Question

A population consists of three housing units, where the number of rooms in each unit is 2, 3 and 4.

1. a) If a random sample without replacement of 2 housing units will be picked from the population, what is the sampling distribution of the sample median?

2. b) If a random sample with replacement of 2 housing units will be picked from the population, what is the sampling distribution of the sample mean number of rooms? What about the sampling distribution of the sample mean?

Answer 1

We have a population with 3 units i.e. 2,3 and 4.

a) We have to select a random sample of 2 units without replacement. So, number of samples are

So, samples are

The sampling distribution of obervations are 2,2 and 3. These are the summary of the median of the distributions

summary(2,2,3)
Min. 1st Qu. Median Mean 3rd Qu. Max.
2 2 2 2 2 2

b) The sampling distribution of the mean with the replacement is given by and the number of samples will be

The sample mean of the samples are

I have used R and the following are the observations and mean.

samples mean
2 2 2
2 3 2
  2 4 2
3 2 3
3 3 3
  3 4 3
  4 2 4
4 3 4
4 4 4

summary of the sample mean are

Min. 1st Qu. Median Mean 3rd Qu. Max.
2 2 3 3 4 4