Final answer:
The optimal sample size for estimating a population mean with a tolerance margin of error of 68, a confidence level of 95%, and a population standard deviation of 100.4 is approximately 18.
Step-by-step explanation:
Calculating the Optimal Sample Size
To calculate the optimal sample size for estimating a population mean with a known standard deviation, we use the formula for the error bound of the mean (EBM). Given a tolerated margin of error of 68, a desired confidence level of 95%, and an estimated population standard deviation (sigma) of 100.4, we need to find the appropriate z-score for a 95% confidence level, which is approximately 1.96 (corresponding to 95% of the data within the bounds of a normal distribution).
The formula for the EBM in this context is EBM = z * (sigma/sqrt(n)), where n is the sample size. Rearranging to solve for n, we get n = (z * sigma / EBM)^2.
Plugging in the values, we have n = (1.96 * 100.4 / 68)^2. Performing the calculations:
n = (1.96 * 100.4 / 68)^2
= (196 * 1.504 / 68)^2
= (4.292)^2
= 18.427
≈ 18
We round the result up to the nearest whole number since the sample size must be an integer. Therefore, the optimal sample size is approximately
18
.