Final answer:
The population mean for the dataset 2, 3, 4, 5 is 3.5. Despite the sample size or the distribution of the population, the Central Limit Theorem ensures that the mean of sample means will approximate the population mean if the sample size is large enough.
Step-by-step explanation:
To calculate the population mean (μ) of the data set 2, 3, 4, 5, we use the formula:
μ = (sum of all values) / (number of values)
For our example: μ = (2 + 3 + 4 + 5) / 4 = 14 / 4 = 3.5
Therefore, the population mean is 3.5. As the student is considering all possible distinct samples of size two with replacement, it's essential to note that each number in the population has an equal chance of being selected in each draw, making each draw a Bernoulli trial since sampling is done with replacement.
In terms of sampling, when the sample size (n) is sufficiently large, according to the Central Limit Theorem, the distribution of the sample means will approximate a normal distribution even if the population distribution is not normal. This theorem also states that the mean of the sample means will be equal to the population mean.
However, in the sample provided by the student (1, 1, 1, 2, 2, 3, 4, 4, 4, 4, 4), the calculation of the mean is straightforward. The mean (sample mean) is the sum of all the numbers divided by the count of numbers, resulting in:
Sample mean = (1+1+1+2+2+3+4+4+4+4+4) / 11 = 30 / 11 = approximately 2.7272