Question

In: Statistics and Probability

Listed below are amounts of court income and salaries paid to the town justices for a...

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 _____

Solutions

Expert Solution

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 )


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