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Central Limit Theorem-CLT MM4D1. Using simulation, students will develop the idea of the central limit theorem. Key points • The central Limit theorem states that for a given large sample size, if the shape of the population is unknown, the distribution of sample means is normal. • The central limit theorem is important because for any population, it says the sampling distribution of the sample mean is approximately normal, regardless of the sample size. • According to the central limit theorem, the sampling distribution of the mean can be approximated by the normal distribution as the sample size gets large enough. Examples 1. A bottling company uses a filling machine to fill plastic bottles with a popular cola. The bottles are supposed to contain 300 ml. In fact, the contents vary according to a normal distribution with mean µ = 303 ml and standard deviation σ = 3 ml. a. What is the probability that an individual bottle contains less than 300 ml? b. Now take a random sample of 10 bottles. What are the mean and standard deviation of the sample mean contents x-bar of these 10 bottles? c. What is the probability that the sample mean contents of the 10 bottles is less than 300 ml? Solution 1. a) use z=(x-µ)/σ) & Table of negative Z-score z=(300-303)/3 = -1.00 p= 0.1587, b) mean: 303, stdev: 3/sqrt(10) = 0.94868 c) z=(x-µ)/(σ/sqrt(10) & Table of negative Zscore z=(300-303)/0.94868 = -3.16 p=0.0008 (1 in 1250 -- very unlikely).