Question

7. Assume a population of 46​, 51​, 53​, and 59. Assume that samples of size n=2 are randomly selected with replacement from the population. Listed below are the sixteen different samples. Complete parts

​(a​) through (c​).

Sample x1        x2

1          46        46

2          46        51

3          46        53

4          46        59

5          51        46

6          51        51

7          51        53

8          51        59

9          53        46

10        53        51

11        53        53

12        53        59

13        59        46

14        59        51

15        59        53

16. 59        59

a.Find the median of each of the sixteen​ samples, then summarize the sampling distribution of the medians in the format of a table representing the probability distribution of the distinct median values. Use ascending order of the sample medians.

Social Median	Probability	Social Median	Probability
(69,92,46)		(78.5,105,52.5)
(71.5,97,48.5)		(80.5,53,106)
(99,49.5,74)		(110,55,82)
(76,102,51)		(56,84.5,112)
(76.5,52,104)		(88.5,118,59)

                   ​(Type integers or simplified fractions. Use ascending order of the sample​ medians.)

b. Compare the population median to the mean of the sample medians. Choose the correct answer below.

A. The population median is equal to the mean of the sample medians.

B. The population median is not equal to the mean of the sample medians​ (it is also not half or double the mean of the sample​ medians).

C. The population median is equal to double the mean of the sample medians.

D. The population median is equal to half of the mean of the sample medians.

c. Do the sample medians target the value of the population​ median? In​ general, do sample medians make unbiased estimators of population​ medians? Why or why​ not?

A. The sample medians target the population​ median, so sample medians are unbiased​ estimators, because the mean of the sample medians equals the population median.

B. The sample medians target the population​ median, so sample medians are biased​ estimators, because the mean of the sample medians equals the population median.

C. The sample medians do not target the population​ median, so sample medians are unbiased​ estimators, because the mean of the sample medians does not equal the population median.

D. The sample medians do not target the population​ median, so sample medians are biased​ estimators, because the mean of the sample medians does not equal the population median.

Answer 1

sample median           probability

(46+46)/2 = 69                   2/16 = 0.125

(46 +51)/2 = 48.5               2/16 = 0.125

= 49.5                               0.125

    52.5                                  0.125

48.5                                   0.125

51                                       0.0625

52                                       0.0625

55                                       0.125

49.5                                     0.125

52                                       0.125

55                                       0.0625

56                                       0.125

52.5                                    0.125

55                                      0.125

56                                        0.125

59                                       0.0625

median of the population = 46​, 51​, 53​, and 59

median = (51+53)/2 = 52

mean of sample medians = 861/16 =53.81

b)

Option B : The population median is not equal to the mean of the sample medians​ (it is also not half or double the mean of the sample​ medians).

c)

Option C : The sample medians do not target the population​ median, so sample medians are unbiased​ estimators, because the mean of the sample medians does not equal the population median.