All possible samples of size n are selected from a population and the mean of each sample is determined. What is the mean of the sample mean?
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A. B. C. D. E.C
If we selected all possible samples from the population, then the sample mean will be equal to the population mean. Only when we cannot select all possible samples will be a difference.
The mean of the sample means, also known as the expected value of the sample mean or the expected value of the sampling distribution of the sample mean, can be estimated in advance.
To understand why, let's consider the concept of sampling distribution. When we take multiple random samples of the same size from a population, calculate the mean of each sample, and create a frequency distribution of those sample means, we obtain what is known as the sampling distribution of the sample mean.
According to the central limit theorem, when the sample size is sufficiently large (typically greater than 30) and the population is not extremely skewed or has outliers, the sampling distribution of the sample mean will be approximately normally distributed, regardless of the shape of the population. Moreover, the mean of the sampling distribution of the sample mean will be equal to the population mean.
Therefore, the correct answer is C. The mean of the sample mean is exactly the same as the population mean.
It's important to note that this result holds true for large sample sizes due to the central limit theorem. For small sample sizes, the sampling distribution may not be exactly normally distributed, and the mean of the sample means may deviate from the population mean. However, without additional information on the sample size, we assume it is large enough for the central limit theorem to apply.