Final Answer:
False. while the normal distribution is a common occurrence due to the CLT, it doesn't mandate a mean of 0 and standard deviation of 1 for the sampling distribution of x¯ in all cases.
Explanation:
The statement that the sampling distribution of x¯ (x-bar) must be a normal distribution with a mean of 0 and a standard deviation of 1 is false. The Central Limit Theorem (CLT) establishes that the sampling distribution of the sample mean tends to be normally distributed under certain conditions, particularly when the sample size is sufficiently large, regardless of the original population's distribution.
However, it doesn't guarantee a mean of 0 and standard deviation of 1 for every scenario. The CLT indicates that as sample sizes increase, the distribution of sample means converges toward a normal distribution, but the mean and standard deviation of this distribution will not always be 0 and 1, respectively.
The mean and standard deviation of the sampling distribution of the sample mean are influenced by the characteristics of the population being sampled and the sample size.
The Central Limit Theorem emphasizes that with larger sample sizes, the sampling distribution of the sample mean becomes more normal, showcasing the bell-shaped curve typical of a normal distribution.
However, this distribution's mean and standard deviation are subject to variation depending on the original population's distribution and sample size. Therefore, while the normal distribution is a common occurrence due to the CLT, it doesn't mandate a mean of 0 and standard deviation of 1 for the sampling distribution of x¯ in all cases.