Question

Is honey a cough remedy?Refer to the Archives of Pediatrics and Adolescent Medicine(December 2007) study of honeyas a remedy for coughing, Exercise 2.31 (p. 86). The coughingimprovement scores (as determined by the children’s parents)for the patients in the over-the-counter cough medicinedosage (DM) group, honey dosage group, and control groupare reproduced in the accompanying table.

Honey Dosage	12, 11, 15, 11, 10, 13, 10, 4 ,15, 16, 9, 14, 10, 6, 10, 8, 11, 12, 12, 8, 12, 9, 11, 15, 10, 15, 9, 13, 8, 12, 10, 8, 9, 5, 12
DM Dosage	4, 6, 9, 4, 7, 7, 7, 9, 12 ,10, 11, 6, 3, 4, 9, 12, 7, 6, 8, 12, 12, 4, 12, 13, 7, 10
No Dosage (Control)	13, 9, 4, 4, 10, 15, 9, 5, 8, 6, 1, 0, 8, 12, 8, 7, 7, 1, 6, 7, 7, 12, 7, 9, 7, 9, 5, 11, 9, 5, 6, 8, 8, 6, 7, 10, 9, 4, 8, 7, 3, 1, 4, 3

a.Find the standard deviation of the improvement scores for the honey dosage group.

b.Find the standard deviation of the improvement scores for the DM dosage group.

c.Find the standard deviation of the improvement scores for the control group.

d.Based on the results, parts a–c, which group appears to have the most variability in coughing improvement scores? The least variability?

Short answer

a. Standard Deviation = 2.85

b. Standard Deviation = 3.037

c. Standard Deviation = 2.976

d. Most Variable = DM dosage group

Least Variable = Honey dosage group

Step-by-step solution

Finding the standard deviation for honey dosage group

$$\begin{gathered} \begin{array}{rcl}\mathbf{Mean}& \mathbf{=}& \frac{\mathbf{Sumofallscores}}{\mathbf{No}\mathbf{.}\mathbf{ofscores}}\\ & \mathbf{=}& \frac{\mathbf{12}\mathbf{+}\mathbf{11}\mathbf{+}\mathbf{15}\mathbf{+}\mathbf{11}\mathbf{+}\mathbf{10}\mathbf{+}\mathbf{13}\mathbf{+}\mathbf{10}\mathbf{+}\mathbf{4}\mathbf{+}\mathbf{15}\mathbf{+}\mathbf{16}\mathbf{+}\mathbf{9}\mathbf{+}\mathbf{14}\mathbf{+}\mathbf{10}\mathbf{+}\mathbf{6}\mathbf{+}\mathbf{10}\mathbf{+}\mathbf{8}\mathbf{+}\mathbf{11}\mathbf{+} \\ \mathbf{12}\mathbf{+}\mathbf{12}\mathbf{+}\mathbf{8}\mathbf{+}\mathbf{12}\mathbf{+}\mathbf{9}\mathbf{+}\mathbf{11}\mathbf{+}\mathbf{15}\mathbf{+}\mathbf{10}\mathbf{+}\mathbf{15}\mathbf{+}\mathbf{9}\mathbf{+}\mathbf{13}\mathbf{+}\mathbf{8}\mathbf{+}\mathbf{12}\mathbf{+}\mathbf{10}\mathbf{+}\mathbf{8}\mathbf{+}\mathbf{9}\mathbf{+}\mathbf{5}\mathbf{+}\mathbf{12} \\ }{\mathbf{35}}\\ & \mathbf{=}& \frac{\mathbf{375}}{\mathbf{35}}\\ & \mathbf{=}& \mathbf{10}\mathbf{.}\mathbf{71}\\ & & \end{array} \end{gathered}$$

x	$(\mathrm{x}-\overline{\mathrm{x}})$	$(\mathrm{x}-\overline{\mathrm{x}}{)}^2$
12	1.29	1.66
11	0.29	0.08
15	4.29	18.40
11	0.29	0.08
10	-0.71	0.50
13	2.29	5.24
10	-0.71	0.50
4	-6.71	45.02
15	4.29	18.40
16	5.29	27.98
9	-1.71	2.92
14	3.29	10.82
10	-0.71	0.50
6	-4.71	22.18
10	-0.71	0.50
8	-2.71	7.34
11	0.29	0.08
12	1.29	1.66
12	1.29	1.66
8	-2.71	7.34
12	1.29	1.66
9	-1.71	2.92
11	0.29	0.08
15	4.29	18.40
10	-0.71	0.50
15	4.29	18.40
9	-1.71	2.92
13	2.29	5.24
8	-2.71	7.34
12	1.29	1.66
10	-0.71	0.50
8	-2.71	7.34
9	-1.71	2.92
5	-5.71	32.60
12	1.29	1.66
Sum	0	277.14

