Sampling distribution

Question

Suppose our population consists of the values (2, 3, 5, 7, 11, 13, 17), that is, our population is the first 7 prime numbers (one of each).

a) If we take a simple random sample of size k, where k is an even number, then find the median. What is the sample space?

b) Explain why, for this population, the sampling distribution of the sample median will never be a normal distribution, regardless of the sample size.



Answer 1

The sample space of the median will include all of the values of the distribution (consider all values in the sample to have that value for example) as well as all pairs (consider equal numbers to have equal numbers of each value of the pair and no others). This will include all possible sets of values on both sides of the middle.

Also, we can see that the only way the average of the middle values can be a non-whole number will be if 2 is one of the numbers (all other numbers are odd). From 2 and an odd, we get (2 1/2, 3 1/2, 4 1/2, 6 1/2, 7 1/2, 9 1/2) We have all original values - 2, 3, 5, 7, 11, 13, 17; we can look, then, for the missing whole numbers

(3+5)/2 = 4

(5+7)/2 = 6

(3 + 13)/2 = 8

(7 + 11)/2 = 9

(7 + 13)/2 = 10

(11+13/2) = 12

(11 + 17)/2 = 14

(13 + 17)/2 = 15

We cannot get 16, as it is not on the list, and the only number higher on the list is 17, but it would have to average with 15 to get 16, and 15 is not on the list.

Thus, our sample space is all integers between 2 and 17 except for 16 and the listed 1/2 numbers:

(2, 2 1/2, 3, 3 1/2, 4, 4 1/2, 5, 6, 6 1/2, 7, 7 1/2, 8, 9, 9 1/2, 10, 11, 12, 13, 14, 15, 16)

A distribution defined on these values will, of course, not be normal;

the critical thing is that, unlike with the mean, the distribution will not approach the normal distribution in the limit, either.