1. Central Limit Theorem-CLT MM4D1. Using simulation, students will develop the idea of the central limit theorem.

2. 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.

3. 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?

4. 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 Z- score z=(300-303)/0.94868 = -3.16 p=0.0008 (1 in 1250 -- very unlikely).