Final answer:
The sampling distribution of the sample mean for a normally distributed population with a mean of 72 and standard deviation of 12, based on samples of size 9, will also be normally distributed with a mean of 72 and a standard deviation of 4.
Step-by-step explanation:
If a population is known to be normally distributed with a mean (μ) of 72 and a standard deviation (σ) of 12, the characteristics of the sampling distribution for the sample mean (x-bar) based on a random sample of size 9 selected from the population can be described using the Central Limit Theorem (CLT).
Since the sample size is small, typically a t-distribution would be used, however, since the population standard deviation is known and the population is normal, we can use the normal distribution for the sampling distribution.
According to the CLT, the sampling distribution of the sample mean will be normally distributed with a mean equal to the population mean (μ = 72) and a standard deviation equal to the population standard deviation divided by the square root of the sample size (σ/√n = 12/√9 = 12/3 = 4). Therefore, the sampling distribution for x-bar will have a mean of 72 and a standard deviation of 4.
This means that if you were to take many samples of size 9 from this population, the sample means would form a distribution that is approximately normal, centered around 72, and with less variability than the original population due to the reduced standard deviation of the sampling distribution.