Question

PLEASE HELP WITH MATH.....
Three randomly selected households are surveyed. The numbers of people in the households are 1, 4 and 10. Assume that the samples of size n=2 are randomly selected with replacement from the population of 1, 4 and 10. Listed below are the nine different samples:

1,1 1,4 1,10 4,1 4,4 4,10 10,1 10,4 10,10

Find the variance of each of the nine samples: (find s^2 and probability)

s^ 2 is 4.5, 0, or 1 ? Whats the probability?
s^2 is 9, 4.5, 2.3 ? Whats the probability?
s^2 is 18, 72, or 9? Whats the probability?
s^2 is 20.3, 81 or 40.5? Whats the probability?

Answer 1

We have the following cases to consider (1,1), (1,4), (1,10) ,(4,4) (4,10), (10,10). Well we know three of our samples will have a variance of 0. The variance for the case when we have 1 and 4 is 4.5; likewise, the variance for when we have 10 and 4 is 18 and the variance for when we have 1 and 10 is 40.5. The possible variances are 0, 4.5, 18, and 40.5. The probability of having a variance of 0 is 3/9. For a variance of 4.5 its 2/9, for 18 its 2/9 and lastly, for a variance of 40.5 its 2/9. As you can see, 3/9 + 2/9 + 2/9 + 2/9 = 1.