Concept:Use mean and variance formulas to compute sums and apply the change due to replacement.Explanation:Given n=10, initial mean xˉ1=10, variance σ12=2.Sum of observations ∑xi=n×xˉ1=100.Variance formula: σ12=n∑xi2−(xˉ1)2.So ∑xi2=n(σ12+(xˉ1)2)=10(2+100)=1020.After replacing α with β, new mean xˉ2=10.1.Thus 10100−α+β=10.1⇒100−α+β=101⇒β−α=1.New variance σ22=1.99.Using σ22=10∑xi2−α2+β2−(xˉ2)2.Substitute: 1.99=101020−α2+β2−(10.1)2.(10.1)2=102.01 so 101020−α2+β2=1.99+102.01=104.Thus 1020−α2+β2=1040⇒β2−α2=20.Factor: (β+α)(β−α)=20.Using β−α=1, we get β+α=20.Hence α+β=20.Answer:20 (Option D).