INTRODUCTORY STATISTICS
Chapter 7 THE CENTRAL LIMIT THEOREM
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SEC. 7.2: THE CENTRAL LIMIT THEOREM FOR
AVERAGES
Not every distribution is normal…
How do we make a normal distribution from a situation that isn’t necessarily?
Use the Central Limit Theorem!
This says that if we keep drawing samples and taking their mean, the means
will form their own normal distribution, even if the original data was not
normal.
EXAMPLE
Let’s consider women’s heights.
We assume these heights, taken as a population, are normally
distributed with a certain mean ( 𝜇 = 65 inches) and a certain
standard deviation ( 𝜎 = 3 inches). We called the random variable for
height X. Instead of 𝜇 inches, we could write more precisely 𝜇𝑥
inches, and we could also write 𝜎𝑥 inches.
Now, let’s look at groups of 10 women and find the mean 𝑥 of their
heights. If we look at lots of different groups of 10, the means create
their own distribution called a sampling distribution.
MEAN AND STANDARD DEVIATION
If we look at the mean of the means (𝜇𝑥 ), no matter how many
samples we take, this will be the same as 𝜇𝑥 . (𝜇𝑥 = 𝜇𝑥 ) Why?
The standard deviation will be different though. This is a measure of
variation. How unusual would it be to find a woman who is at least 6
ft. tall with the original parameters? (𝑋~𝑁(65,3))
𝑃 𝑋 ≥ 72 = 𝑁𝑜𝑟𝑚𝑎𝑙𝑐𝑑𝑓 72, ∞, 65,3 ≈ 0.01
How unlikely would it be to find a group of 10 women whose average
height is 6 ft. or more?
Far more unlikely. As we take more and more samples, the average of
the averages will get closer and closer to together, meaning the
standard deviation goes down.
STANDARD DEVIATION OF SAMPLE MEANS
In general, 𝜎𝑥 =
𝜎𝑥
.
𝑛
So, in our example, if we took 100 groups of 10 women, the standard
deviation, also called the standard error of the mean would be
3
3
𝜎𝑥 =
=
= 0.3
10
100
As we take more samples, we get less variation.
EXAMPLE:
The average life of a certain type of motor is 10 years, with a standard
deviation of 2 years. Assume that the lives of the motors follow a
normal distribution. 36 engines are randomly sampled.
a) State in words, X=
b) 𝑋~_ _, _
c) State in words, 𝑋 =
d) 𝑋~_ _, _
e) What is the probability that the average of the 36 engines is more
than 10 years?
f)
Find the 70th percentile of the distribution of the averages of the
36 engines.
PRACTICE
Home prices in a certain area are normally distributed with a mean of
400 and a standard deviation of 17 (both in thousands of dollars). A
realtor looks at a sample of 25 homes.
a) State in words, X=
b) 𝑋~_ _, _
c) State in words, 𝑋 =
d) 𝑋~_ _, _
e) What is the probability that the average of the 25 homes is less
than 390?
f)
What is probability that the average of the 25 homes is between
390 and 410?
g) Find the lower limit of the upper quartile (𝑄3 ) of the distribution of
the averages of the 25 homes.
ANOTHER EXAMPLE
Salaries at a business are uniformly distributed between $12 per hour
and $20 per hour. The company randomly samples 50 of its
employees and records their salary.
a) State in words, X=
b) 𝑋~_ _, _ . Find the mean and standard deviation of X.
c) State in words, 𝑋 =
d) 𝑋~_ _, _
e) Find the IQR (Interquartile Range) for the distribution of averages.
ONE MORE PRACTICE
Commute times in a city are normally distributed with a mean of 20
minutes and a standard deviation of 3 minutes. A random survey of
100 commuters is taken.
a) State in words, X=
b) 𝑋~_ _, _
c) State in words, 𝑋 =
d) 𝑋~_ _, _
e) What is the probability that the average of the 100 commutes is
more than 30 minutes?
f)
What is 50th percentile of commutes?
SEC. 7.3: CENTRAL LIMIT THEOREM FOR SUMS
The Central Limit Theorem also applies to sums of samples.
No matter the type of distribution, if you keep drawing larger and
larger samples and taking their sums, the sums form their own normal
distribution (the sampling distribution), which approaches a normal
distribution as the sample size increases.
MEAN AND STANDARD DEVIATION
If the mean of X is 𝜇𝑥 , then the mean of the sums is 𝑛 ∙ 𝜇𝑥 .
If the standard deviation of X is 𝜎𝑥 , then the standard deviation of the
sums is 𝑛 ∙ 𝜎𝑥 .
So the distribution of the sums is written
𝑋~𝑁( 𝑛 ∙ 𝜇𝑥 , 𝑛 ∙ 𝜎𝑥 ).
EXAMPLE
Suppose the weight of bananas customers buy at a grocery store is
normally distributed with a mean of 1.3 lbs. and a standard deviation
of 0.4 lbs. The grocery store samples groups of 20 customers.
a) State in words, X=
b) 𝑋~_ _, _
c) State in words,
d)
𝑋=
𝑋 ~_ _, _
e) What is the probability that the total weight of bananas is over 30
lbs.?
f)
Find the probability that the total weight is between 15 and 20 lbs.
g) 80% of the sample totals will be less than what weight?
PRACTICE
A commuter spends an average of 15 minutes on a train ride with a
standard deviation of 2 minutes. Over a year, the commuter has to
travel 220 days.
a) a) State in words, X=
b) 𝑋~_ _, _
c) State in words,
d)
𝑋=
𝑋 ~_ _, _
e) Find the probability that on a single day, the commuter spends
more than 20 minutes on the train.
f)
Find the 90th percentile of single train rides.
g) Find the 90th percentile for the total time the commuter spends on
the train in a year.
SEC. 7.4: USING THE CENTRAL LIMIT THEOREM
It is important for you to understand when to use the central limit
theorem (clt).
If you are being asked to find the probability of the mean, use the clt
for the mean. If you are being asked to find the probability of a sum or
total, use the clt for sums.
If you are being asked to find the probability of an individual value,
do not use the clt. Use the distribution of its random variable.
LAW OF LARGE NUMBERS
The law of large numbers says that if you take samples of larger and
larger size from any population, then the mean 𝑥 of the sample tends
to get closer and closer to μ.
From the central limit theorem, we know that as n gets larger and
larger, the sample means follow a normal distribution. The larger n
gets, the smaller the standard deviation gets. (Remember that the
𝜎
standard deviation for 𝑋 is .)
𝑛
This means that the sample mean 𝑥 must be close to the population
mean μ. We can say that μ is the value that the sample means
approach as n gets larger.
The central limit theorem illustrates the law of large numbers.
EXAMPLE
In an upcoming election, it is determined that 55% of voters favor a
certain bill. There are 1000 voters.
a) What is X in words?
b) Write the distribution of X.
c) What is the probability that 590 voters will vote yes?
d) What is the probability that only 450 voters will vote yes?
ONE LAST EXAMPLE
A certain computer is known to have an average running life of 4
years with a standard deviation of 1.2 years that is normally
distributed. A sample of 25 computers is taken.
a) Define X and write the distribution of X.
b) What is the probability an individual computer has a running life of
5 years.
c) Define 𝑋 and write its distribution.
d) What is the probability that the average of the sample is less than
3.5 years?
e) Find the 90th percentile of an individual computer’s life.
f)
Find the 90th percentile of the average computer’s life.