The central limit theorem states that the sampling distribution can be approximated by which type of distribution if the sample size is large?

The central limit theorem is a fundamental concept in statistics that states that, regardless of the original population distribution, the sampling distribution of the sample means will approach a normal distribution as the sample size becomes large. This means that if you take sufficiently large random samples from a population and calculate their means, those means will tend to form a normal distribution centered around the true population mean.

This property is particularly important because it allows statisticians to use normal probability techniques when making inferences about population parameters, even if the underlying population is not normally distributed, provided that the sample size is large enough (typically n ≥ 30 is a common rule of thumb). The normal distribution has well-known properties that facilitate statistical analysis and hypothesis testing, making it a critical aspect of inferential statistics.

Understanding the implications of the central limit theorem helps in designing studies and interpreting data, providing a bridge between sample data and broader conclusions about populations.