Final answer:
The sampling distribution of the sample mean for a sample size of 84 from a population with mean 23 and standard deviation 7 will be approximately normal, thanks to the Central Limit Theorem. The sample mean equals the population mean, and the standard error is calculated by dividing the population standard deviation by the square root of the sample size.
Step-by-step explanation:
When selecting a sample of 84 measurements from a population with a mean of 23 and a standard deviation of 7, the shape of the sampling distribution of the sample mean will be approximately normal. This is due to the Central Limit Theorem, which states that if the sample size (n) is sufficiently large, the distribution of the sample means will be approximately normal, regardless of the shape of the population distribution. The mean of the sample means will be the same as the population mean (23 in this case), and the standard error of the mean will be the population standard deviation (7) divided by the square root of the sample size (84).
The sampling distribution of the sample mean will be approximately normal even if we do not assume that the population distribution is normal since the sample size is large (n = 84). This is in accordance with the Central Limit Theorem.