Question

Listed below are amounts of court income and salaries paid to the town justices for a certain town. All amounts are in thousands of dollars. Find the​ (a) explained​ variation, (b) unexplained​ variation, and​ (c) indicated prediction interval. There is sufficient evidence to support a claim of a linear​ correlation, so it is reasonable to use the regression equation when making predictions. For the prediction​ interval, use a 99​% confidence level with a court income of ​$800,000

Court Income Justice Salary

60 34

394 49

1550 94

1144 59

272 44

252 61

102 25

151 27

26 20

a.) find explained variation ______

b.) unexplained variation ______

c.) indicated prediction interval _____

Answer 1

X :-  Court Income

Y :- Justice Salary

ΣX = 3951
ΣY = 413
ΣX * Y = 269029
ΣX2 = 4041441
ΣY2 = 23285

r = 0.877

Coefficient of Determination
R2 = r2 = 0.77

Part a)

Explained variation = 0.77* 100 = 77%

Part b)

Unexplained variation = 1 - 0.77* 100 = 23%

Part c)

Sxx =Σ (Xi - X̅ )2 = 2306952
Syy = Σ( Yi - Y̅ )2 = 4332.8889
Sxy = Σ (Xi - X̅ ) * (Yi - Y̅) = 87722

X̅ = Σ (Xi / n ) = 3951/9 = 439
Y̅ = Σ (Yi / n ) = 413/9 = 45.8889

Equation of regression line is Ŷ = a + bX
b = ( n Σ(XY) - (ΣX* ΣY) ) / ( n Σ X2 - (ΣX)2 )
b = ( 9 * 269029 - 3951 * 413 ) / ( 9 * 4041441 - ( 3951 )2)
b = 0.038

a =( ΣY - ( b * ΣX ) ) / n
a =( 413 - ( 0.038 * 3951 ) ) / 9
a = 29.1959
Equation of regression line becomes Ŷ = 29.1959 + 0.038 X

Predictive Confidence Interval of
Ŷ = 29.1959 + 0.038X
Ŷ = 59.5959

t(0/2) = t(0.01/2) = 3.499
X̅ = (Xi / n ) = 3951/9 = 439

99% Predictive confidence interval is ( 14.4182 < < 104.7736 )