In a study to estimate a population mean, i start with a sample size 10 and calculate the 95% confidence interval. Suppose i am not satisfied with the size of the confidence interval and want to reduce it by half. Give an estimate of sample size that i need.
To reduce the width of a confidence interval in half, you have to multiply the sample size by 4 (in effect quadrupling the sample size).
To estimate the population mean with a smaller confidence interval, you would need a larger sample size. However, it is important to note that reducing the confidence interval by half would require a fourfold increase in the sample size. Let's go through the steps to estimate the required sample size.
1. Identify the initial sample size and confidence interval:
- Sample Size (n1) = 10
- Confidence Interval = 95%
2. Determine the desired reduction in the confidence interval:
- We want to reduce the confidence interval by half (i.e., reduce it to 50% of the original size).
3. Calculate the adjustment factor:
- Adjustment Factor (AF) = desired reduction in confidence interval / (100% - desired reduction in confidence interval)
- AF = 50% / (100% - 50%) = 0.5 / 0.5 = 1
4. Apply the adjustment factor to the initial sample size:
- Required Sample Size (n2) = Sample Size (n1) * Adjustment Factor (AF)
- n2 = n1 * AF
- n2 = 10 * 1 = 10
5. Multiply n2 by the square of the adjustment factor to achieve the desired reduction in the confidence interval:
- Required Sample Size = n2 * (AF^2)
- Required Sample Size = 10 * (1^2)
- Required Sample Size = 10
In this case, the estimated sample size needed to reduce the confidence interval by half is still 10. However, it is worth noting that obtaining a smaller confidence interval often requires a larger sample size.