Multiple choice

The mean and standard deviation of a random variable $x$ is given by $5$ and $3$ respectively. The standard deviation of $2-3x$ is _____

  1. $-7$
  2. $81$
  3. $34$
  4. $9$
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D Correct answer
Explanation

Standard deviation is invariant to constant shifts (the '2' in 2-3x). For a linear transformation y = ax + b, SD(y) = |a| * SD(x). SD(2-3x) = |-3| * 3 = 9.

AI explanation

Using the properties of standard deviation, multiplying a random variable by a constant multiplies the standard deviation by the absolute value of that constant. Therefore, multiplying the random variable by 3 gives a new standard deviation of 3 times 3, which is 9. Adding a constant, like subtracting the random variable from 2, shifts the data but does not change its spread, leaving the standard deviation at 9.

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