Suppose a data set x1,x2,…,xn has mean xˉ and variance σ2. Each observation is transformed in the following two ways.
Case I: yi=xi+k where k is a constant.
Case II: zi=cxi where c is a non-zero constant.
(i) Show that the variance of the transformed data {yi} remains the same as σ2. Clearly state each algebraic step.
(ii) Prove that the variance of the transformed data {zi} becomes c2σ2. Give complete reasoning.
(iii) A class recorded the weekly pocket money (in rupees) of 8 students as: 90, 95, 100, 110, 115, 120, 125, 135. The variance of this data is 250. Using the results of parts (i) and (ii), calculate the new variance if (a) each amount is increased by ₹20, and (b) each amount is doubled.