Applications of the CLT

Statistics · Axiom Academy

LESSON Applications of the CLT Using the Central Limit Theorem to find probabilities about sample means and solve practical statistical problems The Central Limit Theorem establishes that for a population with mean μ and standard deviation σ , the sampling distribution of the sample mean has these properties: Mean of sampling distribution: μ x̄ = μ Standard deviation (standard error): σ x̄ = σ/√n Shape: Approximately normal when n ≥ 30 This animation shows how individual samples cluster around the population mean, with spread determined by the standard error. To find probabilities about sample means, we convert them to z-scores using the standard error. This transforms our problem into one we can solve using the standard normal distribution. Step 1: Calculate standard error P(x̄ > 105) = P(z > 2.0) = 0.0228 The animation demonstrates how we transform the sampling distribution to the standard normal distribution for probability calculations. Let's apply the CLT to a real-world scenario: A manufacturing company produces light bulbs with a mean lifetime of 1200 hours and standard deviation of 150 hours. Step 2: Calculate standard error σ x̄ = 150/√100 = 150/10 = 15 hours P(-1.33 < z < 1.33) = 0.9082 - 0.0918 = 0.8164

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