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?
Thanks so much @Manikandan G R
and to you,
Your understanding is on the right track, but let's clarify a few points for better comprehension so tat somebody reading this later on can easily understand
Mean of Sample Means and Population Mean
When you take multiple samples from a population and calculate the mean of each sample, the mean of these sample means tends to be close to the actual population mean. This is a principle known as the Central Limit Theorem (CLT). The CLT states that, given a sufficiently large sample size, the sampling distribution of the sample mean will be approximately normally distributed, regardless of the shape of the population distribution, and its mean will be close to the population mean.
Purpose of Confidence Intervals (CIs)
Confidence intervals are used to estimate a population parameter (like the mean) based on sample data. The idea behind constructing a CI around a sample mean isn't just to capture a range of possible sample means. Rather, it's to estimate a range within which the true population mean is likely to lie, with a certain level of confidence (like 95% or 99%).
Interpreting Confidence Intervals
When you construct, say, a 95% confidence interval around a sample mean, it means that if you were to take many samples and construct a CI for each, approximately 95% of these intervals would contain the true population mean. It's important to note that the CI either contains the population mean or it doesn't; the 95% confidence level refers to the long-run proportion of CIs that will contain the population mean if the same sampling process is repeated numerous times.
Large Samples and Accuracy
Larger sample sizes generally lead to narrower confidence intervals, assuming the level of confidence is kept constant. This means you get a more precise estimate of the population mean. However, the width of the CI also depends on the variability of the data; more variable data leads to wider intervals.