MATH 10: Elementary Statistics and Probability
Chapter 7: The Central Limit Theorem
Tony Pourmohamad
Department of Mathematics
De Anza College
Spring 2015
The Central Limit Theorem
Using the Central Limit Theorem
Objectives
By the end of this set of slides, you should be able to:
1
Understand what the central limit theorem is
2
Recognize the central limit theorem problems
3
Apply and interpret the central limit theorem for means
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The Central Limit Theorem
Using the Central Limit Theorem
The Central Limit Theorem
• The Central Limit Theorem (CLT) is one of the most powerful and
useful ideas in all of statistics
• For this class, we will consider two application of the CLT:
1
2
CLT for means (or averages) of random variables
CLT for sums of random variables
• Let’s start with an example, courtesy of Professor Mo Geraghty
http://nebula2.deanza.edu:16080/˜mo/holistic/clt.swf
• Try exploring the following website to better understand the CLT
http://spark.rstudio.com/minebocek/CLT_mean/
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The Central Limit Theorem
Using the Central Limit Theorem
The Central Limit Theorem
• So what is happening in the CLT video?
100 Samples
20
Frequency
10
2
0
0
1
Frequency
3
30
4
10 Samples
2.5
3.0
3.5
4.0
4.5
5.0
1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0
10,000 Samples
1000
Frequency
500
150
0
50
0
Frequency
250
1500
1,000 Samples
2
3
4
5
2
3
4
5
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The Central Limit Theorem
Using the Central Limit Theorem
The Central Limit Theorem -- Basic Idea
• Imagine there is some population with a mean µ and standard
deviation σ
• We can collect samples of size n where the value of n is "large
enough"
• We can then calculate the mean of each sample
• If we create a histogram of those means, then the resulting
histogram will look close to being a normal distribution
• It does not matter what the distribution of the original population is,
or whether you even know it. The important fact is the the
distribution of the sample means tend to follow the normal
distribution
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The Central Limit Theorem
Using the Central Limit Theorem
The Central Limit Theorem -- More Formally
• Suppose that we have a large population with with mean µ and
standard deviation σ
• Suppose that we select random samples of size n items this
population
• Each sample taken from the population has its own average X̄ .
• The sample average for any specific sample may not equal the
population average exactly.
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The Central Limit Theorem
Using the Central Limit Theorem
The Central Limit Theorem -- More Formally Continued
• The sample averages X̄ follow a probability distribution of their own
• The average of the sample averages is the population average:
µx̄ = µ
• The standard deviation of the sample averages equals the
population standard deviation divided by the square root of the
sample size
σ
σx̄ = √
n
• The shape of the distribution of the sample averages X̄ is normally
distributed if the sample size is large enough
• The larger the sample size, the closer the shape of the distribution
of sample averages becomes to the normal distribution
• This is the Central Limit Theorem!
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The Central Limit Theorem
Using the Central Limit Theorem
The Central Limit Theorem -- Case 1
• IF a random sample of any size n is taken from a population with a
normal distribution with mean and standard deviation σ
• THEN distribution of the sample mean has a normal distribution
with:
µx̄ = µ
and
σ
σx̄ = √
n
and
X̄ ∼ N (µx̄ , σx̄ )
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The Central Limit Theorem
Using the Central Limit Theorem
The Central Limit Theorem -- Case 1
X ~ N(10, 2)
µ
X ~ N(10, 2
50)
µ
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The Central Limit Theorem
Using the Central Limit Theorem
The Central Limit Theorem -- Case 2
• IF a random sample of sufficiently large size n is taken from a
population with ANY distribution with mean µ and standard
deviation
• THEN the distribution of the sample mean has approximately a
normal distribution with:
µx̄ = µ
and
σ
σx̄ = √
n
and
X̄ ∼ N (µx̄ , σx̄ )
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The Central Limit Theorem
Using the Central Limit Theorem
The Central Limit Theorem -- Case 2
X ~ N(10, 2)
µ
X ~ N(µ, σ
n)
µ
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The Central Limit Theorem
Using the Central Limit Theorem
The Central Limit Theorem -- Recap
• 3 important results for the distribution of X̄
1
The mean stays the same
µx̄ = µ
2
The standard deviation gets smaller
σ
σx̄ = √
n
3
If n is sufficiently large, X̄ has a normal distribution where
X̄ ∼ N (µx̄ , σx̄ )
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The Central Limit Theorem
Using the Central Limit Theorem
What is Large n?
