Final answer:
The sampling distribution of x, when taking a random sample size of 16 from a normally distributed population with mean 444 and standard deviation 32, is Normal with mean 444 and standard deviation 8. This is option 'b' and can be derived using the Central Limit Theorem.
Step-by-step explanation:
When drawing a random sample of size 16 from a normally distributed population with a known mean (µ444) and standard deviation (µ32), the sampling distribution of the sample mean (µx) can be described by the Central Limit Theorem. According to this theorem, if the sample size (n) is large enough, the distribution of the sample means will be approximately normal. Since our population distribution is already normal, even for a sample size of 16, the sampling distribution of the sample mean will also be normal.
The mean of the sampling distribution (µx) will be the same as the population mean (µ444), which means that, on average, sample means are expected to equal the population mean. However, the standard deviation of the sampling distribution (also known as the standard error) is equal to the population standard deviation (µ32) divided by the square root of the sample size (µnµ16). Therefore, the correct standard error would be 32/√16, which is 32/4, yielding a standard error of 8.
Consequently, the sampling distribution of x would be described as: Normal with mean 444 and standard deviation 8, which corresponds to option 'b' from the set of choices provided. Therefore, the answer to the student's question would be option b: Normal with mean 444 and standard deviation 8.