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 below a certain value, we can use a formula known as the power analysis formula. This formula allows us to calculate the sample size required based on the desired level of significance (α), the desired power (1-β), the current variance, and the minimum effect size.
In this case, the current variance is given as the %RSD (relative standard deviation) of analysis, which is 15.3%. To convert this to a decimal form, we divide it by 100: 15.3% / 100 = 0.153.
The desired variance is 200 (since we want to reduce it below this value).
We can now use the power analysis formula to calculate the required sample size:
n = [ (Zα/2 + Zβ) * (σ / δ) ] ^ 2
Where:
- n: required sample size
- Zα/2: critical value for the desired level of significance (α)
- Zβ: critical value for the desired power (1-β)
- σ: current standard deviation (calculated by multiplying the current variance with the mean)
- δ: minimum effect size (equal to the current variance minus the desired variance)
Let's calculate the required sample size using the given information:
1. Zα/2: The critical value for the desired level of significance (α) depends on the threshold chosen. Let's assume it to be 0.05 (95% confidence level). The corresponding value for Zα/2 can be found using a Z-table or a statistical calculator. For a confidence level of 0.05 (two-tailed test), Zα/2 ≈ 1.96.
2. Zβ: The critical value for the desired power (1-β) depends on the chosen power and test type. Assuming a power of 0.8 (80%) and a two-tailed test, the corresponding value for Zβ can be found using a Z-table or a statistical calculator. Let's assume Zβ ≈ 0.84 for a power of 0.8.
3. σ: The current standard deviation can be calculated by multiplying the current variance (0.153) with the mean (133.5 ng/L). σ = 0.153 * 133.5 = 20.4355.
4. δ: The minimum effect size is the difference between the current variance and the desired variance: δ = 0.153 - 0.2 = -0.047.
Now we can substitute these values into the power analysis formula:
n = [ (Zα/2 + Zβ) * (σ / δ) ] ^ 2
= [ (1.96 + 0.84) * (20.4355 / -0.047) ] ^ 2
≈ [ 2.8 * (-434.4681) ] ^ 2
≈ (-1216.4785) ^ 2
≈ 1,480,513.39
Therefore, approximately 1,480,514 samples will need to be taken to reduce the variance to under a value of 200, assuming the average estrogen level remains the same.
It's important to note that such a large sample size might not be practical or feasible in a real-world scenario. Therefore, you may need to consider alternative approaches, such as reducing measurement variability or optimizing sampling strategies, to achieve an acceptable level of precision within a more reasonable sample size.