Question

A survey of 16 energy drinks noted the caffeine concentration of each drink in milligrams per ounce. The results are given in the table below.

Concentration of caffeine (mg/oz)

9.1 8.9 7.4 7.8 7.7 8.9 8.9 9.0 9.0 9.1 8.2 9.0 9.2 8.1 8.6 9.0

Find the mean and sample standard deviation of these data. Round to the nearest hundredth.

A. mean

B. sample standard deviation

Answer 1

Solution:

The formulas for mean, variance, and standard deviation for sample are given as below:

Sample Mean = X̄ = ∑ X/n

Sample Variance = S² = ∑[ (X - mean)^2]/(n - 1)

Sample Standard deviation = S = Sqrt(S²) = Sqrt(Variance)

The calculation table is given as below:

No.	X	(X - mean)^2
1	9.1	0.231601562
2	8.9	0.079101563
3	7.4	1.485351563
4	7.8	0.670351563
5	7.7	0.844101563
6	8.9	0.079101563
7	8.9	0.079101563
8	9	0.145351563
9	9	0.145351563
10	9.1	0.231601562
11	8.2	0.175351563
12	9	0.145351563
13	9.2	0.337851562
14	8.1	0.269101563
15	8.6	0.000351563
16	9	0.145351563
Total	137.9	5.064375

From above table, we have

n = 16

Sample Mean = X̄ = 137.9/16 = 8.61875

Sample mean = 8.62

Sample Variance = S² = 5.064375/(16 - 1) = 0.337625

Sample Standard deviation = S =sqrt(0.337625) = 0.581055075

Sample Standard deviation = 0.58