Questions: The mean for the sampling distribution of X is the same as the mean of the underlying random variable X. True False

Solution Steps

The statement is true. The mean of the sampling distribution of the sample mean $\bar{X}$ is equal to the mean of the population distribution of the random variable $X$. This is a fundamental property of the sampling distribution of the sample mean.

The statement claims that the mean for the sampling distribution of $\bar{X}$ is the same as the mean of the underlying random variable $X$. This can be expressed mathematically as: $$E(\bar{X}) = E(X)$$

According to the Central Limit Theorem, the mean of the sampling distribution of the sample mean $\bar{X}$ is equal to the mean of the population distribution $X$. Therefore, we have: $$E(\bar{X}) = \mu$$ where $\mu$ is the mean of the population.

Since the statement holds true based on the properties of sampling distributions, we conclude that the answer to the question is indeed true.

Final Answer