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	32	53	28	50	25
y	2	2	7	5	9	3

Complete parts (a) through (e), given Σx = 204, Σy = 28, Σx² = 7998, Σy² = 172, Σxy = 1132, and r ≈ 0.859.

(a) Draw a scatter diagram displaying the data.

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(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  = 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	=
	=	+  x

(d) Graph the least-squares line. Be sure to plot the point (x, y) as a point on the line.

(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 = 37 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	2	32	256	4
	32	2	64	1024	4
	53	7	371	2809	49
	28	5	140	784	25
	50	9	450	2500	81
	25	3	75	625	9
Total	204	28	1132	7998	172

r = 0.859

Part c)

X̅ = Σ( Xi / n ) = 204/6 = 34
Y̅ = Σ( Yi / n ) = 28/6 = 4.67

Equation of regression line is Ŷ = a + bX


b = 0.169
a =( Σ Y - ( b * Σ X) ) / n
a =( 28 - ( 0.1695 * 204 ) ) / 6
a = -1.096
Equation of regression line becomes Ŷ = -1.096 + 0.169 X

part e)

Coefficient of Determination
R2 = r2 = 0.738
Explained variation = 0.738* 100 = 73.8%
Unexplained variation = 1 - 0.738* 100 = 26.2%

part f)

When X = 37
Ŷ = -1.096 + 0.169 X
Ŷ = -1.096 + ( 0.169 * 37 )
Ŷ = 5.16