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 17 35 53 28 50 25 y 2 4 6 5 9 3 Complete parts (a) through (e), given Σx = 208, Σy = 29, Σx2 = 8232, Σy2 = 171, Σxy = 1157, and r ≈ 0.855. (a) Draw a scatter diagram displaying the data. (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 = 30 hundred jobs, how many are predicted to be entry level jobs? (Round your answer to two decimal places.) hundred jobs

Answer 1

X	Y	XY	X²	Y²
17	2	34	289	4
35	4	140	1225	16
53	6	318	2809	36
28	5	140	784	25
50	9	450	2500	81
25	3	75	625	9
Ʃx =	Ʃy =	Ʃxy =	Ʃx² =	Ʃy² =
208	29	1157	8232	171

Sample size, n =	6
x̅ = Ʃx/n = 208/6 =	34.6666667
y̅ = Ʃy/n = 29/6 =	4.83333333
SSxx = Ʃx² - (Ʃx)²/n = 8232 - (208)²/6 =	1021.33333
SSyy = Ʃy² - (Ʃy)²/n = 171 - (29)²/6 =	30.8333333
SSxy = Ʃxy - (Ʃx)(Ʃy)/n = 1157 - (208)(29)/6 =	151.666667

a) Scatterplot:

b) Ʃ x = 208

Ʃ y = 29

Ʃ xy = 1157

Ʃ x² = 8232

Ʃ y² = 171

Correlation coefficient, r = SSxy/√(SSxx*SSyy) = 151.66667/√(1021.33333*30.83333) = 0.855

c) x̅ = Ʃx/n = 208/6 = 34.6666667

y̅ = Ʃy/n = 29/6 = 4.83333333

Slope, b = SSxy/SSxx = 0.14849869

y-intercept, a = y̅ -b* x̅ = -0.31462141

Regression equation :

ŷ = -0.315 + (0.148) x

d) Least regression line:

e) Coefficient of determination, r² = (SSxy)²/(SSxx*SSyy) = (151.66667)²/(1021.33333*30.83333) = 0.730

r² = 0.730

Explained = 73.0%

Unexplained = 27.0%

f) Predicted value of y at x = 30

ŷ = -0.315 + (0.148) * 30 = 4.14 hundred jobs