A common symptom of otitis media in young children is the pr

A common symptom of otitis media in young children is the prolonged presence of fluid in the middle ear, known as middle-ear effusion. The presence of fluid may result in temporary hearing loss and interfere with normal learning skills in the first 2 years of life. One hypothesis is that babies who are breastfed for at least 1 month build up some immunity against the effects of the infection and have less prolonged effusion than do bottle-fed babies. A small study of 24 pairs of babies is set up, in which the babies are matched on a one-to-one basis according to age, sex, socioeconomic status, and type of medications taken. One member of the matched pair is a breastfed baby, and the other member is a bottle-fed baby. The outcome variable is the duration of middle-ear effusion after the first episode of otitis media. Using the Wilcoxon signed-rank test, test the hypothesis that the duration of effusion is the same among breastfed babies and bottle-fed babies.

Breastfed

Bottle-fed

Breastfed

Bottle-fed

Breastfed

Bottle-fed

Breastfed

Bottle-fed

20

18

58

223

52

39

65

77

11

35

7

7

14

15

10

12

3

7

39

57

12

21

7

8

24

182

17

76

30

28

19

16

7

6

17

186

7

8

34

28

28

33

12

29

15

27

25

20

Breastfed	Bottle-fed	Breastfed	Bottle-fed	Breastfed	Bottle-fed	Breastfed	Bottle-fed
20	18	58	223	52	39	65	77
11	35	7	7	14	15	10	12
3	7	39	57	12	21	7	8
24	182	17	76	30	28	19	16
7	6	17	186	7	8	34	28
28	33	12	29	15	27	25	20

Solution

Null Hypothesis, H₀: Any difference between breastfed and bottlefed is due to chance.

Alternative Hypothesis, H₁: Any difference between breastfed and bottlefed is not due to chance.

The value of the test statistics is the minimum of the sum of positive ranks and negative ranks.

Hence here, t=61.

From Wilcoxon Signed-Ranks Table, for n = 23 and alpha = 0.05, t critical = 73.

Since, t < t critical, we fail to reject H₀ and conclude that any difference between breastfed and bottlefed is due to chance.

Breastfed	Bottlefed	Diff	Abs Diff	Rank of Abs Diff	Positive Ranks	Negative Ranks
20	18	2	2	6	6
11	35	-24	24	19		19
3	7	-4	4	9		9
24	182	-158	158	21		21
7	6	1	1	2.5	2.5
28	33	-5	5	10.5		10.5
58	223	-165	165	22		22
7	7	0
39	57	-18	18	18		18
17	76	-59	59	20		20
17	186	-169	169	23		23
12	29	-17	17	17		17
52	39	13	13	16	16
14	15	-1	1	2.5		2.5
12	21	-9	9	13		13
30	28	2	2	6	6
7	8	-1	1	2.5		2.5
15	27	-12	12	14.5		14.5
65	77	-12	12	14.5		14.5
10	12	-2	2	6		6
7	8	-1	1	2.5		2.5
19	16	3	3	8	8
34	28	6	6	12	12
25	20	5	5	10.5	10.5
					61	215