What happens to the shape of the sampling distribution as the sample size increases?

As the sample size increases, the shape of the sampling distribution of the sample mean approaches a normal distribution. This principle is a consequence of the Central Limit Theorem, which states that, regardless of the original distribution of the population, the distribution of the sample means will tend to be normal if the sample size is sufficiently large. This occurs because larger samples provide a more accurate estimate of the population mean, resulting in reduced variability and a bell-shaped curve.

The Central Limit Theorem is particularly powerful because it allows statisticians to use normal distribution properties to make inferences about population parameters even when the underlying population distribution is not normal. Thus, as the sample size grows, you would observe the distribution becoming increasingly normal, culminating in a clear approximation to a normal distribution when the sample size is large enough.