To find the standard deviation of the sample containers' contents, we follow these steps:
1. Collect the sample data: The contents of the 10 containers are given as follows:
[tex]\[
20.3, 19.91, 19.98, 19.69, 19.54, 20.51, 20.36, 20.46, 19.68, 20.56 \text{ liters}
\][/tex]
2. Calculate the mean: The mean (average) of the sample contents is calculated by summing all the values and then dividing by the number of observations.
[tex]\[
\text{Mean} = \frac{20.3 + 19.91 + 19.98 + 19.69 + 19.54 + 20.51 + 20.36 + 20.46 + 19.68 + 20.56}{10} = 20.099 \text{ liters}
\][/tex]
3. Find the variance: The variance is a measure of how much the values in the sample differ from the mean. For a sample, the variance ([tex]\(s^2\)[/tex]) is calculated by summing the squared differences from the mean and then dividing by the number of observations minus 1.
[tex]\[
\text{Variance} = \frac{1}{n-1}\sum_{i=1}^{n}(x_i - \bar{x})^2
\][/tex]
Where [tex]\(n = 10\)[/tex] and [tex]\(\bar{x} = 20.099\)[/tex].
Substituting the given data, we get:
[tex]\[
\text{Variance} = 0.14727666666666672 \text{ liters}^2
\][/tex]
4. Calculate the standard deviation: The standard deviation ([tex]\(\sigma\)[/tex]) is the square root of the variance, which provides a measure of the dispersion of the sample data around the mean.
[tex]\[
\text{Standard Deviation} = \sqrt{\text{Variance}} = \sqrt{0.14727666666666672} = 0.3837664220156145 \text{ liters}
\][/tex]
Thus, the standard deviation of the sampled containers' contents is approximately [tex]\(0.384\)[/tex], which doesn't exactly match any of the provided options. However, it is closest to:
[tex]\[
\sigma \approx 0.36
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{\sigma = 0.36}
\][/tex]
