Practical Use of CLT?
I understand that CLT holds when we have a large number of large samples. Because then we can assume normality of the sampling distribution of the mean. But how would this benefit us in real life? In real life, we only take 1 large sample of the population not multiple samples and take the mean of the sample means. What's the practical use of the CLT?
Thanks for your Question.
The most common use of the CLT is in inferential statistics, specifically when constructing confidence intervals and conducting hypothesis tests. When we estimate population parameters, such as the mean, we typically use a sample and acknowledge that our estimate contains some uncertainty. The CLT allows us to quantify that uncertainty.
Let's say we want to know the average weight of all people in a city. We draw one large sample and calculate the sample mean. But we recognize that this sample mean is just one of many possible sample means we could have gotten. With the CLT, we know that if we hypothetically repeated this process many times, the distribution of those sample means would be normal. Therefore, we can construct a confidence interval around our one sample mean to estimate where the true population mean likely falls.
In essence, the practical use of the CLT is less about physically taking multiple samples from a population, but more about providing a theoretical basis that enables us to estimate, model, and make inferences from the single large sample we have drawn.
Just to clear my doubts of my understanding. We know that the mean of the samples' means is the actual population mean so is that the reason why we create CIs around our large sample mean? To get a range of the sample means that its part of so we get a higher chance of including the actual population from the sample means?