Question

An economist is studying the job market in Denver area neighborhoods. Let x represent the total number of jobs in a given neighborhood, and let y represent the number of entry-level jobs in the same neighborhood. A sample of six Denver neighborhoods gave the following information (units in hundreds of jobs). Complete parts (a) through (e), given Ex 197, Sy 28, x2 7399, Sy2 - 174, Exy 1109, and r 0.944 (a) Draw a scatter diagram displaying the data. Flash Player version 10 or higher is required for this question. You can get Flash Player free from Adobe's website. (b) Verify the given sums Ex, Ey, x2, y2, Exy, and the value of the sample correlation coefficient r. (Round your value for r to three decimal places.) (c) Find x, and y. Then find the equation of the least-squares line a + bx. (Round your answers for x and y to two decimal places. Round your answers for a and b to three decimal places.) (d) Graph the least-squares line. Be sure to plot the point (x, y) as a point on the line. 30405060 020

Answer 1

Answer

Here by the problem, an economist is studying the job market in Denver area neighbourhoods. Let x represent the total number of jobs in a given neighborhood, and let y represent the number of entry level jobs in the same neighborhood.

A sample of n=6 Denver neighborhoods gave the information on x and y as follows,

x	y
15	1
31	3
48	7
28	5
50	9
25	3

(a) Based on this data the scatter diagram is as below,

(b) Now we verify the given information as below,

	x	y	x^2	y^2	xy
	15	1	225	1	15
	31	3	961	9	93
	48	7	2304	49	336
	28	5	784	25	140
	50	9	2500	81	450
	25	3	625	9	75
Total	197	28	7399	174	1109

So we can observe that

$\sum x=197;\sum y=28;\sum x^2=7399;\sum y^2=174;\sum xy=1109$

So the correlation coefficient be,

$r=\frac{\sum xy-\frac{1}{n}(\sum x)(\sum y)}{\sqrt{\sum x^2-\frac{1}{n}(\sum x)^2}\sqrt{\sum y^2-\frac{1}{n}(\sum y)^2}}$

So putting the values we get,

$r=\frac{1109-\frac{1}{6}(197)(28)}{\sqrt{7399-\frac{1}{6}(197)^2}\sqrt{174-\frac{1}{6}(28)^2}}=0.9444\approx 0.944$

Hence verified

(c) Now we have,

$\bar{x}=197/6=32.83$ and  $\bar{y}=28/6=4.67$

Now if $\hat{y}=a+bx$ be the least square line then we know,

$a=\bar{y}-b\bar{x}$ and $b=\frac{Cov(x,y)}{s_x^2}=\frac{\sum xy -n\bar{x}\bar{y}}{\sum x^2-n\bar{x}^2}=\frac{1109-6(32.83)(4.67)}{7399-6(32.83)^2}=0.2037\approx 0.204$ Hence,

$a=4.67-0.204\times 32.83=-2.023$

SO the regression equation be,

yーー2.023 + 0.2041.

Hence the graph of the least square line be,

(e) Now the coefficient of determination be,

$r^2=(0.944)^2=0.892$

Hence we can say 0.892*100%=89.2% variability is explained.

And (1-0.892)*100%=10.8% variability is unexplained

(f) Now for a neighborhood with x=38 hundred jobs , the number off predicted entry level jobs be,

$\hat{y}=-2.023+0.204\times 38=5.72$

Hence the answer.............

Thank you..........