Energy consumption in 2001, by state, from the major sources x_1 = petroleum, x_
ID: 3206899 • Letter: E
Question
Energy consumption in 2001, by state, from the major sources x_1 = petroleum, x_2 = natural gas, x_3 = hydroelectric power, and x_4 = nuclear electric power is recorded in quadrillions (10^15) of BTUs. The resulting mean and covariance matrix are x = [0.766 0.508 0.438 0.161] S = [0.856 0.635 0.173 0.096 0.635 0.568 0.128 0.067 0.173 0.127 0.171 0.039 0.096 0.067 0.039 0.043]. Using the summary statistics, determine the sample mean and variance of a state's total energy consumption for these major sources. Determine the sample mean and variance of the excess of petroleum consumption over natural gas consumption. Also find the sample covariance of this variable with the total variable in part a).Explanation / Answer
(a) sample mean=(0.766+0.508+0.438+0.161)/4=0.46825
sample variance(total)=
var(x1)+var(x2)+var(x3)+var(x4)+cov(x1,x2)+cov(x1,x3)+cov(x1,x4)+cov(x2,x3)++cov(x2,x4)++cov(x3,x4)
=0.856+0.568+0.171+0.043+0.635+0.173+0.096+0.128+0.067+0.039=2.776
(b) mean(x1-x2)=0.766-0.508=0.258
Var(x1-x2)=var(x1)+var(x2)+cov(x1,x2)=0.856+0.568+0.635=2.059
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