Chapter 5 Project Sampling Distributions and the Central Limit Theorem Name In chapter 5, you learned about the distribution of sample means. In other words, if you were to take a population and sample from it repeatedly, each time you sample there would be a corresponding sample mean [It bar]. We would expect some sample means to be more likely than others. Thus, the distribution of sample means is a probability distribution of the 1: bars obtained from each sample. The Central Limit Theorem is a very important result in statistics. Loosely, it says that the probability distribution of the x bars is approximately normal when the sample size is at least 30. The beauty in the CLT is the fact that you can sample from populations that are not normal, and the distribution of 1; bars from these non-normal populations will be approximately normal! Of course, when we sample from a normal population, the distribution of the 1:: bars is always normal, regardless of the sample size. You may be wondering why we focused so much attention on the distribution of x bars instead of other statistics. After all, each sample has a standard deviation, variance, and other statistics we could study the distributions of. To see why the mean is so important, consider the following taken from Fundamentals ofProbobiiiiy by Saeed Ghahramani: The central limit theorem shows that many natural phenomena obey approximately a standard normal distribution. In natural processes, the changes that take place are often the result of a sum of many insignificant random factors. Although the effect of each of these factors, separately, may possibly be ignored, the effect of their sum in general may not. Therefore, the study of distributions of sums of a large number of independent random variables is essential. in this connection, the central limit theorem shows that