Step 1 :The statement is referring to the concept of the Central Limit Theorem in statistics. According to the Central Limit Theorem, as the size of a sample increases, the distribution of sample means approaches a normal distribution. However, the mean of the distribution of sample means does not change with the increase in sample size. It remains the same as the population mean.
Step 2 :Therefore, the statement is false. The correct statement should be 'As the size of a sample increases, the mean of the distribution of sample means does not change.'
Step 3 :\(\boxed{\text{B. False. As the size of a sample increases, the mean of the distribution of sample means does not change.}}\)