Worked Examples Under the Central Limit Theorem: A Comprehensive Guide

Introduction

The Central Limit Theorem (CLT) is one of the most important concepts in probability and statistics. It states that, given a sufficiently large sample size from a population with a finite level of variance, the mean of samples drawn from that population will be approximately normally distributed, regardless of the population’s distribution. The theorem is widely used in inferential statistics and plays a crucial role in various real-world applications, particularly when dealing with a large number of data points.

Historical Significance of the Central Limit Theorem

Historically, the Central Limit Theorem has been a beacon of hope for statisticians when dealing with large samples from non-normal distributions. Traditionally, when one needed to evaluate the probability ( P(X geq a) ) where ( X sim text{B}(n, p) ) and ( n ) was very large to use binomial tables, the best approach was to approximate the binomial distribution with a normal distribution. The justification behind this approximation lies in the Central Limit Theorem, which guarantees that for large ( n ), the binomial distribution can be closely approximated by a normal distribution with the same mean and variance.

The key component in this approximation is translating the binomial parameters ( n ) and ( p ) into the parameters of a normal distribution. The mean of the binomial distribution is ( np ) and the variance is ( np(1-p) ). Therefore, by applying the CLT, we can convert ( X sim text{B}(n, p) ) into ( frac{X - np}{sqrt{np(1-p)}} sim N(0, 1) ), which is a standard normal distribution. This conversion allows for easier calculation and understanding of probabilities.

Modern Tools and the Decline of Traditional Methods

Advancements in technology have significantly altered the landscape of statistical computation. With the advent of scientific calculators and software like R, Python, and SPSS, the need for manual approximations using the Central Limit Theorem has diminished. These tools now offer robust cumulative binomial functions that provide precise and quick results without the need for manual approximations. As a result, the traditional method of using the CLT to approximate binomial distributions is falling into disuse in many practical settings.

Worked Examples: Applying the Central Limit Theorem

Understanding the application of the Central Limit Theorem is essential for any aspiring statistician or data analyst. Here, we present a few worked examples to illustrate the theorem in action.

Example 1: Quality Control in Manufacturing

Suppose a manufacturing company produces light bulbs, and on average, 10% of the light bulbs produced are defective. The company wants to determine the probability that in a sample of 1000 light bulbs, more than 110 will be defective. Using the Central Limit Theorem, we can approximate this binomial probability with a normal distribution.

The parameters are ( n 1000 ) and ( p 0.1 ). The mean (( mu )) and standard deviation (( sigma )) of the binomial distribution are:

[ mu np 1000 times 0.1 100 ] [ sigma sqrt{np(1-p)} sqrt{1000 times 0.1 times 0.9} sqrt{90} approx 9.49 ]

We now convert the probability to a standard normal distribution:

[ Z frac{X - mu}{sigma} frac{110 - 100}{9.49} approx 1.05 ]

Using a standard normal distribution table or a calculator, we find the probability ( P(Z > 1.05) ). This is approximately 0.1469. Therefore, the probability that more than 110 bulbs in a sample of 1000 are defective is about 0.1469 or 14.69%.

Example 2: Medical Research Studies

In medical research, the Central Limit Theorem is often used to estimate the effectiveness of a new drug. Suppose a study involves 5000 patients, and the success rate of the drug is 60%. The research team wishes to determine the probability that at least 2800 patients will show improvement due to the drug.

The parameters are ( n 5000 ) and ( p 0.6 ). The mean and standard deviation are:

[ mu np 5000 times 0.6 3000 ] [ sigma sqrt{np(1-p)} sqrt{5000 times 0.6 times 0.4} sqrt{1200} approx 34.64 ]

Converting to the standard normal distribution:

[ Z frac{X - mu}{sigma} frac{2800 - 3000}{34.64} approx -0.58 ]

The probability ( P(Z > -0.58) ) is approximately 0.7190. Therefore, the probability that at least 2800 patients will show improvement due to the drug is about 71.90%.

Example 3: Gaming Industry Analysis

In the gaming industry, the Central Limit Theorem can be applied to analyze the distribution of scores in a video game. Suppose a game has 10,000 players, and the mean score is 1500 with a standard deviation of 100. The company wants to determine the probability that a randomly selected player will score more than 1600 points. Using the CLT, we can approximate this with a normal distribution.

The mean and standard deviation of the game’s scores are ( mu 1500 ) and ( sigma 100 ). Converting to the standard normal distribution:

[ Z frac{1600 - 1500}{100} 1 ]

The probability ( P(Z > 1) ) is approximately 0.1587. Therefore, the probability that a randomly selected player will score more than 1600 points is about 15.87%.

Conclusion

The Central Limit Theorem has been a cornerstone in the field of statistics, providing a robust method to approximate the distribution of sample means for large samples. While modern technology has made manual approximations less necessary, understanding the theorem remains crucial for grasping the principles underlying inferential statistics. Whether in manufacturing, medical research, or gaming industries, the CLT offers a powerful tool for analyzing and predicting outcomes based on sample data.