Question

Year	Number of Alternative-Fueled Vehicles in US
2000	394,664
2001	425,457
2002	471,098
2003	533,999
2004	565,492
2005	592,125
2006	634,562
2007	695,766

Do the variables have significant correlation? For full credit, you must show each step of the hypothesis test. Use the 0.05 significance.

Answer 1

Cor, r = Cov(X,Y)/sqrt(V(X)*V(Y))

V(X) = sum(X - mean(X))^2

V(Y) = sum(Y - mean(Y))^2

Cov(X,Y) = sum( (Y - mean(Y))*(X - mean(X)))

n = 8

mean(X) = ( 2000 + 2001 + 2002 +2003 +2004 + 2005+ 2006 + 2007 ) = 16025/8 = 2003.5

V(X) = 42/8 = 5.25

V(Y) = 75592182567.875/ 8 = 9449022820.984375

Cov(X,Y) = 1773906.5/8 = 221738.3125

r = 221738.3125/ sqrt(5.25*9449022820.984375 ) = 0.9956

significance test

Null hypothesis : population correlation value is 0

Alternate hypothesis : population correlation value is not equal to 0

t = r*sqrt((n-2)/(1-r^2)) = 0.9956*sqrt(6/(1-0.9956^2)) = 0.9956*( 6 / 0.00878 ) = 0.9956*26.1414 = 26.026

degrees of freedom = n - 2 = 6

p value at 6 degrees of freedom = 1

Reject H0 if p < alpha

here, p( 1 ) > alpha ( 0.05)

therefore, we failed to reject H

hence, H0 is accepted that population correlation value is 0