Final answer:
To ensure a 95% confidence level that the sample mean is within 2 of the true mean, with a population standard deviation of 14, at least 189 samples are needed.
Step-by-step explanation:
To answer the question: If a population has a standard deviation of 14, what is the minimum number of samples that need to be averaged in order to be 95% confident that the average of the means is within 2 of the true mean, we need to apply the formula for the margin of error in a confidence interval:
E = Z * (σ/sqrt(n))
Where E is the desired margin of error, Z is the z-score corresponding to the desired confidence level (in this case, 95%, which corresponds to a Z-score of approximately 1.96), σ is the standard deviation of the population, and n is the sample size. Rearranging the formula to solve for n gives us:
n = (Z * σ / E)^2
Substituting the known values into the equation:
n = (1.96 * 14 / 2)^2
n = (27.44 / 2)^2
n = (13.72)^2
n = 188.3584
Since we can only have a whole number of samples, we round up to get n = 189. Therefore, we need a minimum of 189 samples to be 95% confident that our sample mean is within 2 units of the true population mean.