Final answer:
The mean of the transformed variable 3X + 27 is 333, and the standard deviation is 99.
Step-by-step explanation:
When considering a random variable X with a mean of 102 and a standard deviation of 33, and looking to calculate the mean and standard deviation of a transformed variable 3X + 27, we apply some basic rules of statistics:
Mean of 3X + 27
To find the mean of the new random variable, we use the formula:
Mean of aX + b = a × (Mean of X) + b
Thus, the mean of 3X + 27 is 3 × 102 + 27 = 333.
Standard Deviation of 3X + 27
The standard deviation of a linear transformation ax + b is the original standard deviation multiplied by the absolute value of a:
Standard Deviation of aX + b = |a| × (Standard Deviation of X)
Hence, the standard deviation of 3X + 27 is |3| × 33 = 99.