Applying the Central Limit Theorem?

📌 CONCEPT: The Central Limit Theorem (CLT) states that the distribution of the mean of a large sample of independent and identically distributed random variables will be approximately normal, regardless of the original distribution of the variables.

📐 RULE / FORMULA: The CLT formula for a sample of n elements is given by Z = (X̄ - μ) / (σ / √n), where Z is the standard normal variable, X̄ is the sample mean, μ is the population mean, σ is the population standard deviation, and n is the sample size.

💡 WORKED EXAMPLE: Suppose we randomly select 5 students from a town with a mean score of 80 and a standard deviation of 5. We want to find the probability that their mean score exceeds 85. Using the CLT formula, Z = (85 - 80) / (5 / √5) = 1. We then refer to the standard normal distribution table to find the probability corresponding to a Z-score greater than 1, which is approximately 0.1587.

⚠️ COMMON MISTAKE: Students often forget to square the standard deviation when calculating the standard error (σ / √n), leading to incorrect Z-scores and probabilities.