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The coefficient of variation of the rainfall for six rain gauge stations in a catchment was found to be 29.54% The optimum number of stations in the catchment for an admissible 10% error in the estimation of mean rainfall will be:
- a)3
- b)6
- c)9
- d)12
Correct answer is option 'C'. Can you explain this answer?
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Understanding Coefficient of Variation
The coefficient of variation (CV) is a statistical measure of the relative variability of a dataset. It is defined as the ratio of the standard deviation to the mean, expressed as a percentage. In this case, the CV of the rainfall data is 29.54%.
Estimation Error and Sample Size
To estimate the mean rainfall with a certain level of precision, the sample size required can be determined using the formula:
Sample Size (n) = (Z^2 * CV^2) / E^2
Where:
- Z = Z-value corresponding to the desired confidence level
- CV = Coefficient of Variation
- E = Acceptable error (margin of error)
For a 95% confidence level, Z is typically taken as 1.96.
Calculating the Optimum Sample Size
1. Coefficient of Variation: CV = 29.54% = 0.2954
2. Desired Error (E): 10% = 0.1
3. Z-value: For 95% confidence, Z = 1.96
Now, substituting these values into the formula:
Sample Size (n) = (1.96^2 * 0.2954^2) / (0.1^2)
Calculating this yields an approximate sample size of 9.
Conclusion
Based on the calculations, the optimum number of stations required for an admissible 10% error in the estimation of mean rainfall is approximately 9. Therefore, the correct answer is option 'C'.
This approach ensures adequate representation of rainfall variability while maintaining the desired accuracy in mean estimation.
The coefficient of variation (CV) is a statistical measure of the relative variability of a dataset. It is defined as the ratio of the standard deviation to the mean, expressed as a percentage. In this case, the CV of the rainfall data is 29.54%.
Estimation Error and Sample Size
To estimate the mean rainfall with a certain level of precision, the sample size required can be determined using the formula:
Sample Size (n) = (Z^2 * CV^2) / E^2
Where:
- Z = Z-value corresponding to the desired confidence level
- CV = Coefficient of Variation
- E = Acceptable error (margin of error)
For a 95% confidence level, Z is typically taken as 1.96.
Calculating the Optimum Sample Size
1. Coefficient of Variation: CV = 29.54% = 0.2954
2. Desired Error (E): 10% = 0.1
3. Z-value: For 95% confidence, Z = 1.96
Now, substituting these values into the formula:
Sample Size (n) = (1.96^2 * 0.2954^2) / (0.1^2)
Calculating this yields an approximate sample size of 9.
Conclusion
Based on the calculations, the optimum number of stations required for an admissible 10% error in the estimation of mean rainfall is approximately 9. Therefore, the correct answer is option 'C'.
This approach ensures adequate representation of rainfall variability while maintaining the desired accuracy in mean estimation.
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The coefficient of variation of therainfall for six rain gauge stationsin a catchment was found to be29.54% The optimum number ofstations in the catchment for anadmissible 10% error in theestimation of mean rainfall will be:a)3b)6c)9d)12Correct answer is option 'C'. Can you explain this answer?
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