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

Transcript text: QUESTION 1 The mean for the sampling distribution of $X$ is the same as the mean of the underlying random variable $X$. True False

Solution

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.

Step 1: Understanding the Statement

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: \[ E(\bar{X}) = E(X) \]

Step 2: Applying the Central Limit Theorem

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.

Step 3: Conclusion

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