Question

Refer to Exercise 5.3. Assume that a random sample of n = 2 measurements is randomly selected from the population.

a. List the different values that the sample median m may assume and find the probability of each. Then give the sampling distribution of the sample median.

b. Construct a probability histogram for the sampling distribution of the sample median and compare it with the probability histogram for the sample mean (Exercise 5.3, part b).

Short answer

a.

Median	Probability
1	0.04
1.5	0.12
2	0.17
2.5	0.20
3	0.20
3.5	0.14
4	0.08
4.5	0.04
5	0.01

b.

Step-by-step solution

Calculation of the medians

a.

The values of the medians of the respective samples taken from Exercise 5.3 have been determined by taking the central value, as shown below.

Sample	Median
1,1	1
1,2	1.5
1,3	2
1,4	2.5
1,5	3
2,1	1.5
2,2	2
2,3	2.5
2,4	3
2,5	3.5
3,1	2
3,2	2.5
3,3	3
3,4	3.5
3,5	4
4,1	2.5
4,2	3
4,3	3.5
4,4	4
4,5	4.5
5,1	3
5,2	3.5
5,3	4
5,4	4.5
5,5	5

Finding the sample distribution

b.

The respective probabilities of the given samples are added to get the final probabilities of the medians, as shown below

Median	Probability
1	0.04
1.5	$\mathbf{0}\mathbf{.}\mathbf{06}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{06}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{12}$
2	$\mathbf{0}\mathbf{.}\mathbf{04}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{09}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{04}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{17}$
2.5	$\mathbf{0}\mathbf{.}\mathbf{04}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{06}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{06}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{04}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{20}$
3	$\mathbf{0}\mathbf{.}\mathbf{02}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{06}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{04}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{06}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{02}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{20}$
3.5	role="math" localid="1658120147164" $\mathbf{0}\mathbf{.}\mathbf{03}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{04}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{04}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{03}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{14}$
4	$\mathbf{0}\mathbf{.}\mathbf{02}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{04}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{02}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{08}$
4.5	$\mathbf{0}\mathbf{.}\mathbf{02}\mathbf{+}\mathbf{0}\mathbf{.}\mathbf{02}\mathbf{=}\mathbf{0}\mathbf{.}\mathbf{04}$
5	0.01

The sample distribution shows that the probabilities of the respective samples are greater than 0 but less than 1.

Elucidation of the histogram

The graph contains probabilities on the y-axis and the values of the respective medians of x from 1 to 5 on the x-axis. From the graph, we can say that 2.5 and 3 show the highest probability, which is 0.20.

Comparison between the probability histograms of the mean and the median

Exercise 5.3 deals with the mean where the probability histogram of the mean has been drawn.On comparing the same with that of the medians, it can be deduced that the probability histograms are exactly the same.