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). x 16 33 50 28 50 25 y 3 3 7 5 9 3 Complete parts (a) through (e), given Σx = 202, Σy = 30, Σx2 = 7754, Σy2 = 182, Σxy = 1162, and r ≈ 0.870. (a) Draw a scatter diagram displaying the data. Selection Tool Line Ray Segment Circle Vertical Parabola Horizontal Parabola Point No Solution Help 5101520253035404550556012345678910 Clear Graph Delete Layer Fill WebAssign Graphing Tool Graph LayersToggle Open/Closed After you add an object to the graph you can use Graph Layers to view and edit its properties. (b) Verify the given sums Σx, Σy, Σx2, Σy2, Σxy, and the value of the sample correlation coefficient r. (Round your value for r to three decimal places.) Σx = Σy = Σx2 = Σy2 = Σxy = r = (c) Find x, and y. Then find the equation of the least-squares line y hat = a + bx. (Round your answers for x and y to two decimal places. Round your answers for a and b to three decimal places.) x = y = y hat = + x (d) Graph the least-squares line. Be sure to plot the point (x, y) as a point on the line. WebAssign Plot WebAssign Plot WebAssign Plot WebAssign Plot (e) Find the value of the coefficient of determination r2. What percentage of the variation in y can be explained by the corresponding variation in x and the least-squares line? What percentage is unexplained? (Round your answer for r2 to three decimal places. Round your answers for the percentages to one decimal place.) r2 = explained % unexplained % (f) For a neighborhood with x = 42 hundred jobs, how many are predicted to be entry level jobs? (Round your answer to two decimal places.) hundred jobs

Answer 1

Part a)

Part b)

	X	Y	X * Y	X2	Y2
	16	3	48	256	9
	33	3	99	1089	9
	50	7	350	2500	49
	28	5	140	784	25
	50	9	450	2500	81
	25	3	75	625	9
Total	202	30	1162	7754	182

r = 0.870

Part c)

X̅ = Σ( Xi / n ) = 202/6 = 33.67
Y̅ = Σ( Yi / n ) = 30/6 = 5

Equation of regression line is Ŷ = a + bX


b = 0.159
a =( Σ Y - ( b * Σ X) ) / n
a =( 30 - ( 0.1594 * 202 ) ) / 6
a = -0.368
Equation of regression line becomes Ŷ = -0.368 + 0.159 X

Part e)

Coefficient of Determination
R² = r² = 0.757
Explained variation = 0.757* 100 = 75.7%
Unexplained variation = 1 - 0.757* 100 = 24.3%

Part f)

When X = 42
Ŷ = -0.368 + 0.159 X
Ŷ = -0.368 + ( 0.159 * 42 )
Ŷ = 6.31