What is the appropriate sample size for a confidence interval around the sample mean if you want a margin of error of 0.8 with 95% confidence and a population standard deviation of 12.
95% = mean ± 1.96 SEm
SEm = SD/√n
.8 = 1.96 (12/√n)
Solve for n.
To determine the appropriate sample size for a confidence interval around the sample mean, you can use the formula:
n = (Z * σ / E)²
Where:
- n is the required sample size
- Z is the Z-score that corresponds to the desired level of confidence (in this case, 95% confidence)
- σ is the population standard deviation
- E is the desired margin of error
In this case, the desired margin of error is given as 0.8, the population standard deviation is 12, and the desired confidence level is 95%.
Step 1: Find the Z-score
To find the Z-score for a 95% confidence level, you can use a Z-table or a statistical calculator. The Z-score for a 95% confidence level is approximately 1.96.
Step 2: Calculate the required sample size
Plug the values into the formula:
n = (1.96 * 12 / 0.8)²
≈ (23.52 / 0.8)²
≈ 29.4²
≈ 864.36
So, the appropriate sample size for a confidence interval around the sample mean with a margin of error of 0.8, 95% confidence, and a population standard deviation of 12 is approximately 865.