Download 6.5 Day 2: The Central Limit Theorem - Miss-Stow-Math

6.3: THE CENTRAL LIMIT
THEOREM
S W B AT U S E T H E C E N T R A L
L I M I T T H E O R E M TO S O LV E
P R O B L E M S I N V O LV I N G
SAMPLE MEANS FOR LARGE
SAMPLES
A PROBLEM WE KNOW…
The average teacher’s salary in North Dakota is
$29,863. Assume a normal distribution with a
standard deviation of $5,100.
What is the probability that a random teacher’s
salary is more than $40,000?
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A NEW PROBLEM…SAMPLE MEANS
The average teacher’s salary in North Dakota is $29,863.
Assume a normal distribution with a standard deviation
of $5,100.
What is the probability that the
mean
for a sample
of 80 teachers’ salaries is greater than $30,000?
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SAMPLING DISTRIBUTION OF SAMPLE MEANS
A distribution using the means computed
from all possible random samples of a
specific size taken from a population.
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SAMPLING ERROR
The difference between the sample measure
and the corresponding population measure
due to the fact that the sample is not a perfect
representation of the population
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PROPERTIES OF THE DISTRIBUTION OF SAMPLE
MEANS
1. The mean of the sample means will be the same as
the population mean
2. The standard deviation of the sample means will be
smaller than the standard deviation of the population,
and it will be equal to the population standard
deviation divided by the square root of the sample
size
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STANDARD ERROR OF THE MEAN
The standard deviation of the sample means
Formula:
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EXAMPLE 1
The average score on Mr. Smith’s Calculus final was a 68.7 with a
standard deviation of 13.4 points. What is the standard error for
the average scores of:
4 students:
9 students:
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THE CENTRAL LIMIT THEOREM
STATES WHEN WE CAN USE A
NORMAL DISTRIBUTION TO
SOLVE PROBLEMS….
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CENTRAL LIMIT THEOREM
As the sample size n increases without limit,
the shape of the distribution of the sample
means taken with replacement from a
population with mean  and standard
deviation  will approach a normal
distribution with mean  and standard
deviation 
n
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CENTRAL LIMIT THEOREM FORMULAS
Used to gain information
about an individual data
value when the variable is
normally distributed
z=
x-m
s
Used to gain information when
applying the central limit
theorem about a sample mean
when the variable is normally
distributed or when the sample
size is 30 or more.
z
x

n
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IN ENGLISH….
If the sample size is larger than 30, a distribution
of sample means can be approximated using
the normal distribution
If the original population is normally distributed,
then the sample means will be normally
distributed for any sample size
As the sample size increases, the sampling
distribution of sample means approaches a
normal distribution
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EXAMPLE 1
The average height of an adult female in the US is 64.5 in with
a standard deviation of 2.5 in. If a sample of 70 females is
taken, what is the probability that
The mean will be more than 65 cm?
The mean will be less than 63.5 cm?
The mean will be between 63.5 cm and 65 cm?
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EXAMPLE 2
A.C. Neilsen reported that children between the ages of
2 and 5 watch an average of 25 hours of television
per week. Assume the variable is normally distributed
and the standard deviation is 3 hours. If 20 children
between the ages of 2 and 5 are randomly selected,
find the probability that the mean of the number of
hours they watch television will be greater than 26.3
hours.
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EXAMPLE 3
The average age of a vehicle registered in the US
is 8 years, or 96 months. Assume the
standard deviation is 16 months. If a random
sample of 36 vehicles is selected, find the
probability that the mean of their age is
between 90 and 100 months.
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EXAMPLE 4:
The average number of pounds of meat that a person consumes per
year is 218.4 pounds. Assume that the standard deviation is 25
pounds and the distribution is approximately normal.
a.
Find the probability that a person selected at random consumes
less than 224 pounds per year
b.
If a sample of 40 individuals is selected, find the probability that
the mean of the sample will be less than 224 pounds per year.
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