True or false: as then number in the sample becomes larger and larger (assuming it means as the sample gets larger and larger), then the mean of the sampling distribution is equal to the population distribution for any normal population.

Is it false? That it would be approximately normal as the sample size increases?

The statement is true. As the sample size increases, the mean of the sampling distribution approaches the mean of the population distribution, assuming that the sample is randomly selected and sufficiently large.

To understand why this is the case, it helps to know the concept of the Central Limit Theorem. The Central Limit Theorem states that as the sample size increases, the sampling distribution of the mean approaches a normal distribution, regardless of the shape of the population distribution. This holds true as long as certain conditions are met, such as the samples being independent and identically distributed with a finite population standard deviation.

In practical terms, this means that if you take multiple random samples of the same size from a population, calculate the mean of each sample, and plot the distribution of those means, the distribution will be approximately normal. Moreover, the mean of this sampling distribution will be close to the population mean.

In summary, the statement is true: as the sample size increases, the mean of the sampling distribution becomes more accurate and tends towards the population mean. Additionally, it is also true that the sampling distribution approaches approximate normality as the sample size increases, thanks to the Central Limit Theorem.