Transcript Central Limit Theorem

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).