Final answer:
Using the Central Limit Theorem, the probabilities of the sample mean being within 16 or 8 units of the population mean μ are both very close to 100%, due to the large distance when compared to the standard error.
Step-by-step explanation:
The question deals with the concept of the Central Limit Theorem (CLT), which is a fundamental principle in statistics used to predict the behavior of sample means when drawn from a population. When a random sample of size 100 is drawn from a population with a known standard deviation of 10, the Central Limit Theorem tells us that the sample means will be approximately normally distributed, especially given the sufficiently large sample size.
In part a, to find the probability that the sample mean will be within 16 of the population mean μ, we calculate a z-score range for the sample mean equidistant from μ at μ ± 16. Since the standard deviation of the sample means (σ/√n) is 1 (10/√100), a z-score of ± 16 corresponds to ± 16 standard deviations from the mean. Given this is well beyond the typical range depicted on a standard normal distribution, the probability should be very close to 100%.
For part b, we approach in a similar manner. The z-score for a distance of 8 from μ is ± 8 standard deviations. Again, using the standard normal distribution, this range is substantial and would also yield a probability close to 100%.