• How large does the sample size n need to be in order to use the
Central Limit Theorem?
• The value of n needed to be a "large enough" sample size
depends on the shape of the original distribution of the individuals
in the population
• If the individuals in the original population follow a normal
distribution, then the sample averages will have a normal
distribution, no matter how small or large the sample size is
• If the individuals in the original population do not follow a normal
distribution, then the sample averages X̄ become more normally
distributed as the sample size grows larger. In this case the sample
averages X̄ do not follow the same distribution as the original
population
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The Central Limit Theorem
Using the Central Limit Theorem
What is Large n? Continued
• The more skewed the original distribution of individual values, the
larger the sample size needed
• If the original distribution is symmetric, the sample size needed can
be smaller
• Many statistics textbooks use the rule of thumb n ≥ 30, considering
30 as the minimum sample size to use the Central Limit Theorem.
But in reality there is not a universal minimum sample size that
works for all distributions; the sample size needed depends on the
shape of the original distribution
• In this class, we will assume the sample size is large enough for the
Central Limit Theorem to be used to find probabilities for X̄
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The Central Limit Theorem
Using the Central Limit Theorem
Calculating Probabilities from a Normal Distribution
• Here is the general procedure to calculate probabilities from the
distribution of the sample mean X̄
1
You are given an interval in terms of x̄, i.e.
P (X̄ < x̄ )
2
Convert to a z score by using
z=
3
x̄ − µ
√
σ/ n
Look up probability in z table that corresponds to z score, i.e.
P (Z < z )
• This is just the same idea we used in Chapter 6!
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The Central Limit Theorem
Using the Central Limit Theorem
Examples
• Look at Handout #5 on the website
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The Central Limit Theorem
Using the Central Limit Theorem
Percentile Calculations Based on the Normal Distribution
• Here is the general procedure to calculate the value x̄ that
corresponds to the P th percentile
1
You are given a probability or percentile desired
2
Look up the z score in table that corresponds to the probability
3
Convert to x̄ by the following formula:
x̄ = µ + z
σ
√
n
• Examples: Look at Handout #5 on the website
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The Central Limit Theorem
Using the Central Limit Theorem
Using Your Calculator
• If you have a graphing calculator, your calculator can calculate
all of these probabilities without using a z table
• If you want to calculate P (a < X̄ < b) follow these steps:
Push 2nd, then DISTR
Select normalcdf() and then push ENTER
√
3 Then enter the following: normalcdf(a , b, µ, σ/ n)
1
2
• Question: If X̄ ∼ N (0, 1), what is the probability P (−1 < X̄ < 1)?
• Solution: normalcdf(−1, 1, 0, 1) = 0.6827 ≈ 68%
• Question: If X̄ ∼ N (10, 2), what is the probability P (7 < X̄ < 9)?
• Solution: normalcdf(7, 9, 10, 2) = 0.2417
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The Central Limit Theorem
Using the Central Limit Theorem
Using Your Calculator
• If you want to calculate P (X̄ < a ) follow these steps:
Push 2nd, then DISTR
Select normalcdf() and then push ENTER
√
99
3 Then enter the following: normalcdf(−10 , a , µ, σ/ n)
1
2
• Question: If X̄ ∼ N (10, 2), what is the probability P (X̄ < 8)?
• Solution: normalcdf(−1099 , 8, 10, 2) = 0.158656
• If you want to calculate P (X̄ > a ) follow these steps:
Push 2nd, then DISTR
Select normalcdf() and then push ENTER
√
99
3 Then enter the following: normalcdf(a , 10 , µ, σ/ n)
1
2
• Question: If X̄ ∼ N (10, 2), what is the probability P (X̄ > 9)?
• Solution: normalcdf(9, 1099 , 10, 2) = 0.691462
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The Central Limit Theorem
Using the Central Limit Theorem
Using Your Calculator
• If you want to calculate the value of X̄ that gives you the P th
percentile then follow these steps:
Push 2nd, then DISTR
Select invNorm() and then push ENTER
3 Then enter the following: invNorm(percentile,µ, σ )
1
2
• Question: If X̄ ∼ N (10, 2), what value of X̄ gives us the 25th
percentile?
• Solution: normalcdf(.25, 10, 2) = 8.65102
√
• Recall: We used the formula x̄ = µ + z σ/ n, so
x̄ = 10 + (−0.67)(2) = 8.66
• We got -0.67 from the z table
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