$\begin{array}{rcl}\mathbf{Variance}& \mathbf{=}& \frac{\mathbf{\sum }\mathbf{(}\mathbf{x}\mathbf{-}\overline{\mathbf{x}}{\mathbf{)}}^{\mathbf{2}}}{\mathbf{n}\mathbf{-}\mathbf{1}}\\ & \mathbf{=}& \frac{\mathbf{277}\mathbf{.}\mathbf{14}}{\mathbf{35}\mathbf{-}\mathbf{1}}\\ & \mathbf{=}& \frac{\mathbf{277}\mathbf{.}\mathbf{14}}{\mathbf{34}}\\ & \mathbf{=}& \mathbf{8}\mathbf{.}\mathbf{15}\\ & & \\ \mathbf{StandardDeviation}& \mathbf{=}& \sqrt{\mathbf{Variance}}\\ & \mathbf{=}& \sqrt{\mathbf{8}\mathbf{.}\mathbf{15}}\\ & \mathbf{=}& \mathbf{2}\mathbf{.}\mathbf{85}\\ & & \end{array}$

Therefore, Standard deviation for honey dosage group is 2.85.

Calculating the standard deviation for DM dosage group

$$\begin{gathered} \begin{array}{rcl}\mathbf{Mean}& \mathbf{=}& \frac{\mathbf{Sumofallscores}}{\mathbf{No}\mathbf{.}\mathbf{ofscores}}\\ & \mathbf{=}& \frac{\mathbf{4}\mathbf{+}\mathbf{6}\mathbf{+}\mathbf{9}\mathbf{+}\mathbf{4}\mathbf{+}\mathbf{7}\mathbf{+}\mathbf{7}\mathbf{+}\mathbf{7}\mathbf{+}\mathbf{9}\mathbf{+}\mathbf{12}\mathbf{+}\mathbf{10}\mathbf{+}\mathbf{11}\mathbf{+}\mathbf{6}\mathbf{+}\mathbf{3}\mathbf{+}\mathbf{4}\mathbf{+} \\ \mathbf{9}\mathbf{+}\mathbf{12}\mathbf{+}\mathbf{7}\mathbf{+}\mathbf{6}\mathbf{+}\mathbf{8}\mathbf{+}\mathbf{12}\mathbf{+}\mathbf{12}\mathbf{+}\mathbf{4}\mathbf{+}\mathbf{12}\mathbf{+}\mathbf{13}\mathbf{+}\mathbf{7}\mathbf{+}\mathbf{10} \\ }{\mathbf{26}}\\ & \mathbf{=}& \frac{\mathbf{211}}{\mathbf{26}}\\ & \mathbf{=}& \mathbf{8}\mathbf{.}\mathbf{115}\\ & & \\ & & \end{array} \end{gathered}$$

x	$(\mathrm{x}-\overline{\mathrm{x}})$	$(\mathrm{x}-\overline{\mathrm{x}}{)}^2$
4	-4.115	16.93
6	-2.115	4.47
9	0.885	0.78
4	-4.115	16.93
7	-1.115	1.24
7	-1.115	1.24
7	-1.115	1.24
9	0.885	0.78
12	3.885	15.09
10	1.885	3.55
11	2.885	8.32
6	-2.115	4.47
3	-5.115	26.16
4	-4.115	16.93
9	0.885	0.78
12	3.885	15.09
7	-1.115	1.24
6	-2.115	4.47
8	-0.115	0.01
12	3.885	15.09
12	3.885	15.09
4	-4.115	16.93
12	3.885	15.09
13	4.885	23.86
7	-1.115	1.24
10	1.885	3.55
Sum	0	230.65

$\begin{array}{rcl}\mathbf{Variance}& \mathbf{=}& \frac{\mathbf{\sum }\mathbf{(}\mathbf{x}\mathbf{-}\overline{\mathbf{x}}{\mathbf{)}}^{\mathbf{2}}}{\mathbf{n}\mathbf{-}\mathbf{1}}\\ & \mathbf{=}& \frac{\mathbf{230}\mathbf{.}\mathbf{65}}{\mathbf{26}\mathbf{-}\mathbf{1}}\\ & \mathbf{=}& \frac{\mathbf{230}\mathbf{.}\mathbf{65}}{\mathbf{25}}\\ & \mathbf{=}& \mathbf{9}\mathbf{.}\mathbf{226}\\ & & \\ \mathbf{StandardDeviation}& \mathbf{=}& \sqrt{\mathbf{Variance}}\\ & \mathbf{=}& \sqrt{\mathbf{9}\mathbf{.}\mathbf{226}}\\ & \mathbf{=}& \mathbf{3}\mathbf{.}\mathbf{037}\\ & & \end{array}$

Therefore, Standard deviation for DM dosage group is 3.03.

