$  

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:  

$y=4.1447 x +44.644$  

Step-by-step explanation:

$m=\frac{S_{xy}}{S_{xx}}$  

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

Where:  

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

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

With these we can find the sums:  

$S_{xx}=\sum_{i=1}^n x^2_i -\frac{(\sum_{i=1}^n x_i)^2}{n}=1593-\frac{77^2}{6}=604.833$  

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

And the slope would be:  

$m=\frac{2506.833}{604.833}=4.1447$  

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

$\bar x= \frac{\sum x_i}{n}=\frac{77}{6}=12.833$  

$\bar y= \frac{\sum y_i}{n}=\frac{587}{6}=97.833$  

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:  

$y=4.1447 x +44.644$