Market researchers are testing customers’ preferences for five products. There are four researchers who are allocated to different groups: researcher R₁ deals with men under 40, R₂ deals with men over 40, R₃ deals with women under 40 and R₄ deals with women over 40. They return their findings as a vector giving the number of customers with first preference for a particular product.

Find the average over the whole sample. The company decides that the main target is older women, so it weights the returns in the ratio 1 : 1 : 2 : 3; find the weighted average.