Transcript Central Limit Theorem

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Take two dice. Roll them together 40 times and
record the sums that you got. Draw a
histogram of your 40 points.
Ms. Morton will generate 50 random numbers
between 1 and 20. Draw a histogram of these
50 data points with bin size 2.
Honors Advanced Algebra
Presentation 1-8
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Central Limit Theorem - Choose a simple
random sample of size n from any population
with mean µ and standard deviation σ. When n
is large (at least 30), the sampling distribution
of the sample mean x is approximately normal
σ
with mean µ and standard deviation .
𝑛
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Central Limit Theorem - Choose a simple random
sample of size n from a large population with
population parameter p having some characteristic
of interest. Then the sampling distribution of the
sample proportion 𝑝 is approximately normal with
mean p and standard deviation
𝑝(1−𝑝)
.
𝑛
This
approximation becomes more and more accurate
as the sample size n increases, and it is generally
considered valid if the population is much larger
than the sample, i.e. np ≥ 10 and n(1 – p) ≥ 10..
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Central Limit Theorem - The CLT allows us to
use normal calculations to determine
probabilities about sample proportions and
sample means obtained from populations that
are not normally distributed.
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As we make a histogram of multiple sample
means, the data approaches a normal curve.
The mean of the means is the same as the mean
of the population. (𝜇 = 𝜇𝑥 )
𝜎2
𝑛
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The variance of the means is equal to
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The standard deviation of the means is equal to
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𝜎
𝑛
The larger the sample size, the more certain we
can be of the mean and the smaller the
standard deviation.
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The time that an A/C technician requires to
perform maintenance on an A/C unit is an
exponential decay distribution. The mean time
is μ = 1 hour and the standard deviation is σ =
1 hour. Your company has a contract to
maintain 70 of these units in an apartment
building. Is it safe to budget 1.1 hours for each
unit or should you budget an average of 1.25
hours?
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The number of flaws per square yard in a type of
carpet material varies with mean 1.6 flaws per
square yard and standard deviation 1.2 flaws per
square yard. The population distribution cannot be
Normal because a count takes only whole-number
values. An inspector studies 200 square yards of
the material, records the number of flaws found in
each square yard, and calculates 𝑥 (the mean
number of flaws per square yard inspected). Use
the central limit theorem to find the approximate
probability that the mean number of flaws exceeds
2 per square yard. Show your work.
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In response to the increasing weight of airline
passengers, the FAA in 2003 told airlines to
assume that passengers average 190 pounds in the
summer, including clothes and carry-on baggage.
But passengers vary, and the FAA did not specify
a standard deviation. A reasonable standard
deviation is 35 pounds. Weights are not Normally
distributed, especially when the population
includes both men and women, but they are not
very non-Normal. A commuter plane carries 20
passengers.
Can you calculate the probability that the total
weight of the passengers on the flight exceeds 4000
pounds?
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The number of traffic accidents per week at an
intersection varies with mean 2.2 and standard
deviation 1.4. The number of accidents in a week
must be a whole number, so the population
distribution is not Normal.
Let 𝑥 be the mean number of accidents per week at
the intersection during a year (52 weeks). What is
the standard deviation of the sample means?
What is the approximate probability that 𝑥 is less
than 2?
What is the approximate probability that there are
fewer than 100 accidents at the intersection in a
year?