Question

Let x be per capita income in thousands of dollars. Let y be the number of medical doctors per 10,000 residents. Six small cities in Oregon gave the following information about x and y. x 8.3 9.3 10.2 8.0 8.3 8.7 y 9.9 18.1 20.6 10.2 11.4 13.1 Complete parts (a) through (e), given Σx = 52.8, Σy = 83.3, Σx2 = 468, Σy2 = 1255.59, Σxy = 750.81, and r ≈ 0.974. (a) Draw a scatter diagram displaying the data. Selection Tool Line Ray Segment Circle Vertical Parabola Horizontal Parabola Point No Solution Help 123456789101112345678910111213141516171819202122 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) Suppose a small city in Oregon has a per capita income of 9.2 thousand dollars. What is the predicted number of M.D.s per 10,000 residents? (Round your answer to two decimal places.) M.D.s per 10,000 residents

Answer 1

Part a)

Part b)

	X	Y	X * Y	X2	Y2
	8.3	9.9	82.17	68.89	98.01
	9.3	18.1	168.33	86.49	327.61
	10.2	20.6	210.12	104.04	424.36
	8	10.2	81.6	64	104.04
	8.3	11.4	94.62	68.89	129.96
	8.7	13.1	113.97	75.69	171.61
Total	52.8	83.3	750.81	468	1255.59

r = 0.974

Part c)

X̅ = Σ( Xi / n ) = 52.8/6 = 8.8
Y̅ = Σ( Yi / n ) = 83.3/6 = 13.88

Equation of regression line is Ŷ = a + bX


b = 5.289
a =( Σ Y - ( b * Σ X) ) / n
a =( 83.3 - ( 5.2887 * 52.8 ) ) / 6
a = -32.657
Equation of regression line becomes Ŷ = -32.657 + 5.289 X

Part e)

Coefficient of Determination
R² = r² = 0.948
Explained variation = 0.948* 100 = 94.8%
Unexplained variation = 1 - 0.948* 100 = 5.2%

Part f)

When X = 9.2
Ŷ = -32.657 + 5.289 X
Ŷ = -32.657 + ( 5.289 * 9.2 )
Ŷ = 16