An extended study is being designed to sample the release of estrogens into the environment from a hog farrowing farm in North Carolina. An initial set of samples (n = 6) showed that the %RSD of analysis was 15.3%, for an average estrogen level of 133.5 ng/L in sediment outside of the barn. How many samples will need to be taken to reduce the variance to under a value of 200, if it is assumed that the average estrogen level will remain the same?
To determine the number of samples needed to reduce the variance to a specific value, we can use a formula involving the Coefficient of Variation (CV) and the desired level of variance.
The Coefficient of Variation (CV) is a measure of the relative variability in the data and can be calculated as the ratio of the standard deviation to the mean, expressed as a percentage.
In this case, the %RSD (relative standard deviation) is given as 15.3%, which is equivalent to CV = 15.3 / 100 = 0.153.
The formula to calculate the required number of samples can be derived as follows:
n = ( (Z * CV) / (desired CV) )^2
Where:
- n is the number of samples required
- Z is the Z-score corresponding to the desired level of confidence (in this case, we can use Z = 1 to achieve a desired level of variance)
- CV is the current Coefficient of Variation
- desired CV is the desired Coefficient of Variation
Given that the current CV is 0.153 and the desired variance is 200, let's calculate the required number of samples:
n = ( (1 * 0.153) / (sqrt(200 / 133.5)) )^2
First, let's calculate sqrt(200 / 133.5):
sqrt(200 / 133.5) ≈ 1.100
Plugging this value into the formula:
n = ( (1 * 0.153) / 1.100 )^2
= (0.153 / 1.100)^2
= 0.139^2
≈ 0.0193
Rounding up to the nearest whole number, the required number of samples is approximately 1.
Therefore, to reduce the variance to under a value of 200, only one additional sample would be required, assuming the average estrogen level remains the same.