Solution to Homework 8

Problem #1
n=9
∑X=36
∑Y=39.4
∑X²=204
∑Y²=232.98
∑XY=216.3

S_XX=60
S_YY=60.50
S_XY=58.7

r = S_XY/√(S_XXS_YY) = 58.7/√(60·60.5) = 0.974

Problem #2
n=9
∑Z=399.6
∑Y=39.4
∑Z²=26819.64
∑Y²=232.98
∑ZY=2471.37

S_ZZ=9077.4
S_YY=60.50
S_ZY=722.01

r = S_XY/√(S_XXS_YY) = 722.01/√(9077.4·60.5) = 0.974

Problem 3
n=9
∑X=36
∑Y=39.4
∑X²=204
∑XY=216.3

S_XX=60
S_XY=58.7

b₁ = S_xy/S_xx = 58.1/60 = 0.9783.

b₀ = -b₁ = 39.4/9 - 0.9783·36/9 = 0.4644

so we have y=0.46+0.98x

Problem 4
n=9
∑X=36
∑Y=39.4
∑Y²=232.98
∑XY=216.3

S_YY=60.50
S_XY=58.7

b₁ = S_xy/S_yy = 58.7/60.5 = 0.9703
Note: now the dependent variable is x, so we devide by S_yy

b₀ = -b₁ = 36/9 - 0.9703·39.4/9 = -0.2478

so we have x=-0.25+0.97y

Problem 5
To add a line to a graph we need two points on the line.

First line:
x=0, y=0.46+0.98·0=0.46, point (0,0.46)
x=8, y=0.46+0.98·8=8.3, point (0,8.3)

Second line:
y=1, x=-0.25+0.97·1=0.72, point (0.72,1)
y=8, x=-0.25+0.97·8=7.51, point(7.51,8)

Now

Of course we already know one point that is on both lines: (,) = (4,4.38) and we could have used that one as well.