I only have solutions to parts (i) and (ii).
W=2X+3.
(i) E(W)=2µ+3, because all X data are shifted by the same amount in dataset W, so the mean is similarly shifted:
X={ x₁ x₂ x₃ ... }, µ=∑x/n
W={ 2x₁+3 2x₂+3 2x₃+3 ... }, E(W)=∑(2x+3)/n=2∑x/n+3n/n=2µ+3.
(ii) Var(W)=4Var(X)=4σ²:
Var(X)=∑(x-µ)²/n=(1/n)[∑x²-2µ∑x+nµ²]=
(1/n)[∑x²-(2µ)(nµ)+nµ²]=(1/n)[∑x²-nµ²]=
∑x²/n-µ².
Var(W)=∑w²/n-(2µ+3)²=
∑(2x+3)²/n-(2µ+3)²=
(4∑x²+12∑x+9n)/n-(4µ²+12µ+9)=
(4∑x²+12nµ+9n)/n-(4µ²+12µ+9)=
4∑x²/n+12µ+9-4µ²-12µ-9=
4(∑x²/n-µ²)=4Var(X)=4σ².