Question 1191018
Here's the breakdown of how to calculate the variance and standard deviation of Z for each case:

**Key Principles:**

* **Variance of a constant:** Var(c) = 0
* **Variance of a scalar multiple:** Var(aX) = a²Var(X)
* **Variance of a sum/difference (independent variables):** If X and Y are independent (correlation is 0), Var(X + Y) = Var(X) + Var(Y) and Var(X - Y) = Var(X) + Var(Y)
* **Standard Deviation:** SD(Z) = sqrt(Var(Z))

We're given Var(X) = 3² = 9 and Var(Y) = 2² = 4.

**A) Z = 33 - 8X**

* Var(Z) = Var(33 - 8X) = (-8)²Var(X) = 64 * 9 = 576
* SD(Z) = sqrt(576) = 24

**B) Z = 11X - 6**

* Var(Z) = Var(11X - 6) = 11²Var(X) = 121 * 9 = 1089
* SD(Z) = sqrt(1089) = 33

**C) Z = X + Y**

* Var(Z) = Var(X + Y) = Var(X) + Var(Y) = 9 + 4 = 13
* SD(Z) = sqrt(13) ≈ 3.61

**D) Z = X - Y**

* Var(Z) = Var(X - Y) = Var(X) + Var(Y) = 9 + 4 = 13
* SD(Z) = sqrt(13) ≈ 3.61

**E) Z = -2X + 2Y**

* Var(Z) = Var(-2X + 2Y) = (-2)²Var(X) + 2²Var(Y) = 4 * 9 + 4 * 4 = 36 + 16 = 52
* SD(Z) = sqrt(52) ≈ 7.21