How can we use a single sample mean to make a decision about a population mean?

In order to make inferences involving means, we need to understand one of the most fundamental ideas to inferential statistics: The Central Limit Theorem. The Central Limit Theorem allows us to make conclusions about the shape of the distribution of a sample mean even if we don’t know about the shape of the distribution of the original population. In fact, if the sample size is large enough, we can use normal distribution calculations to find the probability of observing a specific range of values.

To get a sense of how powerful this is consider the following example. The distribution on the left is baseball player salaries. Notice that the distribution is heavily skewed to the right. If we selected 40 salaries at random, what is the probability that the sample mean will be larger than 2 million dollars? The distribution on the right shows 10,000 sample means based on samples of size n = 40. Since the shape of the distribution is approximately normal, we can use normal distribution calculations to answer this question.

Being able to use sampling distributions and understanding the powerful nature of The Central Limit Theorem are the last steps before our final modules on Confidence Intervals and Hypothesis Testing.