In a given village there are only three sizes of families: families with 2 members, families with 4 members and families with 6 members. The proportion of families with 2,4 and 6 members are roughly equal. A poll is conducted in this village wherein a person is chosen at random and asked about his/her family size. The average family size computed by sampling 1000 such persons from the village would be closest to

Let us assume there are a total of 3X families. Since the proportion of families with 2,4, and 6 members is roughly equal, we will assume there are X families of each size. 

=> Total number of people in village = 2X + 4X + 6X = 12X

Probability that the person selected is from a 2-membered family = $$\dfrac{2X}{12X}=\dfrac{1}{6}$$

Probability that the person selected is from a 4-membered family = $$\dfrac{4X}{12X}=\dfrac{1}{3}$$

Probability that the person selected is from a 6-membered family = $$\dfrac{6X}{12X}=\dfrac{1}{2}$$

The average family size will be given by summing up the products of number of family members and the probability of that membered family being chosen.

=> Average family size = $$\dfrac{\frac{1}{6}\times2+\frac{1}{3}\times4+\frac{1}{2}\times6}{\frac{1}{3}+\frac{1}{3}+\frac{1}{3}}=\dfrac{14}{3}=4.667$$