Final answer:
The sampling distribution of the sample mean will be normal only if the population distribution is normal or the sample size is sufficiently large. Changing the sample size affects the standard deviation of the sampling distribution of the sample means. When testing a single population mean, use a z-test if the variance is known or a t-test if the variance is unknown and the sample size is small in a normal population.
Step-by-step explanation:
When the sample size is reduced from 36 to 16, the sampling distribution of the sample mean will still be normal based on the Central Limit Theorem, only if the original population is normal or the sample size is sufficiently large. This is due to the fact that the Central Limit Theorem ensures the sampling distribution of the sample mean approaches a normal distribution as the sample size increases, even if the population distribution is not normal.
Using the formula for the standard deviation of the sampling distribution of the sample means (the standard error of the mean), which is the population standard deviation divided by the square root of the sample size (√n), we can observe the effect of changing sample size. For example, if a population standard deviation is given and you have a sample size (n), the formula would be Population Standard Deviation / √n. This calculation is essential when conducting a hypothesis test for a population mean.
Furthermore, if you conduct a hypothesis test for a single population mean and the population variance is known, you would use the z-test. But if the population variance is unknown and the sample size is small (e.g., n=10) and the population is assumed to be normal, you would use the t-test.