To find the standard deviation of the sampled containers' contents, follow these steps:
1. Gather the data:
The contents of the 10 containers are:
[tex]\[ 20.84, 19.62, 19.74, 19.72, 20.58, 20.85, 20.99, 19.62, 20.38, 20.06 \][/tex]
2. Calculate the mean (average) of the data:
First, sum up all the values, then divide by the number of observations.
[tex]\[ \text{Mean} = \frac{20.84 + 19.62 + 19.74 + 19.72 + 20.58 + 20.85 + 20.99 + 19.62 + 20.38 + 20.06}{10} \][/tex]
3. Calculate each value's deviation from the mean:
Subtract the mean from each individual data point to get the deviations.
4. Square each deviation:
This step ensures that all deviations are positive and emphasizes larger deviations.
5. Calculate the variance:
Sum all the squared deviations and divide by the number of observations minus one (N-1, because this is sample variance).
[tex]\[ \text{Variance} = \frac{\sum (\text{deviation}^2)}{10-1} \][/tex]
6. Calculate the standard deviation:
The standard deviation is the square root of the variance.
Given steps, calculations for the mean, deviations, variance, and finally standard deviation yield:
[tex]\[ \sigma \approx 0.552 \][/tex]
So, the correct option that closely matches our calculated standard deviation value is:
D. [tex]\(\sigma=0.53\)[/tex]
