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 5 of the true mean?
A) 61
B) 12
C) 31
D) 6

Question

Quizplus · Accepted Answer

To determine the minimum number of samples needed to be within a certain margin of error (in this case, 5) with a given level of confidence (95% in this case), we use the formula for the margin of error in estimating a population mean, which is $E = z \frac{\sigma}{\sqrt{n}}$, where $E$ is the margin of error, $z$ is the z-score corresponding to the desired confidence level (for 95% confidence, $z = 1.96$), $\sigma$ is the population standard deviation, and $n$ is the sample size. Rearranging the formula to solve for $n$, we get $n = \left(\frac{z\sigma}{E}\right)^2$. Plugging in the given values ($z = 1.96$, $\sigma = 14$, $E = 5$), we get $n = \left(\frac{1.96 \times 14}{5}\right)^2 = \left(\frac{27.44}{5}\right)^2 = \left(5.488\right)^2 \approx 30.1$. Since we can't have a fraction of a sample, we round up to the nearest whole number, which is 31.

If a Population Has a Standard Deviation of 14, What