Concept:
Statistics - Properties of Variance. When each observation is multiplied by a constant , the new variance becomes times the original variance.
Step 1: Define the original variance and its formula.
Let the original set of observations be denoted as . The variance of these observations, , is given as 5. The formula is , where is the mean of the original observations.
Step 2: Define the new set of observations.
If each observation is multiplied by 2, we get a new set of observations .
Step 3: Determine the new mean.
The new variance is , where is the new mean. Since every term is doubled, the new mean is also doubled: .
Step 4: Substitute and simplify the new variance expression.
Substitute into the equation: . Factoring out the 4 gives us: .
Step 5: Calculate the final numerical value.
Recognize that the remaining sum is the original variance. Therefore, . Since , we calculate . $ $