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).

3. If the population is normal, then the theorem holds true even for samples smaller than 30.

4. 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.

5. So, the sample mean is a good estimator of the population mean, since its distribution has the same mean (is unbiased) and its standard deviation gets smaller with larger sample sizes.

6. This is why we can use the sample mean to make inferences about the population mean even when the population distribution is not normal.