4.5.1 Central Limit Theorem

Overview

• Purpose: The Central Limit Theorem (CLT) explains how the sampling distribution of the sample mean (x̄) becomes approximately normal regardless of the population's original distribution, given a sufficiently large sample size.

Key Concepts

• Sampling Distribution of x̄:
• Even if the population distribution is not normal, the distribution of the sample mean x̄ will approximate a normal distribution as the sample size increases.
• This approximation becomes effective when the sample size (n) is 30 or more.

Practical Implications

• Normal Approximation:
• For any population with mean (μ) and standard deviation (σ), the mean of x̄ is μ and the standard deviation (σ_x̄) of x̄ is σ/√n, known as the standard error.
• As n increases, σ_x̄ decreases, which means the sampling distribution of x̄ becomes more concentrated around μ.

Central Limit Theorem Application

• Statistical Inference:
• CLT allows for making probabilistic statements about how close x̄ is to μ using the normal distribution model.
• This is crucial for hypothesis testing and constructing confidence intervals in practical business scenarios.

Example Application

• Visual Illustration:
• Various population shapes (Normal, Uniform, Exponential, Arbitrary) show that as n increases to 30 or more, the sampling distribution of x̄ closely resembles a normal distribution.

Conclusion

• Significance in Business Statistics:
• Understanding and applying CLT helps in decision-making based on sample data, facilitating accurate predictions and strategic planning in uncertain conditions.