The sampling distribution of sample mean for a large population is approximately normal if the sample size is

The sampling distribution of the sample mean for a large population is approximately normal if the sample size is large enough, regardless of the shape of the population distribution. This is known as the Central Limit Theorem.

Here are the steps to understand this:

1. The Central Limit Theorem states that if you have a population with mean μ and standard deviation σ and take sufficiently large random samples from the population with replacement, then the distribution of the sample means will be approximately normally distributed.

2. This will hold true regardless of whether the source population is normal or skewed, provided the sample size is sufficiently large (usually n > 30). If the population is normal, then the theorem holds true even for samples smaller than 30.

3. In other words, if the sample size is large enough, the distribution of the sample mean is approximately normal, regardless of the shape of the population distribution.

4. So, the sample mean is normally distributed with mean equal to the population mean, and its standard deviation (known as the standard error) is the population standard deviation divided by the square root of the sample size.

5. As the sample size increases, the standard error decreases and the shape of the distribution becomes more and more like a normal distribution.

6. This is a key foundation of inferential statistics and hypothesis testing. It allows us to make generalizations from samples to populations.