Three randomly selected households are surveyed. the numbers of people in the households are 1,3,8 assume that samples of size n =2 are randomly selelcted with replacement from the population of 1,3,8. listed below are the nine different sampls

1,1 1,3 1,8 3,1 3,3 3,8 8,1 8,3 8,8

find the variance of each of the nine samples then summerize the sampling distribution of the variances in the format of a table representing the probability distribution of the distinct variance values

s^2 probability

3.3

1

6.3

49

I assumed that 3 households are sampled.
Now, is the number in each of these households equal to 1, 3 and 8 people?

Next, the question says, samples of n=2 are randomly selected (with replacement) from these 3.

So are sample space is
11
13
31
33
18
81
38
88
83
88

This is done 9 times.
Is that how you understand the problem?
When they select the household with only 1 person, the variance is 0
When they select the household with 8 people, they variance will be from a discrete uniform distribution

Similarly for household with 3 individuals.