Final answer:
The Central Limit Theorem indicates that the distribution of sample means is approximately normal and that the mean of these sample means equals the population mean. For a population comprised of children aged 3, 4, and 5, the mean of all possible sample means of pairs of children equals the population mean.
Step-by-step explanation:
Central Limit Theorem and Sample Means
When discussing the Central Limit Theorem for sample means, we recognize that for a population with any distribution, if the sample size (n) is large enough, the distribution of the sample means will be roughly normal. The important part is that the mean of the sample means will equal the population mean, and the standard error of the mean will be the population standard deviation divided by the square root of the sample size.
In the context of the student's question, let's consider a population of 3 children with ages: 3, 4, and 5. To find the distribution of all sample means of samples of size 2, we would pair the children in all possible ways, calculate the means of these pairs, and then determine if the mean of these sample means equals the population mean.
The possible pairs and their means would be:
- (3,4): Mean = 3.5
- (3,5): Mean = 4
- (4,5): Mean = 4.5
The mean of all sample means would be (3.5 + 4 + 4.5) / 3 = 4, which is indeed the population mean (3 + 4 + 5) / 3 = 4.