Final answer:
The standard deviation of the transformed variable z, defined as 3y − 12, is calculated by multiplying the coefficient of the transformation (3) by the standard deviation of y (4), which results in a standard deviation of 12 for z.
Step-by-step explanation:
If you have a new random variable z defined as 3y − 12, where y is a random variable with a mean of 36 and a standard deviation of 4 and assuming that x and y are independent, then the transformation affects only the mean and the standard deviation of y. The transformation does not affect variable x since it is independent of y.
The formula for the standard deviation of z after the transformation is σz = |a|σy, where σy is the standard deviation of y, and a is the coefficient of y in the transformation. In this case, a=3.
Therefore, the new standard deviation of the transformed variable z is given by:
σz = |3|σy
= 3 × 4
= 12
The standard deviation of the new random variable z is 12.