Let's solve the problem step-by-step to find the mean and the sample standard deviation of the caffeine concentrations. We will round our results to the nearest hundredth.
Step 1: List the data
The caffeine concentrations in milligrams per ounce are as follows:
[tex]\[ 9.1, 8.9, 7.4, 7.7, 7.7, 8.7, 8.9, 8.9, 9.0, 9.1, 8.3, 9.1, 9.1, 8.3, 8.6, 8.9 \][/tex]
Step 2: Calculate the mean
The mean (average) is calculated by summing up all the values and then dividing by the number of values.
Sum the values:
[tex]\[ 9.1 + 8.9 + 7.4 + 7.7 + 7.7 + 8.7 + 8.9 + 8.9 + 9.0 + 9.1 + 8.3 + 9.1 + 9.1 + 8.3 + 8.6 + 8.9 \][/tex]
Divide the sum by the number of values (16 in this case):
[tex]\[ \text{Mean} = \frac{\text{Sum of all values}}{16} \][/tex]
Thus, the calculated mean (rounded to the nearest hundredth) is:
[tex]\[ \boxed{8.61} \][/tex]
Step 3: Calculate the sample standard deviation
Sample standard deviation is a measure of the amount of variation or dispersion in a set of values.
First, we calculate the variance. Variance is the average of the squared differences from the mean. For a sample, the formula is:
[tex]\[ s^2 = \frac{\sum (x_i - \bar{x})^2}{n-1} \][/tex]
where [tex]\( x_i \)[/tex] are the data points, [tex]\( \bar{x} \)[/tex] is the mean, and [tex]\( n \)[/tex] is the sample size.
The squared differences from the mean for each value are calculated, summed, and then divided by [tex]\( n-1 \)[/tex].
Finally, the square root of the variance gives the sample standard deviation:
[tex]\[ s = \sqrt{s^2} \][/tex]
The calculated sample standard deviation (rounded to the nearest hundredth) is:
[tex]\[ \boxed{0.56} \][/tex]
Summary:
- The mean caffeine concentration is [tex]\( \boxed{8.61} \)[/tex] mg/oz.
- The sample standard deviation of the caffeine concentration is [tex]\( \boxed{0.56} \)[/tex] mg/oz.
