Explain Central Limit Theorem with an example

Question

Explain Central Limit Theorem in simple terms and provide a concrete example.

Codemia · Accepted Answer

The Central Limit Theorem (CLT) states that when independent random variables are added, their standardized sum tends toward a normal distribution, regardless of the original distribution of the variables. This is crucial in statistics because it allows us to make inferences about population parameters using sample statistics.

Mathematically, if we have a random variable $ X $ with a mean $ \mu $ and standard deviation $ \sigma $, and we take a sufficiently large sample size $ n $, the distribution of the sample mean $ ar{X} $ will approach a normal distribution:

$$ ar{X} \sim N\left(\mu, \frac{\sigma^2}{n}\right) $$

This means as $ n $ increases, $ \bar{X} $ becomes normally distributed.

To illustrate the Central Limit Theorem, let's consider a practical example. Suppose we have a population of students' test scores from a large school that follows a uniform distribution from 0 to 100. We want to analyze the average test scores of students by taking samples from this population.

1. **Define the parameters**: 
   - Population Mean, $ \mu = 50 $
   - Population Standard Deviation, $ \sigma = \sqrt{\frac{(100-0)^2}{12}} \approx 28.87 $
   - Sample Size, $ n = 30 $

2. **Apply the Central Limit Theorem**: Since $ n $ is large enough, we expect the sampling distribution of the sample mean $ \bar{X} $ to be normally distributed with:
   - Mean, $ \mu_{\bar{X}} = \mu = 50 $
   - Standard Deviation (Standard Error), $ \sigma_{\bar{X}} = \frac{\sigma}{\sqrt{n}} = \frac{28.87}{\sqrt{30}} \approx 5.26 $

Let's say we take a random sample of 30 students and calculate their average test score. Assume the sample mean $ \bar{X} = 55 $.

To determine how likely this sample mean is under the null hypothesis (that the population mean is 50), we calculate the z-score:

$$ z = \frac{\bar{X} - \mu}{\sigma_{\bar{X}}} = \frac{55 - 50}{5.26} \approx 0.95 $$

Using the z-table, we find the probability associated with $ z = 0.95 $, which gives us a p-value of approximately 0.171. Since this p-value is greater than common significance levels (like 0.05), we do not reject the null hypothesis.

We conclude that the average test score of 55 is not statistically significant.

In practical terms, the result indicates that while the sample mean of 55 is higher than the population mean of 50, this difference is not statistically significant at the 0.05 level. This suggests that the observed difference could be due to random chance rather than a true difference in population means.

It’s important to remember that statistical significance does not imply practical significance; even if the result were statistically significant, we would need to consider whether the difference is meaningful in a real-world context.

The Central Limit Theorem connects to various statistical concepts:
1. **Confidence Intervals**: Since we can assume the sampling distribution is normal, we can construct confidence intervals for the population mean based on sample means.
2. **Hypothesis Testing**: Understanding CLT helps in hypothesis testing, particularly in determining p-values for sample means.
3. **Bayesian vs Frequentist Inference**: While CLT is fundamental in frequentist statistics, Bayesian methods approach inference differently, focusing on prior distributions and updating beliefs with new data.
4. **Practical Implications**: Knowledge of the CLT is essential for ensuring that sample sizes are adequate for making reliable inferences about populations, especially in fields like quality control and market research.

Explain Central Limit Theorem with an example

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