Download Central Limit Theorem

Warm-Up
• Take one die. Roll it 10 times and record the numbers that you
get. Add your information to the frequency table on the
board.
• Record the average of the 10 rolls. Repeat 9 more times
(finding the average of each 10 rolls). Add to the frequency
table on the board.
Central Limit
Theorem
Honors Advanced Algebra
Presentation 1-8
Central Limit Theorem
• 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
Central Limit Theorem
• 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.
Central Limit Theorem
• 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 )
• The variance of the means is equal to
𝜎2
𝑛
• The standard deviation of the means is equal to
𝜎
𝑛
• The larger the sample size, the more certain we can be of the
mean and the smaller the standard deviation.
Example
• 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?
Problem 1
• 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.
Problem 2
• 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?
Problem 3
• 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?
Central Limit Theorem
• 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..