Question

Three randomly selected households are surveyed. The numbers of people in the households are 4, 5, and 9. Assume that samples of size n = 2 are randomly selected with replacement from the population consisting of 4, 5 and 9. Listed below are the nine different samples.

4,4     4,5      4,9    5,4   5,5     5,9   9,4    9,5   9,9

A. ) Find the median of each nine samples then summarize the sampling distribution of the medians in the format of a table representing the probability

B.) Compare the population median to the mean of the sample medians

C.) Do the sample medians target the value of the population median? In general do sample medians make good estimators of population medians? Why or why not?

Answer 1

In above question

The population includes 3 values that is 4 ,5, 9 that is N = 3

We select samples of size n = 2 using with replacement

Here there are 9 samples with each of size 2

A.)

We construct the following table for calculating mean of the sample medians

Sample observations	Sample mean	Sample mean in ascending order
4	4	4	4
4	5	4.5	4.5
4	9	6.5	4.5
5	4	4.5	5
5	5	5	6.5
5	9	7	6.5
9	4	6.5	7
9	5	7	7
9	9	9	9

We find the value of population median using values 4, 5, 9 here N = 3

here the values are already in ascending order, so

Population median = value of (N + 1)/2 th observation

= value of (3 + 1)/2 th observation

= value of 2th observation

Population median = 5

we have to find Sample median using value in last column in the above table N = 9

In last column we arrange the values in ascending order

Sample median = value of (N + 1)/2 th observation

= value of (9+ 1)/2 th observation

  = value of 5th observation

Sample median = 6.5

B.)Population median = 5 and Sample median = 6.5

which is not eqaul for above data.

C.) here the sample median is not target the value of the population median.

Generally, sample median make good estimators of population median

the sample median is a good estimator of the population mean when the population mean and population median are equal.

If the population mean and population median are different, then the sample median estimates the population median and will likely not do a good job of estimating the population mean.