## Sample mean and covariance matrix problem

Energy consumption, by state, from the major sources

$x_{1}$=petroleum
$x_{2}$=natural gas
$x_{3}$=hydroelectric power
$x_{4}$=nuclear electric power

Let

$\bar{x}=\left[ \begin{array}{c} 0.766 \\ 0.508 \\ 0.438 \\ 0.161 \end{array} \right]$

and

$S\left[ \begin{array}{cccc} 0.856 & 0.635 & 0.173 & 0.096 \\ 0.635 & 0.568 & 0.128 & 0.067 \\ 0.173 & 0.127 & 0.171 & 0.039 \\ 0.096 & 0.067 & 0.039 & 0.043 \end{array} \right]$

a) Using the summary statistics, determine the smaple mean and variance of the state's total engery consumption for these major sources.

b) Determine the smaple mean and variance of the excess of petroleum consumption over natural gas consumption.

I am not sure that I understand the questions. At first I thought all I had to do for (a) was look at the first component of the sample mean vector and then look at first row and first column of the sample covariance matrix. However, the question is asking for the sample mean and variance of a state's total energy consumption. This leads me to beleive I am thinking of the original data matrix incorrectly.

I would imagine the orginal data matrix would have four columns corresponding to each energy type and each column would have 50 rows representing each state. So, if I look at just the first row, I am looking at the four energy measurments on the first state and so on.

So, since I am looking for the total consumption for just one state, I need to find a way to find the the average of one row and find its corresponding variance too.

How would I approach a and b?