Finding the standard deviation for honey dosage group

$$\begin{gathered} \begin{array}{rcl}\mathbf{Mean}& \mathbf{=}& \frac{\mathbf{Sumofallscores}}{\mathbf{No}\mathbf{.}\mathbf{ofscores}}\\ & \mathbf{=}& \frac{\mathbf{5}\mathbf{+}\mathbf{8}\mathbf{+}\mathbf{6}\mathbf{+}\mathbf{1}\mathbf{+}\mathbf{0}\mathbf{+}\mathbf{8}\mathbf{+}\mathbf{12}\mathbf{+}\mathbf{8}\mathbf{+}\mathbf{7}\mathbf{+}\mathbf{7}\mathbf{+}\mathbf{1}\mathbf{+}\mathbf{6}\mathbf{+}\mathbf{7}\mathbf{+}\mathbf{7}\mathbf{+}\mathbf{12}\mathbf{+}\mathbf{7}\mathbf{+}\mathbf{9}\mathbf{+}\mathbf{5}\mathbf{+} \\ \mathbf{6}\mathbf{+}\mathbf{8}\mathbf{+}\mathbf{8}\mathbf{+}\mathbf{6}\mathbf{+}\mathbf{7}\mathbf{+}\mathbf{10}\mathbf{+}\mathbf{9}\mathbf{+}\mathbf{4}\mathbf{+}\mathbf{8}\mathbf{+}\mathbf{7}\mathbf{+}\mathbf{3}\mathbf{+}\mathbf{1}\mathbf{+}\mathbf{4}\mathbf{+}\mathbf{8}\mathbf{+}\mathbf{7}\mathbf{+}\mathbf{3}\mathbf{+}\mathbf{1}\mathbf{+}\mathbf{4}\mathbf{+}\mathbf{3} \\ }{\mathbf{37}}\\ & \mathbf{=}& \frac{\mathbf{223}}{\mathbf{37}}\\ & \mathbf{=}& \mathbf{6}\mathbf{.}\mathbf{027}\\ & & \end{array} \end{gathered}$$

x	$(\mathrm{x}-\overline{\mathrm{x}})$	$(\mathrm{x}-\overline{\mathrm{x}}{)}^2$
5	-1.027	1.05
8	1.973	3.89
6	-0.027	0.0007
1	-5.027	25.27
0	-6.027	36.32
8	1.973	3.89
12	5.973	35.67
8	1.973	3.89
7	0.973	0.94
7	0.973	0.94
1	-5.027	25.27
6	-0.027	0.0007
7	0.973	0.94
7	0.973	0.94
12	5.973	35.67
7	0.973	0.94
9	2.973	8.83
5	-1.027	1.05
6	-0.027	0.0007
8	1.973	3.89
8	1.973	3.89
6	-0.027	0.0007
7	0.973	0.94
10	3.973	15.78
9	2.973	8.83
4	-2.027	4.11
8	1.973	3.89
7	0.973	0.94
3	-3.027	9.16
1	-5.027	25.27
4	-2.027	4.11
8	1.973	3.89
7	0.973	0.94
3	-3.027	9.16
1	-5.027	25.27
4	-2.027	4.11
3	-3.027	9.16
Sum	0	318.973

$\begin{array}{rcl}\mathbf{Variance}& \mathbf{=}& \frac{\mathbf{\sum }\mathbf{(}\mathbf{x}\mathbf{-}\overline{\mathbf{x}}{\mathbf{)}}^{\mathbf{2}}}{\mathbf{n}\mathbf{-}\mathbf{1}}\\ & \mathbf{=}& \frac{\mathbf{318}\mathbf{.}\mathbf{973}}{\mathbf{37}\mathbf{-}\mathbf{1}}\\ & \mathbf{=}& \frac{\mathbf{318}\mathbf{.}\mathbf{973}}{\mathbf{36}}\\ & \mathbf{=}& \mathbf{8}\mathbf{.}\mathbf{86}\\ & & \\ \mathbf{StandardDeviation}& \mathbf{=}& \sqrt{\mathbf{Variance}}\\ & \mathbf{=}& \sqrt{\mathbf{8}\mathbf{.}\mathbf{86}}\\ & \mathbf{=}& \mathbf{2}\mathbf{.}\mathbf{97}6\\ & & \end{array}$

Therefore, Standard deviation for control group is 2.976.

Identifying the least and the most variable group

The most variable group is the DM dosage group as it has the highest standard deviation out of the 3.

The least variable group is the honey dosage group with the lowest standard deviation value.